High School Engineering & Design

26 September 1908: chlorine goes into Jersey City's water

On 26 September 1908 a chlorination plant at the Boonton Reservoir in New Jersey went on line. It dosed an average of 40 million gallons of water a day, about 150 million litres, on its way to the taps of Jersey City. The man who wanted it was John L. Leal, a physician responsible for the city's water. The man he hired that June to build it was George W. Fuller, a sanitary engineer. Water from that source has been chlorinated without a break ever since, the longest record of disinfection of any drinking-water supply.

Look at who did what. That chlorine kills bacteria was not news in 1908, so Leal did not need a discovery. He needed a machine that would put the right amount of a chlorine compound into every litre of a river-sized flow, all day and all night: strong enough to kill the germs that spread typhoid fever, weak enough that people would drink the water happily. The dose had to be measured, the equipment had to keep running, and the whole thing had to cost what a city would pay. In a court case between the city and its water company, the ruling accepted chlorine as a treatment that made the water "pure and wholesome" to drink. Water chlorination and other treatment then spread across the United States, and the Centers for Disease Control and Prevention lists it among the public health practices that cut waterborne disease in the twentieth century.

That is the work of an engineer in a single example. Take something known to be true, and make it happen reliably, at scale, safely, for a price, with measurements that prove it is still working next Tuesday.

8 October 1958: the pacemaker that lasted three hours

On 8 October 1958, at the Karolinska Institute in Solna, Sweden, the surgeon Åke Senning implanted a pacemaker designed by Rune Elmqvist, a physician who had turned engineer. An artificial cardiac pacemaker is a small electrical device that sends a timed pulse to the heart when the heart's own timing fails. The patient was Arne Larsson. The device failed after three hours. A second one, put in next, lasted two days.

Larsson lived until 2001, when he died at 86. In those 43 years he received 26 pacemakers, and he outlived both the man who designed the first one and the surgeon who put it in. In the United States, Wilson Greatbatch, who had graduated in electrical engineering from Cornell University in 1950, built implantable pacemakers that went into patients from April 1960, after long testing in animals. The first of his patients lived a further 18 months. Greatbatch's company went on to make the lithium-iodide cell that became the standard pacemaker battery, because it stored a lot of energy for its size, lost very little charge while it sat unused, and could be trusted.

Read the numbers again: three hours, then two days. The idea of pacing a heart with electricity worked on the first day. What took decades was making the device last inside a body whose heart beats roughly 100,000 times a day: a battery that keeps going for years, a sealed case that body fluids cannot get into, leads that survive being flexed with every beat. Most of engineering is that second part.

What engineers actually spend their days doing

The Bureau of Labor Statistics (BLS), the U.S. government agency that tracks jobs, publishes the typical duties of each occupation in its Occupational Outlook Handbook. Read three of its lists side by side and the job looks very different from the lone inventor in a garage.

• Civil engineers, the BLS says, analyze plans, survey reports and maps; analyze the results of tests on soil and building materials to find their strength; prepare cost estimates for equipment, materials and labor; submit permit applications to local, state and federal agencies; and manage construction, repair and maintenance.
• Mechanical engineers design or redesign devices using analysis and computer-aided design; investigate equipment failures to diagnose what went wrong and recommend remedies; develop and test prototypes; analyze the test results and change the design as needed; and oversee manufacturing.
• Electrical and electronics engineers determine what a product or system is required to do; inspect and test the reliability and safety of devices; prepare documentation covering the requirements and maintenance; and supervise manufacturing or construction.

Count the verbs: analyze, estimate, test, investigate, document, supervise. Design is on every list, but it sits among checking, measuring and writing things down. Look at the mechanical list in particular. It runs in a loop: build a prototype, test it, study the results, change the design. That loop is the design process, and this course will run it on one problem from start to finish.

A good deal of the job is writing: requirements, test reports, permit applications, maintenance manuals. A brilliant design described in a report nobody can follow is a design nobody can build, check or repair. You will write a lot in this course for that reason, and every activity ends with a written result.

The disciplines, with real numbers

Engineering splits into disciplines, each defined by the kind of problem it takes on. The table uses the BLS figures published in 2026. Median pay is the middle salary: half the people in the job earned more and half earned less. The job counts are for 2025.

Read down the job column before the pay column. Civil and industrial engineering together account for 745,700 jobs, and civil engineering alone employs about seventeen times as many people as chemical engineering (380,600 divided by 21,900 is 17.4). Size matters when you are choosing: a large discipline has employers in every region, a small one may cluster where its industry is. Then read the entry route. Every engineering row lists a bachelor's degree as the typical entry-level education; the technician row lists an associate's degree, usually two years. For the whole group of architecture and engineering occupations, the BLS gives a median of $99,520 in May 2025, against $50,980 for all occupations.

Real projects ignore the boundaries in the table. Fuller was a sanitary engineer, which today would be called civil or environmental. Elmqvist trained as a physician. Greatbatch was an electrical engineer whose most famous work was a medical device. The disciplines are ways of training people, not walls between them.

Key idea: the disciplines differ in the problems they take on, but every one of them runs the same loop of requirements, design, test and change, and every one of them is judged by whether the thing works for somebody else.

Engineer, scientist, technician: three jobs on one project

Go back to Boonton. A scientist asks what is true and why: does chlorine kill the bacteria in this water, and how? An engineer asks what will work within limits: can we dose 40 million gallons a day, safely, reliably, at a price the city will pay, starting this year? A technician keeps the answer working: measuring the dose, calibrating the equipment, repairing the pump at three in the morning. The three need each other. In the engineering technician's hands the plant either runs or it does not.

One difference deserves attention because it surprises people. An engineer is often required to act before the science is complete. Fuller could not wait for a full theory of how chlorine attacks every microbe; the city was drinking the water now. Engineers handle that uncertainty in two ways you will practise in this course: they test, measuring what their design really does rather than what it should do, and they build in margin, designing a part to carry more than it will ever be asked to. You will meet that margin as the factor of safety when you calculate stress.

How an engineer earns a licence

In the United States a qualified engineer can become a licensed Professional Engineer (PE). The National Council of Examiners for Engineering and Surveying (NCEES), which writes the exams, describes three steps: generally a bachelor's degree from a program accredited by ABET; in most states, four years of verified work experience; and two exams, the Fundamentals of Engineering (FE) exam, usually taken near graduation, and the PE exam. NCEES states the purpose plainly: licensure exists to protect the public by ensuring a baseline of proven qualification.

A licence is not needed for every engineering job. The BLS notes, for example, that licensure is not required for entry-level mechanical engineers, that a PE licence allows higher levels of leadership and independence, and that mechanical engineers who sell their services to the public must be licensed in every state and the District of Columbia. Engineering degree programs are built on mathematics and the physical sciences, so the high school courses that keep the door open are the ones you would guess: as much mathematics as your school offers, physics and chemistry, and enough writing practice to produce a report someone else can follow.

The problem this course will carry from start to finish

Here is a problem many people know from their own homes. Two people share a bedroom. One has to be asleep by 8:30 p.m.; the other has reading and homework until 10. The ceiling light is all or nothing, and a phone screen is a poor reading lamp. It sounds small. It has everything a real engineering problem needs: a user who is not you (the sleeper, who never asked to be part of your project), a physical quantity you can measure (light where it is wanted and light where it is not), limits on money, power and safety, and more than one reasonable answer.

This course treats it as a real design job. In the next lesson you turn the complaint into a written brief with constraints and criteria. In lesson 3 you score four concepts in a decision matrix. In lesson 4 you build the winner out of card and test it. After that the reading light keeps coming back: you sketch its hood, check the fit of its joints, choose a material for its arm, stop its base tipping over, calculate the resistor for its LED, work out its energy use for a year, write a program that switches it on at dusk, cost it at ten thousand units, and weigh it over its whole life. The last lesson shows its complete design report, the model for your own capstone project.

The kit for the whole course is deliberately cheap: a ruler with millimetre marks, a kitchen scale that reads to 1 gram, card from cereal boxes, drinking straws, masking tape, a small battery pack, a few LEDs and resistors, and a multimeter if you can borrow one. Nothing in this course asks you to work on mains electricity. The wall socket, anything plugged into it, and the inside of any plug-in device are off limits; every circuit here runs from batteries or a USB power bank.

Common misconceptions

• "Engineers are people who work on engines." The word is older than the engine. It comes from the Latin ingenium, cleverness, and ingeniare, to contrive or devise. Engines got their name from the same root.
• "Engineering is mostly inventing new things." The BLS duty lists are dominated by analysis, testing, cost estimates, documentation and supervision. Invention happens, but most engineering time goes into making a known idea work reliably, like Fuller's dosing plant.
• "Scientists discover and engineers just apply what scientists found." Engineers produce knowledge that nobody had before, and much of it comes from testing and failure. Nobody knew how long an implanted pacemaker would last until the first one stopped after three hours.
• "You need a four-year degree to work in engineering at all." Engineering technicians and technologists typically enter with an associate's degree, and the BLS counted 70,900 civil engineering technicians alone in 2025.

What to carry forward

Engineering is the work of making something known to be true happen reliably, at scale, safely and for a price, and proving it with measurements. George Fuller's chlorination plant and Rune Elmqvist's pacemaker show the pattern: the idea worked first, and the engineering was everything that made it keep working. The BLS duty lists show what the job is day to day: analysis, design, prototypes, tests, cost estimates, permits, documentation and supervision, with a design loop of prototype, test, study, change at the centre. The disciplines, from civil with 380,600 jobs to chemical with 21,900, differ in the problems they take on. A bachelor's degree is the usual entry for engineers and an associate's degree for technicians, and licensure as a PE, through the FE exam, experience and the PE exam, exists to protect the public.

Key idea: an engineer's product is never just an idea. It is a thing that keeps working for someone else, together with the calculations, tests and documents that show it will. That is the standard every activity in this course asks you to meet.

Sources

1. U.S. Bureau of Labor Statistics. (2026). Architecture and engineering occupations. Occupational Outlook Handbook. U.S. Department of Labor. BLS
2. U.S. Bureau of Labor Statistics. (2026). Mechanical engineers and Civil engineers. Occupational Outlook Handbook. U.S. Department of Labor. BLS
3. Centers for Disease Control and Prevention. (1999). Achievements in public health, 1900-1999: Control of infectious diseases. Morbidity and Mortality Weekly Report, 48(29). CDC
4. National Council of Examiners for Engineering and Surveying. (n.d.). Licensure. NCEES. NCEES
5. Wikipedia contributors. (2026). John L. Leal. In Wikipedia, The Free Encyclopedia. Wikipedia

"Your light keeps me awake"

"Your light keeps me awake." Five words, said into a pillow at 9:40 on a school night by someone who has to be up at 6:30. The person at the desk has two hours of homework left and one ceiling light, which is on for both of you or off for both of you. Most engineering starts exactly here: not with an idea, but with somebody unhappy about something.

Now try to design something from that sentence. You cannot, and it is worth seeing why. How much light counts as keeping someone awake? How much does the reader need? Could the answer cost $200, or plug into the wall? Is a sleep mask an answer? Is moving the homework to the kitchen an answer? The complaint does not say. A complaint tells you that something is wrong. A design brief says what right would look like, precisely enough that two people could test the same design and agree on whether it passed.

This lesson turns the complaint into a brief, one part at a time, and ends with the complete brief that the next three lessons build on. First, look at a brief that people spent nearly sixty years trying to meet.

Four ships on the rocks, and a brief with a number in it

On the evening of 22 October 1707, five ships of a British fleet under Admiral Sir Cloudesley Shovell struck the rocks off the Isles of Scilly, at the south-western tip of England. Within hours four had sunk, and more than 1,400 men, the admiral among them, were dead. The Scilly naval disaster has been blamed on several things at once: the navigators' inability to work out where they were, errors in their charts, and poor compasses. The first was the old, hard problem. A navigator could find latitude, the position north or south, from the height of the Sun at noon. Finding longitude, the position east or west, was another matter, because at sea there was nothing reliable to measure it against.

"Ships get lost" had been a complaint for centuries. In May 1714 merchants and sea captains petitioned Parliament, and on 9 July 1714 Parliament passed the Longitude Act. Read what it asked for:

• A result, not a method. A practical way of finding longitude at sea.
• A number. £10,000 if the error was no more than one degree, £15,000 for no more than 40 minutes of arc, £20,000 for no more than half a degree.
• A test. Under the rules for the longitude rewards, a method would be tried on a voyage from Britain to a port in the West Indies, about six weeks at sea, and had to keep its longitude within those limits.
• A judge. The Act set up the Board of Longitude to decide claims.

Put the number into distance and time and you can see how hard it was. One degree of longitude is 60 nautical miles at the equator, and a nautical mile is 1.852 km, so half a degree is 30 nautical miles, and 30 × 1.852 km = 55.6 km. Earth turns 360 degrees in 24 hours, which is 360 ÷ 24 = 15 degrees an hour, or one degree every 4 minutes. A ship's clock that kept the time of the home port could give longitude from the difference between home time and local noon, and half a degree meant an error of 0.5 × 4 = 2 minutes. To win £20,000 with a clock, the clock had to gain or lose less than 2 minutes over six weeks on a rolling, damp ship, through swings of temperature that change the size of metal parts and so the rate at which a clock runs.

The Act did not say "build a better clock", and that silence mattered. Astronomers worked on the lunar distance method, which reads the time from the position of the Moon among the stars; the British government later granted £3,000 to the widow of the German astronomer Tobias Mayer for his tables of the Moon. John Harrison, an English joiner and clockmaker, took the other road and built a series of marine timekeepers. His fourth, the watch now called H4, dated 1759, was tested on a voyage to Jamaica in 1761-62 and found to be just 5 seconds slow on arrival. Five seconds of time is 5 × 15 ÷ 3,600 = 0.021 degrees, or 1.25 minutes of arc: roughly one nautical mile, against a prize limit of thirty. The Board suspected luck and demanded a second trial. Harrison received £10,000 in 1765, and in 1773, at the age of 80, a further £8,750 from Parliament.

A brief that said "improve navigation" would have given the Board nothing to measure. A brief that said "build a better clock" would have shut out the Moon tables. The Longitude Act named the result, put a number on it, said how it would be tested and who would judge, and left the method open. That is the whole shape of a good brief.

The point: a brief states the result a user needs, with a number and a test attached, and leaves the method open. The rest of this lesson applies that sentence to a bedroom.

The five parts of a brief

Engineering organisations use different names for them, but a working brief carries five things. The table gives each one the question it answers, and shows it in the 1714 brief and in ours.

Step 1: a need statement that does not smuggle in a solution

The commonest mistake in a first brief is to write down the answer. "Design a desk lamp with a hood" sounds like a brief, but it has already decided that the answer is a lamp, on a desk, with a hood. Every other answer has been thrown away before anyone thought about it. The guide to writing requirements in the NASA systems engineering handbook puts the rule in one line: requirements should state what is needed, not how to provide it; state the problem, not the solution.

So write the need in terms of the result:

A reader needs enough light on a page to read and write from 8:30 to 10:30 p.m., while a sleeper about 2 m away receives almost none of it.

Count the answers that statement still allows: a hooded desk lamp; a small light clipped to the book; a headband light with a narrow beam; a folding screen between desk and bed; a light built into a clipboard; even a tablet whose own screen lights the page. Some of these are poor ideas, and that is fine. Lesson 3 is where you choose, and you cannot choose among ideas the brief has already thrown out.

Step 2: everyone the design touches

The reader is the obvious user. The sleeper is the important one. The sleeper will never switch the light on, never carry it, never pay for it, and the design still succeeds or fails on the pillow. Engineers call the wider group stakeholders: everyone the design affects, whether or not they use it. For the reading light they are the reader, 13 to 17 years old, working at a desk or sitting up in bed; the sleeper, asleep from 8:30 p.m. with their head about 2 m from the desk; whoever pays for the lamp and its batteries; and whoever tidies the room, who will want it off the floor and out of the way.

A list like this is not a formality. Each name on it turns into a line of the brief. The sleeper gives you the light limit at the pillow. The person paying gives you the cost limit. The reader, fumbling in the dark, gives you the requirement about finding the switch.

Step 3: constraints, the lines no design may cross

A constraint is a limit that every design must satisfy. It is pass or fail: a design that meets it earns no extra credit for meeting it by more, and a design that breaks it is out, however well it does on everything else. For the reading light:

1. C1 Power. The light shall run from batteries or a USB power bank only. Reason: nothing in this course connects to mains electricity, and a lamp that plugs into the wall would need wiring skills and safety approval outside the course.
2. C2 Cost. Materials for one lamp shall cost no more than $15. Reason: a budget a family would accept for fixing a sleep problem. Lesson 21 returns to cost when thousands are made.
3. C3 Size. The lamp shall stand within a shelf area of 30 cm by 20 cm. Reason: that is the measured free space on the bedside shelf.
4. C4 Touch temperature. No surface the user can touch shall exceed 40 °C after 2 hours of use in a 20 °C room. Reason: the lamp sits beside a bed and will be grabbed in the dark.

Every one of those numbers is a decision, and each has its reason written beside it. A different team in a different house could defend $20 or a 25 cm shelf. What makes a number useful is that it is written down with its reason, so that if the reason changes, the number can be changed openly rather than quietly.

Step 4: criteria, and the numbers that make them testable

A criterion (plural criteria) measures how well a design does, where more of something, or less, is better. Criteria are what lesson 3's decision matrix will score. Each one needs a target and a way to measure it, and the trouble is that the first words that come to mind are the ones nobody can measure. NASA's checklist lists words to keep out of requirements because they cannot be verified, among them easy, sufficient, safe, adequate, user-friendly, fast, small and light-weight. Almost all of them turn up in a typical first brief for a reading light.

Where do the numbers come from? The amount of light landing on a surface is its illuminance, measured in lux: one lux is one lumen of light per square metre. Typical office lighting runs from about 320 to 500 lux. The lights of a family living room, in one Australian survey from 1998, gave about 50. A full moon on a clear night gives between 0.05 and 0.3. For reading and writing on paper, 300 lux is a sensible target just below the office range, and a phone light meter can read it.

The page size decides where to measure. An A4 sheet is 210 mm by 297 mm, so its diagonal is the square root of (210² + 297²) = 363.7 mm; a US Letter sheet, 215.9 mm by 279.4 mm, has a diagonal of 353.1 mm. The pool of useful light has to be about 36 cm across or the corners of the page go dim, which is why R1 is measured at five points and not one.

R3 comes from the use. From 8:30 to 10:30 p.m. is 2 hours, and ten evenings between battery changes is 10 × 2 = 20 hours. R5 is a judgement about what can be pushed around a shelf with one hand; lesson 10 checks whether a lamp that light will tip over.

Working one number out: how much light reaches the pillow?

R2 is the requirement the whole design turns on, and a short calculation shows why before anything is built. For a small lamp shining equally in all directions, the inverse-square law says illuminance falls with the square of the distance: twice as far, a quarter of the light; five times as far, a twenty-fifth. The reason is geometry. The same light spreads over the surface of a sphere, and a sphere's area grows with the square of its radius.

Suppose the lamp gives 300 lux on the page at 0.40 m. The pillow is 2.0 m away, which is 2.0 ÷ 0.40 = 5 times as far. With nothing in the way:

E at the pillow = 300 lux × (0.40 m ÷ 2.0 m)² = 300 lux × 0.04 = 12 lux

Twelve lux is 12 ÷ 0.3 = 40 times the light of the brightest full moon, falling on the sleeper's face. To meet R2 the design has to cut the direct light toward the pillow by a factor of 12, which means stopping at least 11 parts in 12, about 92 percent, of the light heading that way.

That is an estimate, and you should know its limits. A real LED is not a point shining equally in all directions, and walls and ceilings bounce light around a room. But look at what one line of arithmetic has already told you. No bare bulb can pass, however dim or bright, because any lamp bright enough for the page is 12 times too bright at the pillow. Shielding is not a finishing touch to the design. It is the problem.

A common slip is to scale with the distance instead of its square: 300 lux × 0.40 ÷ 2.0 = 60 lux. That overstates the pillow light fivefold, and it would send you off to design a much heavier hood than you need.

Step 5: say how each requirement will be checked

A requirement nobody can check is a wish. NASA's checklist asks of every requirement whether the system can be tested, demonstrated, inspected or analysed to show that it meets it, and those four words are the standard verification methods:

• Test: measure it under set conditions. R1, R2 and C4 are tests: a light meter at five points, a light meter at the pillow with the lamp on and off, a thermometer after two hours.
• Demonstration: show that it works in use, without detailed measurement. R4 is a demonstration: a person who has never seen the lamp switches it on in the dark while you time them.
• Inspection: look at it or read it. C1 is an inspection: there is no mains plug, and the power bank's label says USB.
• Analysis: calculate it. C2 is an analysis of the receipts, and R3 can be analysed from the battery's capacity and the lamp's current before it is ever tested, which you will do in lesson 17.

The Board of Longitude worked this way. When Harrison's watch came back from Jamaica 5 seconds slow, the Board did not trust a single result and asked for another voyage. A test repeated under real conditions is how an engineer tells a design that works from a design that got lucky.

The brief, written out

Here is the complete brief for the reading light, in the shall form NASA's guide uses for requirements. The next three lessons use it: lesson 3 scores four concepts against R1 to R5, and lesson 4 tests a card prototype against it.

Notice what is not in it. There is no hood, no LED, no clip, no colour. Those are answers, and the brief is the question.

Common misconceptions

• "A good brief describes the product you want." That describes one solution. A brief describes the result, so that ideas nobody has had yet can still compete. The Longitude Act never mentioned a clock, and both a clock and a set of Moon tables were paid for.
• "Constraints and criteria are two names for the same thing." A constraint is pass or fail: $15.01 is out, and $9 earns nothing more than $14. A criterion is better or worse: 400 lux beats 310. Mixing them up wrecks a decision matrix, because a concept that breaks a constraint can still score well on the criteria and win.
• "Light fades in proportion to distance." From a small source it fades with the square of the distance. Five times as far gives a twenty-fifth of the light, not a fifth.
• "The numbers in a brief have to be facts." Most are decisions. What they need is a reason written beside them, so that they can be argued with and, when the reason changes, changed in the open.

The short version

A complaint says something is wrong; a brief says what right looks like, well enough that two people can test a design and agree on the result. A brief has five parts: a need statement that names the result and not a solution, the users and stakeholders the design touches, constraints that every design must pass, criteria that tell better from worse, and a verification method for each, by test, demonstration, inspection or analysis. The Longitude Act of 1714 had all five: a result, a number (half a degree, which is 30 nautical miles at the equator or 2 minutes of time), a six-week test voyage, a Board to judge, and no method named, which is why both a clock and the Moon tables won money. For the reading light, one inverse-square calculation, 300 lux × (0.40 ÷ 2.0)² = 12 lux at the pillow, showed before anything was built that keeping light off the sleeper is the heart of the problem.

The point: write the problem, not the solution, and put a number and a test on every line. A brief like that tells you when you have finished; a brief without them lets the argument run forever.

Sources

1. NASA. (2016). Appendix C: How to write a good requirement. In NASA systems engineering handbook (NASA/SP-2016-6105 Rev2). NASA
2. Royal Museums Greenwich. (n.d.). What made the search for longitude so important? Royal Museums Greenwich
3. Wikipedia contributors. (2026). Longitude Act. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). John Harrison. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Wikipedia contributors. (2026). Lux. In Wikipedia, The Free Encyclopedia. Wikipedia

Four sketches and an argument that goes round in circles

Four sketches lie on the desk, each one an answer to the brief from lesson 2. A is a small light that clips to the cover of the book on a short flexible neck. B is a lamp on a hinged arm with a deep hood around its LED, standing on the shelf. C is a light on a headband with a narrow beam. D is a folding card screen, 60 cm tall, standing on the floor between the desk and the bed, with any battery lamp behind it.

The reader likes C, because the light goes wherever your eyes go. The sleeper likes D, because nothing gets past a wall. Both reasons are good, and an argument between them can go round for an evening without moving, because each person keeps naming the one thing their favourite does best. What neither has done is put every concept against every requirement at once, with an agreed idea of how much each requirement matters. That is the job of a decision matrix. This lesson builds one step by step, then changes one fact and watches the answer change.

Franklin's two columns

The idea is older than engineering as a profession. In 1772 Lord Shelburne, prompted by Benjamin Franklin and Richard Price, asked the chemist Joseph Priestley to direct the education of his children and act as his general assistant. That same year Franklin described, in a letter to Priestley, how he made hard decisions himself. He wrote the reasons for and against in two columns, estimated how much each one weighed, and struck out reasons of about equal weight on opposite sides, or groups of reasons that balanced each other, until one column was left dominant. Versions of it are still taught, under names such as the decisional balance sheet.

Franklin's method handles one choice at a time, yes or no. An engineer usually has several concepts and several requirements, and needs numbers that somebody else can check. The weighted decision matrix keeps Franklin's two big ideas, write everything down and give each reason a weight, and puts them in a grid. It runs in six steps.

Step 1: screen out anything that breaks a constraint

Constraints are pass or fail, so they come first and they are never scored. Check each concept against C1 to C4 from the brief.

D is out. That can feel unfair, because D blocks light better than anything else on the desk. But the constraint was written for a reason, the free space available, and a screen filling the gap between desk and bed breaks it. If the team believes the constraint is wrong, the honest move is to go back and change the brief, openly, with a reason. What the team may not do is let D compete while it breaks the rules. At the end of the lesson you will see how close D would have come to winning if nobody had checked.

Step 2: set the weights before anyone scores anything

A weight says how much a criterion matters compared with the others. The simplest way to set weights is to share out 100 points among the criteria, and to write a reason beside each share.

Set the weights before anyone scores a concept, and keep the record. Choose weights after the scores are known and it becomes far too easy, usually without noticing, to pick the weights that make your favourite win. The chapter on decision analysis in NASA's systems engineering handbook lists the rank of each criterion by its importance among the things a team settles when it defines its criteria.

Step 3: write down what each score means

Each concept gets a score from 1 to 5 on each criterion. A score is only useful if two people would give the same concept the same number, and that needs a written scale. NASA's handbook calls these operational definitions of the scale. Here are the reading light's.

At the concept stage most scores are estimates, made from sketches, quick card mock-ups and the numbers printed on parts. That is normal. The estimates carry uncertainty, which is exactly why step 6 exists.

Step 4: score, multiply, add

Each cell below shows a concept's score on a criterion, then the score multiplied by the weight. The weighted total is the sum of those products. The best possible total is 5 × 100 = 500, so the last row turns each total into a percentage of that maximum. The scores are illustrative estimates for this worked example.

Work B's column once by hand to see the arithmetic: 5 × 25 + 4 × 30 + 5 × 20 + 5 × 15 + 2 × 10 = 125 + 120 + 100 + 75 + 20 = 440. The reasons behind the scores matter as much as the numbers. A's light sits close to the page, which makes it bright, but only small AAA cells fit in a clip, so its running time scores 3. B has room in its base for a USB power bank and a big switch, but the base is heavy, so it scores 2 on mass. C's beam lights whatever the reader looks at, and swings toward the bed every time the reader turns their head, which is the one thing the brief exists to prevent.

Step 5: read the matrix, not just the total

B wins by 65 points out of 500. The matrix also says where the others lose, if you subtract B's products from A's, criterion by criterion: page light −25, pillow light 0, running time −40, switch −30, mass +30. A loses mostly on running time and the switch. That is useful, because it points at a hybrid. A clip-on light with its batteries in a pack on a cable and a large switch on the pack might score 5 on R3 and R4, while the extra mass drops R5 from 5 to 4: 375 + 40 + 30 − 10 = 435, within 5 points of B. A hybrid is a new concept, and it goes through steps 1 to 4 like the others.

C loses 60 points to B on the pillow light alone. No amount of running time or lightness makes up for a beam that sweeps toward the sleeper, and the weights say so.

What matters here: the total picks a leader, but the rows tell you why it leads and what a rival would have to fix. Read both.

Step 6: push on the answer and see whether it moves

The weights are judgements, and the scores are estimates. A sensitivity check asks how far they would have to move before the ranking changed. A is lighter than B, so try moving weight from running time to mass, a few points at a time:

A only draws level when mass is weighted 23 and running time 7, so that mass counts more than three times as much as running time. Nobody writing the brief argued that, so B's lead is robust. NASA's handbook frames the question the same way: could the uncertainties credibly change the ranking? Here they could not. Had A drawn level at, say, R3 18 and R5 12, the honest conclusion would have been that the matrix cannot choose, and the next step would be to build both.

A second run: one changed fact flips the result

Now change one fact and run the procedure again. Suppose the reader says that most reading happens sitting up in bed, not at the desk. The weights do not change, because the brief's priorities have not changed. What changes is how B performs. A lamp on the shelf, about a metre from a book held in bed, lights the page weakly: R1 falls from 5 to 2. To reach the bed at all it must be aimed across the room, and its LED is hidden from the pillow only if it is aimed with care: R2 falls from 4 to 3. A and C go wherever the book or the head goes, so their scores stand.

B's new total: 2 × 25 + 3 × 30 + 5 × 20 + 5 × 15 + 2 × 10 = 50 + 90 + 100 + 75 + 20 = 335. A still scores 375, so A now wins by 40. Run step 6 again and you find that B only draws level if running time is weighted 28 and mass 2, which nobody would defend, so this result is robust too. The same method, the same weights, a different fact, a different winner.

For this course's reader, whose homework is done at the desk, the first run stands and B, the hooded lamp, goes forward to be prototyped in lesson 4. But the second run found a gap in the brief: R1 never said where the page would be. A good matrix sends you back to the brief as often as it sends you forward.

A faster first pass: the Pugh matrix

When there are many rough concepts and no numbers yet, a lighter tool helps. Stuart Pugh (1929-1993), a British product designer, developed a method of concept selection that compares every concept with one chosen reference, the datum, and marks each criterion + (better), S (same) or − (worse). With B as the datum:

A Pugh matrix is quick and needs no weights, which is its strength early on and its weakness later: it counts a small gain in mass the same as a large gain at the pillow. Its best use is to spot which strengths of weak concepts could be combined into a better one, then to run it again with the new leader as the datum. Once real numbers exist, switch to the weighted matrix.

What the matrix would have done without step 1

Go back to D, screened out in step 1. Score it anyway: 5 × 25 + 5 × 30 + 5 × 20 + 4 × 15 + 1 × 10 = 125 + 150 + 100 + 60 + 10 = 445. D would have beaten B by 5 points. A team that skipped the constraint check would have picked a design that cannot go where it has to go, by a margin of 5 in 500, 1 percent, far smaller than the uncertainty in any estimated score. Both of the matrix's rules are in that one example: constraints first, and never trust a tiny margin.

Common misconceptions

• "The matrix makes the decision." People make it; the matrix makes their reasoning visible. NASA's handbook says that when a team recommends an option other than the top scorer it should explain why the lower score is preferred, and that the criteria themselves, not only the weights, should then be looked at again.
• "A higher total always means a better design." Only if the gap is larger than the uncertainty in the scores. Totals of 402 and 398 from estimated scores are a tie, and the next step is a test.
• "Equal weights are the neutral, fair choice." Equal weights are a judgement too: they say that 10 grams of mass matter as much as light on the sleeper's face. With equal weights of 20, A scores 380 and B 420, a different margin from the brief's real priorities.
• "If the result looks wrong, adjust the weights until it looks right." That turns the matrix into a costume for a decision already made. If the result looks wrong, check the scores and the scale definitions, and if a weight really was wrong, change it with a written reason and rerun everything.

Putting it together

A weighted decision matrix turns an argument about favourites into a record anyone can check. The procedure runs in six steps: screen every concept against the constraints and drop those that fail; set weights that share out 100 points, with reasons, before anything is scored; write down what each score from 1 to 5 means; score, multiply by the weights and add; read the rows as well as the totals; and push on the weights and scores to see whether the ranking moves. For the reading light, the hooded lamp B won with 440 of 500 against the clip-on light's 375, and held its lead unless mass counted more than three times as much as running time. One changed fact, reading in bed, dropped B to 335 and put A ahead, and exposed a gap in the brief. The screen, D, would have won by 5 points if the constraint check had been skipped. Franklin's two columns and Pugh's datum matrix are the same idea at lighter weight.

What matters here: the matrix is only as honest as its weights, scales and facts, so fix those first, write them down, and treat any margin smaller than your uncertainty as a tie to be settled by building and testing.

Sources

1. NASA. (2016). 6.8 Decision analysis. In NASA systems engineering handbook (NASA/SP-2016-6105 Rev2). NASA
2. Pugh, S. (1990). Total design: Integrated methods for successful product engineering. Addison-Wesley.
3. Wikipedia contributors. (2026). Decision-matrix method. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Decisional balance sheet. In Wikipedia, The Free Encyclopedia. Wikipedia

Kitty Hawk, summer 1901: a glider that would not lift its pilot

In the summer of 1901 Wilbur and Orville Wright, two brothers who built and sold bicycles in Dayton, Ohio, took their second glider to the sand hills near Kitty Hawk, North Carolina. They had worked out its size with the best formula and the best published data they could find. On paper, flown as a kite in a 15 mile-per-hour wind with its wings tilted 5 degrees into the wind, it would lift its own 100 pounds plus a 150-pound pilot: 250 pounds in all.

In the wind it could barely lift itself, even with the wings tilted much more steeply. NASA's account of the tests puts it at about a third of the lift the brothers had predicted. The glider did fly, frequently, sometimes 300 feet in a single glide, but nothing like the way the numbers said it would. On the way home, Wilbur remarked that man would not fly in a thousand years.

Look at where the brothers stood. They had a prediction, a prototype and a test, and the test disagreed with the prediction by a factor of about three. Something in the chain was wrong: the glider, the way they flew it, or the numbers they had designed it with. Guessing would not tell them which. This lesson follows how they found out, and then runs the same loop, far more cheaply, on a card prototype of the reading light.

Two borrowed numbers

The brothers had designed the 1901 glider with the lift equation of the day:

L = k × V² × A × C_L

Here L is the lift in pounds, V the airspeed in miles per hour, A the wing area in square feet, and C_L the lift coefficient, a number describing how well a particular wing shape lifts at a particular angle. The fourth symbol, k, was the Smeaton coefficient, named after the eighteenth-century engineer John Smeaton: the pressure force on a flat plate one foot square moving through the air at one mile per hour. Published values ran from 0.0027 to 0.005. The German glider pioneer Otto Lilienthal had used 0.005, and the Wrights took that value. Their lift coefficients came from Lilienthal's tables too.

Two of the four numbers in their prediction were borrowed from someone else's experiments. The speed and the area they could measure themselves. When a prediction built on borrowed numbers fails, the borrowed numbers are the first suspects, but suspicion is not evidence. The brothers needed a way to measure k and C_L for their own wings.

A cheaper question: the wind tunnel in the bicycle shop

They could have built a third full-size glider and waited a season to try it at Kitty Hawk. Instead they built a much cheaper prototype, not of the aircraft but of the question. From September to December 1901, in their shop, they ran a wind tunnel of their own: a simple open design, with a fan pushing air through an enclosed square box about 6 feet long. Into the airflow went model wings made from strips of 20-gauge steel, cut, hammered, filed and soldered into shape. They made between one and two hundred models for quick first tests, then chose about 30 of the best designs for detailed study. To measure the forces they built two balances, one for lift and one for drag, from bicycle spokes and scrap metal.

The tunnel answered both questions. The Smeaton coefficient was closer to 0.0033 than to 0.005, a figure modern aerodynamicists have confirmed as very accurate. And Lilienthal's lift tables were correct for the wings he had measured, but could not be applied to a wing of very different shape. His wings were short in span; the Wrights' were long, thin rectangles. His data described his wings.

The size of the first error is worth working out. Lift is proportional to k, so a prediction made with 0.005 instead of 0.0033 was too high by a factor of 0.005 ÷ 0.0033 = 1.52. Every lift prediction the brothers made in 1901 was about 50 percent too optimistic before anything else went wrong, and the borrowed lift coefficients made it worse.

1902: a prototype designed on their own numbers

The 1902 glider was designed from the tunnel data. Its wing spanned 32 feet 1 inch (9.78 m) and had an aspect ratio, span divided by the front-to-back depth of the wing, of about 6.5 to 1. Between 20 September and late October 1902 the brothers made more than a thousand flights with it, some lasting up to 26 seconds and covering more than 600 feet. After they added a movable rudder, they made their first truly controlled turns on 9 October. On 17 December 1903, with an engine built by their shop mechanic, Charlie Taylor, and wooden propellers the brothers designed from more wind tunnel tests, Orville made the first powered flight of the Wright Flyer: 120 feet in 12 seconds.

Now read the story as a list of prototypes, each built to answer one question. The 1900 and 1901 gliders asked: does a glider designed with the published numbers lift what they predict? (No.) The tunnel asked: what are the right numbers for our kind of wing? (k near 0.0033, and C_L measured on wings like ours.) The 1902 glider asked: does a wing designed with our numbers fly and turn? (Yes, a thousand times.) Each question was answered by the cheapest thing that could answer it, and a model wing of hammered steel costs far less than a glider and a season on the sand.

Remember: a prototype is a question in physical form. Build the cheapest one that can answer the question you actually have, and trust your own measurements over borrowed ones that were made on something else.

What a prototype is for

A prototype is a model built to learn something about a design before the design is final. Prototypes differ in fidelity, how closely they resemble the real thing. A card hood taped around a phone torch is low fidelity: it looks nothing like a finished lamp. But it can answer the question that decided lesson 3's matrix, whether a hood can keep light off the pillow while lighting the page, and it can answer it tonight for the cost of a cereal box.

No single prototype answers everything, so choose each one by its question:

The Wrights' tunnel belongs in the same table. Its question was "what are k and C_L for our wings?", and it could say nothing at all about steering a glider in a gust.

Planning the test before cutting any card

A test is only useful if its result can be compared with something, and the brief supplies that. The card hood is built to test two requirements: R1, at least 300 lux at five points of an A4 page, and R2, a rise of no more than 1 lux at the pillow. The plan fixes everything else so that the only thing changing between versions is the hood:

• Room lights off and the door closed, so that no other light reaches the meter.
• The same phone for every reading, lying flat, with the five page points and the pillow position marked with tape.
• The torch 0.40 m above the page, as in lesson 2's calculation, and the pillow 2.0 m away.
• Every reading taken three times, keeping the middle one; the pillow read with the light on, then off.

A little geometry first says whether the idea can work at all. The corners of an A4 page are 363.7 ÷ 2 = 182 mm from its centre, so at 0.40 m below the light, the useful beam must spread at least as far as the angle whose tangent is 0.182 ÷ 0.40 = 0.455, about 24 degrees from straight down. In this illustrative room the pillow is 2.0 m away and about 0.5 m lower than the lamp, only about 14 degrees below the horizontal. A hood that stops all light more than about 45 degrees from straight down could light the whole page and still hide the torch from the pillow. So the concept can pass on paper. The test finds out what paper left out.

Four versions of a card hood

Here is a test log for four versions. The numbers are illustrative, of the kind a phone light meter gives, and each version changes exactly one thing from the one before.

V1 failed both requirements, and one failure was a surprise. Standing at the pillow in the dark, the whole cone glowed like a lampshade. Thin card lets light through, and lesson 2's calculation had quietly assumed a hood that blocks light perfectly. The corners were dim for a different reason: a phone torch concentrates its light in the middle of its beam, so the centre got 410 lux and the worst corner 240.

V2 tackled the glow alone. Foil on the outside made the cone opaque, and the pillow rise halved from 4 lux to 2. Because nothing else changed, the log proves that the glow was half of the pillow problem. The other half had to be light escaping some other way.

V3 tackled the corners. A white paper lining scatters light inside the cone, spreading it more evenly, and the dimmest corner rose to 310 lux: R1 passed. The pillow stayed at 2 lux, and looking back from the pillow showed why. Near the rim, at a low angle, the brightly lit white lining was visible from the bed.

V4 added a visor on the bed side of the rim, which hides the lining from the pillow without shading the page. The pillow reading fell to 0, meaning less than this phone could detect, and the page still passed. From V1 to V4 the dimmest corner rose from 240 to 305 lux, by a factor of 305 ÷ 240 = 1.27, and the pillow rise went from 4 lux to below the meter's reach.

Two things in this log are easy to miss. First, if V2 had added the foil and the lining together, the result would have shown the pillow falling and the corners rising with no way of telling which change did what. One change per version is slower on paper and faster in practice. Second, the most useful result was V1's failure. It exposed an assumption, the perfectly opaque hood, that no amount of calculation would have questioned, in the same way that the 1901 glider exposed the borrowed coefficients.

Iterating like this, build, test, learn, change one thing, test again, is what engineers call iterative design. It is the loop from the Bureau of Labor Statistics duty list in lesson 1 (develop and test prototypes, analyze the results, change the design) running on a desk.

Knowing when to stop

There are three good reasons to stop iterating. The prototype meets the requirements it was built to test, as V4 does for R1 and R2. The tests show the concept cannot meet them, in which case you go back to the decision matrix, where lesson 3's clip-on light is waiting. Or the time and budget run out, in which case you stop and report honestly how far you got. A bad reason is that the prototype has started to look nice.

V4 has done its job, and only its job. It says nothing about R3, R4 or R5, nothing about touch temperature, and nothing about whether a real hood on an arm will stay where it is put. Those need their own prototypes and tests. What it does give the design is a set of numbers to carry forward: a hood 8 cm deep and 14 cm across, opaque outside, matt white inside, with a 4 cm visor on the bed side. Lesson 5 turns those numbers into drawings.

Common misconceptions

• "A prototype is an early version of the finished product." Often it is nothing like the product. The Wrights' steel model wings could not carry anyone, and the card hood will never sit on a shelf. A prototype is judged by whether it answered its question.
• "A failed test means the design failed." A failed test on a cheap prototype is the cheapest information you will ever get. The 1901 glider's failure sent the brothers to build the tunnel that made 1902 work.
• "Changing several things at once saves time." It saves one test and loses the knowledge of which change did what. When the result improves, you cannot tell which change to keep; when it gets worse, you cannot tell which to undo.
• "Published data can be used as they are for any design." Data describe the conditions they were measured in. Lilienthal's tables were right for Lilienthal's wings and wrong for the Wrights'.

Looking back

The Wrights' 1901 glider was designed to lift 250 pounds and could barely lift its own 100. Instead of guessing, the brothers built a cheaper prototype of the question: a wind tunnel, one to two hundred steel model wings, and balances made from bicycle spokes. It showed that the Smeaton coefficient was near 0.0033, not 0.005, which had made every prediction about 1.52 times too high, and that Lilienthal's lift tables did not apply to a long, thin wing. The 1902 glider, designed on their own numbers, made more than a thousand flights. The same loop on a card hood took four versions, each changing one thing: foil stopped the hood glowing and halved the pillow light, a white lining lifted the dimmest corner past 300 lux, and a 4 cm visor hid the lining from the bed. The first version's failure taught the most, because it exposed an assumption no calculation had questioned.

Remember: build rough, test against the brief, change one thing at a time and write every result down. A test that disagrees with your prediction is not bad news; it is the only kind of news that can tell you something you did not already believe.

Sources

1. NASA Glenn Research Center. (n.d.). 1901 wind tunnel tests. In Beginner's guide to aeronautics. NASA
2. NASA Glenn Research Center. (n.d.). Lift equation of the 1900s. In Beginner's guide to aeronautics. NASA
3. Wikipedia contributors. (2026). Wright brothers. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Iterative design. In Wikipedia, The Free Encyclopedia. Wikipedia

A square with a circle in it: what is it?

Here is a drawing of a part, the way it might arrive in a message from someone who needs it made: a square 40 mm on each side, with a circle 20 mm across in the middle.

Before reading on, decide what the object is, and how you would make it.

Here are five objects that all look exactly like this from the front. A thin square plate with a round hole through it. A 40 mm cube with a round hole bored all the way through. A 40 mm cube with a round hole only 5 mm deep. A square plate with a round peg standing out of its face, like a button. A square frame with a round rod standing inside it. The drawing cannot tell them apart, for two reasons. It shows only two of the object's three dimensions, width and height, and says nothing about depth. And a circle drawn with a single line cannot say whether it is the edge of a hole going in or of a peg coming out.

Make the part from this drawing and you would be guessing, and if you guessed wrong, the fault would belong to the drawing as much as to you. Engineering drawings exist to remove guessing. This lesson shows the two ways engineers put a solid object on flat paper: orthographic views, which are exact, and isometric sketches, which anyone can understand at a glance.

Add a second view and the guessing stops

Now add a view from above. Suppose it shows a long, thin rectangle, 40 mm by 6 mm, with two dashed lines running across it, 20 mm apart. Every question has an answer. The object is 6 mm deep, so it is a plate, not a cube. The dashed lines are edges that cannot be seen from above because they are inside the material, and they run all the way from front to back, so the circle is a hole that goes right through. Had the dashed lines stopped halfway, the hole would stop halfway. Had there been a solid bump standing up from the top edge, the circle would have been a peg.

Add a third view, from the right, and the drawing is complete. Here are all three.

This is an orthographic projection. Each view shows what you would see looking straight at one face of the object, as if from very far away: every line of sight is parallel to the others and square to the paper. That is why each view shows two of the three dimensions at their true size, with no shrinking toward the back, and why a ruler laid on any view measures the real object at the drawing's scale.

The rules that turn three views into one drawing

Three views drawn anywhere on a page are three pictures. A few rules make them one drawing that any engineer can read.

• Three standard views. An object has six principal views: front, rear, top, bottom, left and right. Three are usually enough, and the standard three are the front, the top and the right side. Choose as the front view the one that shows the most about the shape with the fewest hidden lines.
• Alignment. The views share dimensions, so they are lined up. The front and top views share the width, so the top view sits directly above the front view. The front and right side views share the height, so the side view sits directly beside it. The top and side views share the depth. Thin projection lines carry a feature from one view to the next: in the diagram, the edges of the hole drop straight down from the top view to the front view.
• Line types. Visible edges are thick continuous lines. Hidden edges are thin dashed lines, and they stop at the edges of the object. The axis of a hole, a peg or any symmetrical feature is a thin chain line, long dash and short dash, which runs slightly beyond the feature.
• One scale, stated. All views use the same scale, written on the drawing: 1:1 is full size, 1:2 half size, 2:1 twice size. Dimensions are written as the real sizes, in millimetres, whatever the scale.

Read the diagram again with those rules. In the top and side views, the dashed lines are the hole's walls, hidden inside the 6 mm of plate. Each view has a centre line through the hole. The top view's width lines up with the front view's width, and the side view's height lines up with the front view's height. Nothing is left to guess.

In short: one view gives two dimensions and leaves the third to guesswork; three aligned views, with hidden edges dashed and axes marked, describe a solid object completely.

Third angle and first angle: the same views in different places

There are two conventions for where the views go, and they put the top and side views on opposite sides of the front view. In third-angle projection, used in the United States under the ASME Y14.3 standard, and also standard in Canada and Japan and preferred in Australia, the top view goes above the front view and the right side view goes to its right, as in the diagram. In first-angle projection, standard across Europe and most of Asia, the top view goes below the front view and the right side view goes to its left. British practice allows either.

Neither convention is better, but mixing them up is dangerous. Read a first-angle drawing as if it were third angle and you take the view that shows the bottom for a view of the top, and a feature on one side of the part ends up on the other, as in a mirror. So every drawing carries a small symbol of a truncated cone, drawn as a trapezoid beside two circles. In the third-angle symbol the short side of the trapezoid points toward the circles; in the first-angle symbol it points away. Before reading any drawing that came from another country, find that symbol.

Isometric: one picture anyone can read

Three orthographic views are exact but take practice to read. For explaining an idea to someone, or thinking one through yourself, engineers sketch in isometric. The object is turned so that its three main directions look equally important, and the three axes on the page are 120 degrees apart: one vertical, and two at 30 degrees above the horizontal, one going left and one going right.

A true isometric projection would shorten every edge along the axes to about 0.816 of its length, the square root of two-thirds, because the edges are tilted away from you. An isometric drawing skips that and draws edges along the axes at full length, so the picture comes out about 22 percent larger (1 ÷ 0.816 = 1.22) and a ruler can be used directly. That is the version to sketch.

To sketch the plate: draw the three axes from one corner; box the object by marking 40 mm along the right-hand axis, 40 mm up the vertical and 6 mm along the left-hand axis, and complete the box; then add features. A circle on a face becomes an ellipse. The quickest way to get it right is to draw, on that face, the square that would just hold the circle; in isometric the square becomes a rhombus, and the ellipse touches the midpoint of each of its four sides. For the 20 mm hole, the ellipse works out at about 24.5 mm along its long axis and 14.1 mm along its short one, but the rhombus method gets those proportions right without any arithmetic. Hidden edges are usually left out of an isometric sketch. Through the hole in the diagram you can see part of its back edge, 6 mm behind the front face.

Isometric has limits. Only edges parallel to the three axes are true length; a diagonal across a face is not, and right angles are drawn as 60 or 120 degrees. It is a picture for understanding, and the orthographic views remain the drawing a part is made from.

Sketching the reading light's hood

Lesson 4 ended with numbers from the card prototype that passed its tests: a hood 80 mm tall, 40 mm across at the top where the light sits and 140 mm across the rim, with a visor 40 mm deep around the half of the rim that faces the bed. Those numbers are useless to anyone else until they are drawn. Here are the three views, in third angle, with the bed to the right.

Read the three views against each other and the hood shows why one view is never enough. The front view shows a trapezoid, 40 mm across the top and 140 mm across the rim, with the visor hanging below only its right half, because from the front you see the visor's near end at the centre line and its curve running round to the right. The right side view shows the visor across the full width of the rim, because from the bed's side the whole visor faces you. The top view shows two circles, and no visor at all: the visor hangs straight down beneath the rim, so from above it sits exactly behind the rim's circle. One of the three views tells you nothing about the visor, and a maker working from that view alone would build a hood without one, and fail R2 on the first night.

The drawing also fixes the scale question. At 1:1 each view needs about 140 mm of paper, which crowds an A4 sheet. At 1:2 the rim is 70 mm on paper, the front view with its visor is (80 + 40) ÷ 2 = 60 mm tall, and the three views fit in a block about 160 mm by 150 mm with 20 mm gaps, with room left for dimensions and a title block. The dimensions are still written as 140, 80 and 40, the real sizes, with the scale stated once.

Common misconceptions

• "One good view from a clever angle is enough." Any single view loses one dimension, and a single circle cannot say whether it is a hole or a peg. The hood's top view hides its visor completely.
• "Hidden lines are clutter that can be left out." In orthographic views they carry information you cannot get any other way: whether a hole goes right through, stops halfway or is hollow behind a face.
• "Every country puts the views in the same places." The United States uses third-angle projection and most of Europe and Asia use first angle, which put the top and side views on opposite sides of the front view. The projection symbol says which one a drawing uses.
• "An isometric drawing is a perspective drawing." In perspective, parallel edges meet at vanishing points and far things are drawn smaller. In isometric, parallel edges stay parallel and every edge along an axis is drawn to the same scale, which is exactly what makes it measurable.

What you now know

A single view of a solid shows two of its three dimensions and cannot tell a hole from a peg, so the square with a circle in it could be five different objects. An orthographic drawing fixes that with three views, front, top and right side, drawn with parallel lines of sight so that each shows true sizes. The views line up because they share dimensions: width between front and top, height between front and side, depth between top and side. Visible edges are thick, hidden edges dashed and axes drawn as chain lines, at one stated scale. The United States and Japan use third-angle projection, with the top view above the front; most of Europe and Asia use first angle, with the top view below, and the truncated-cone symbol says which. An isometric sketch sets its axes 120 degrees apart, draws edges along them at full length, and turns circles into ellipses that touch the midpoints of a rhombus. The reading light's hood, drawn at 1:2, showed its visor in the front and side views and not at all in the top view.

In short: draw every object you mean someone else to make in three aligned views with its hidden edges, state the scale and the projection, and use isometric to explain it, never to replace the views.

Sources

1. Corradi Dell'Acqua, E., & Mohammadi, J. (n.d.). 2.2: Orthographic projections. In Introduction to engineering drawing and design. Engineering LibreTexts, Illinois Institute of Technology. LibreTexts
2. Corradi Dell'Acqua, E., & Mohammadi, J. (n.d.). 2.4: Hidden edges in orthographic projections. In Introduction to engineering drawing and design. Engineering LibreTexts, Illinois Institute of Technology. LibreTexts
3. Wikipedia contributors. (2026). Multiview orthographic projection. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Isometric projection. In Wikipedia, The Free Encyclopedia. Wikipedia

Two holes, 80 mm apart, give or take half a millimetre

Punch a sheet of paper with an office two-hole punch made to the international standard ISO 838, and the standard has already told the punch what to do. Each hole is 6 mm across, plus or minus 0.5 mm. The centres of the two holes are 80 mm apart, plus or minus 0.5 mm. Each centre sits 12 mm from the edge of the paper, plus or minus 1 mm. Those plus-or-minus figures are tolerances. No punch cuts a hole of exactly 6.000 mm, and the standard does not ask one to. It says how far from 6 a hole may be and still count, so that any hole from 5.5 mm to 6.5 mm is a good one.

That is how a ring binder made by one company can hold paper punched by another company's hole punch, on the other side of the world. Nobody made anything perfect. Everybody made things within agreed limits.

This lesson is about the plus-or-minus. How precisely can a ruler and a kitchen scale measure? How do you say how precise a number is? And how do you choose tolerances for two parts so that they fit together every time, however they come out? The parts are the pin and the hole in the reading light's arm joint, and the lesson works them through in five steps, then changes one fact and watches the fit fail.

What a ruler can and cannot tell you

A 30 cm ruler marked in millimetres has a resolution of 1 mm: that is the smallest step it shows. Reading it by eye, you can usually say which millimetre mark an edge is nearest to, so a common working rule is to give a single ruler reading an uncertainty of plus or minus half a division, 0.5 mm. An edge that lines up near the 297 mark is written 297 ± 0.5 mm.

Two more words need to be kept apart, because they mean different things. OpenStax's College Physics defines accuracy as how close a measurement is to the correct value, and precision as how closely repeated measurements, made under the same conditions, agree with each other. Suppose a sheet of A4 paper, 297 mm long, is measured three times on a ruler whose end has been worn down, and the readings are 301, 301 and 302 mm. They agree closely, so they are precise. They are all about 4 mm too long, so they are not accurate. Repeating the measurement a hundred times would not help, because every reading carries the same error. An error that pushes every reading the same way is a systematic error; one that scatters readings on both sides is a random error. Averaging helps with the second and does nothing for the first.

The National Institute of Standards and Technology (NIST), the United States' measurement laboratory, sorts ways of working out an uncertainty into two kinds: those found by statistics from a series of repeated readings, and those found by other means, such as an instrument's resolution or its maker's specification. At the level of a ruler, that means two questions: how much do my repeated readings scatter, and how finely can this tool read at all?

Step 1: measure the pin more finely than the ruler's marks

The arm joint's pin is a wooden dowel sold as 5 mm. A ruler laid across it reads "about 5", which is useless when the gap around the pin in its hole will be a few tenths of a millimetre. Two tricks get more out of the same ruler.

Wrap it. Wind a thread tightly round the dowel ten times, side by side, mark the thread where the tenth turn ends, unwind it and measure the marked length. Three tries gave 158, 158 and 157 mm; take the middle value, 158 mm. The reading has an uncertainty of about ±1 mm, 0.5 mm at each end of the length, but that is shared among ten turns, so one turn is 15.8 ± 0.1 mm. The distance round a circle is π times its diameter, so the diameter is 15.8 ÷ π = 5.03 mm, with an uncertainty of 0.1 ÷ π = ±0.03 mm. Wrapping ten times has made the ruler ten times more precise.

Correct for the thread. That 5.03 mm is precise, but it is not yet accurate. The middle of the thread runs round a circle bigger than the dowel by one thread thickness, so every wrap reads too big. This is a systematic error: wrapping again gives the same wrong answer. To measure the thread, wind twenty turns side by side on a pencil and measure the width of the band: 4.0 ± 0.5 mm, so one thread is 4.0 ÷ 20 = 0.20 ± 0.03 mm thick. Stacking works the same way as wrapping: measure many identical things together and divide. The pin's diameter is 5.03 − 0.20 = 4.83 mm, and the worst-case uncertainty is the two uncertainties added, 0.03 + 0.03 = 0.06 mm.

Result: pin diameter 4.83 ± 0.06 mm, so a pin anywhere from 4.77 mm to 4.89 mm. The dowel sold as 5 mm is not 5 mm. Its nominal size, the size in its name, is a label; only a measurement says what it actually is.

Step 2: decide what fit the joint needs

When a pin goes through a hole, there are three kinds of fit. In a clearance fit the hole is larger than the pin and the parts slide or turn freely. In a transition fit the hole is fractionally smaller and mild force is needed to assemble them. In an interference fit the hole is smaller than the pin and high force, or heat, is needed; that is how a bearing bush is held in place for good.

The arm joint must turn. What holds the arm at its angle is a wing nut squeezing the two arm pieces together, which is friction and belongs to lesson 13. The pin's only job is to locate the pieces, so it needs a clearance fit, and the design decision is how much clearance: at least 0.10 mm, because a tighter gap can be jammed by a swollen wood fibre or a speck of grit; at most 0.50 mm, because a bigger gap lets the lamp head wobble. Those two numbers are choices with reasons, like the numbers in the brief.

Step 3: give the hole a tolerance

The hole will be drilled in a plywood bracket. Specify it as 5.10 ± 0.10 mm. That tolerance gives two limits: the hole may be as small as 5.00 mm or as large as 5.20 mm, and any hole between them passes inspection. The pin has its limits too, 4.77 mm and 4.89 mm, from step 1.

Step 4: check the two worst cases

A fit is decided by the worst combinations, not the typical one. There are two.

Both worst cases are inside the range the joint needs, so every pin within its limits will fit every hole within its limits. Nobody has to try pins in holes and pick the ones that happen to work. Engineers name the tightest case the maximum material condition, when both parts carry the most material they are allowed, and the loosest the least material condition.

The upshot: a tolerance is a promise about the worst part you will accept. Check the fit at the two worst cases, and if both work, every part in tolerance works.

Step 5: write the numbers down with honest digits

A number written as 4.83 mm claims that the last digit, the hundredths, means something. Written as 4.8 mm it claims only tenths; written as 4.8293 mm, as a calculator would give it, it claims ten-thousandths that no ruler can deliver. The digits that carry real information are the significant figures. College Physics gives the working rules:

• Zeros are significant except when they only hold a place. In 0.043 m the zeros hold places, so it has two significant figures; in 5.00 mm the zeros record that the hundredths were measured, so it has three.
• When multiplying or dividing, give the result the same number of significant figures as the input with the fewest. The calculator gives 15.8 ÷ π = 5.029296...; 15.8 has three significant figures, so write 5.03. And 5.0 V × 2.0 A = 10 W, with two.
• When adding or subtracting, give the result no more decimal places than the least precise input. 12.5 + 0.832 + 3.0 = 16.332 on a calculator, written as 16.3, because 12.5 and 3.0 are known only to tenths.

Counted numbers are exact and do not limit anything: the ten turns of thread and the twenty sheets in a stack are exactly ten and twenty. So twenty sheets of card measuring 8.5 ± 0.5 mm give one sheet as 8.5 ÷ 20 = 0.425 mm, written 0.43 mm to match the two significant figures of 8.5, with an uncertainty of 0.5 ÷ 20 = 0.025, about ±0.03 mm. A single sheet laid against the ruler would have read as 0 or 1 mm, an uncertainty of more than 100 percent. The stack brought it down to about 6 percent.

The same joint on a different machine: why the answer flips

Now change one fact. The team decides to 3D print the bracket instead of drilling plywood. A test print of the bracket, modelled with a 5.10 mm hole, is measured, and its holes come out between 4.70 and 4.90 mm: about 0.30 mm undersize, with a scatter of ±0.10 mm. (These are illustrative numbers from one printer; every printer and material differs, which is exactly why the test print comes first.) Run step 4 again with the same pins:

The drawing has not changed and neither have the pins, but the fit has gone from a guaranteed clearance to a transition: some pins will slide in, and some will jam or split the bracket. There are two fixes. Model the hole 0.30 mm larger, at 5.40 mm, so that it prints at 5.00 to 5.20 mm and the first table applies again. Or print the hole small and open it out with a 5 mm drill; clamp the part and wear eye protection, never hold a small part in your fingers while drilling, and use a power drill only with an adult who knows how. Either way the lesson is the same. A tolerance on a drawing means nothing until you know what your process actually makes.

When tolerances add up

One more case catches almost everyone once. The lamp's arm is made of three card tubes joined end to end, each 100 ± 1 mm long. The assembled arm is 300 mm, and in the worst case all three come out long, or all three short, so the arm is 300 ± 3 mm. Tolerances in a chain add. This is called tolerance stack-up. If the arm must be 300 ± 2 mm, each tube needs a tolerance no wider than 2 ÷ 3 = 0.67 mm, in practice ±0.6 mm, because ±0.7 mm on each would stack to ±2.1 mm.

The reverse lesson matters as much. Every tighter tolerance means a better process or more careful work, and the ISO 838 committee did not ask for holes to be 6.00 ± 0.01 mm because no ring binder needs that. Specify the loosest tolerance the function allows, and prove with the worst cases that it is enough.

Common misconceptions

• "More decimal places means a more accurate answer." Extra digits from a calculator are not measured; they claim a precision the tool never had. Worse, a precise answer can still be wrong, like the worn ruler's 301 mm.
• "Accuracy and precision mean the same thing." Precision is agreement between repeated readings; accuracy is closeness to the true value. The thread-wrap diameter was precise to 0.03 mm and 0.20 mm too big until it was corrected.
• "A 5 mm dowel is 5 mm across." Five millimetres is its nominal size, a name. This one measured 4.83 ± 0.06 mm, and designing the hole for exactly 5.00 would have given a looser fit than intended.
• "If every part is within tolerance, the assembly will be too." In a chain, tolerances add: three parts at ±1 mm make an assembly at ±3 mm.

Recap

ISO 838 gives every punched hole a size of 6 ± 0.5 mm, and that plus-or-minus is what lets parts from different makers work together. A ruler resolves 1 mm, and one reading is good to about ±0.5 mm; accuracy is closeness to the truth, precision is agreement between repeats, and a systematic error, like a worn ruler or a thread's own thickness, spoils accuracy without touching precision. Wrapping and stacking share the ruler's uncertainty over many turns or sheets: ten wraps put the dowel at 15.8 ÷ π = 5.03 mm, and correcting for the 0.20 mm thread gave 4.83 ± 0.06 mm. With the hole at 5.10 ± 0.10 mm, the tightest clearance is 5.00 − 4.89 = 0.11 mm and the loosest 5.20 − 4.77 = 0.43 mm, both inside the 0.10 to 0.50 mm the joint needs. The same drawing on a printer that makes holes 0.30 mm small gives a tightest case of −0.19 mm and a transition fit, fixed by modelling the hole at 5.40 mm. Results keep the significant figures of their least precise input, counted numbers are exact, and tolerances in a chain add.

The upshot: every measurement carries a plus-or-minus and every part a tolerance. Check fits at the two worst cases, measure what your process really makes, and never write more digits than your tools earned.

Sources

1. Urone, P. P., & Hinrichs, R. (2022). 1.3 Accuracy, precision, and significant figures. In College physics 2e. OpenStax. OpenStax
2. National Institute of Standards and Technology. (n.d.). Basic definitions of uncertainty. NIST Physical Measurement Laboratory. NIST
3. Wikipedia contributors. (2026). Engineering tolerance. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Engineering fit. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Wikipedia contributors. (2026). Hole punch. In Wikipedia, The Free Encyclopedia. Wikipedia

23 September 1999: a signal lost 49 seconds early

At 09:00:46 UTC on 23 September 1999, NASA's Mars Climate Orbiter fired its main engine to brake into orbit around Mars. It had been travelling for nine months, since its launch on 11 December 1998, and the burn would carry it behind the planet and out of radio contact for about 21 minutes. The signal disappeared at 09:04:52, 49 seconds earlier than predicted. It never came back.

The spacecraft was built by Lockheed Martin Astronautics in Denver and navigated from NASA's Jet Propulsion Laboratory (JPL) in Pasadena. All through the flight, small thrusters fired to unload its reaction wheels, the spinning flywheels that keep a spacecraft pointing the right way. Each firing nudged the spacecraft's path a little, and the navigators had to model every nudge. A ground program called SM_FORCES worked out the impulse of each firing, the force multiplied by the time it acts, and passed the numbers to the navigation team in a file. The document that fixed the format of that file, its interface specification, said the impulses would be in newton-seconds. SM_FORCES wrote them in pound-force seconds. The navigation software read every number as newton-seconds.

NASA's Mishap Investigation Board issued its first report on 10 November 1999 and named that as the root cause: metric units had not been used in one ground software file. One pound-force is 4.45 newtons, so every thruster nudge was modelled as about 4.45 times weaker than it really was. The spacecraft's solar array stuck out on one side, so sunlight pressing on it built up spin quickly, and the unloading firings happened 10 to 14 times more often than the navigators had expected. Many small errors, all pushing the same way, added up for nine months.

The report gives the numbers. After the last planned course correction, on 15 September, the closest approach to Mars was aimed at 226 km above the surface. Within a week, tracking put it at 150 to 170 km. An hour before the burn, better data suggested it could be as low as 110 km. The lowest the spacecraft was thought able to survive was 80 km. The reconstruction made afterwards, with the thruster data corrected, put the closest approach at 57 km, about 170 km lower than planned. At that depth in the atmosphere the spacecraft was either destroyed or skipped back out into orbit around the Sun.

Here is what makes the case worth studying. Every multiplication in SM_FORCES was right. Every calculation in the navigation software was right. A number crossed from one program to another and lost its unit on the way, and nothing downstream could tell. The board also found that the navigators had seen odd thruster data from April 1999 and reported it only informally; that one method of tracking kept placing the spacecraft closer to the planet, a difference that was never resolved; and that a fifth course correction, which could have raised the path, was discussed shortly before arrival and never carried out.

This lesson is about catching that kind of mistake while it still costs nothing. It starts from three wrong answers, one from the reading light and two from flight, and traces each one to the exact line where it went wrong. Two habits catch all three: writing a unit beside every number, and estimating the size of an answer before trusting it.

A number without its unit is half a fact

Suppose, for illustration, that one firing delivered an impulse of 1.00 pound-force second. The National Institute of Standards and Technology (NIST) gives the conversion factor as 4.448 222 newtons per pound-force, so:

1.00 lbf·s × (4.448 N ÷ 1 lbf) = 4.45 N·s

SM_FORCES wrote 1.00 into the file. The navigation software read 1.00 N·s, and so modelled a push of 1.00 ÷ 4.45 = 0.22 of the real one: 22 percent. The number was correct. The unit it travelled without was the rest of the message. A measurement is a number and a unit together, and either half on its own is not a fact.

Wrong answer 1: a card arm that weighs seven kilograms

Back to the reading light. Lesson 6 built its arm from three card tubes, each 100 mm long. Each tube is rolled from a strip 100 mm by 70 mm: going once round a 20 mm former takes π × 20 = 62.8 mm, and the rest is a 7 mm overlap for glue. Requirement R5 limits the whole lamp to 400 g, so a student checks the arm's mass. They cut an A4-sized piece, 210 mm by 297 mm, from the same cereal box, weigh it on a kitchen scale and get 21 g. Here is their working, exactly as written:

Line 6 is the dangerous one. A wrong number is bad; a wrong number that changes a decision is worse. This one would throw away a material that works.

Tracing it: the line where the units stop cancelling

Write every line again with its units, and treat the units like algebra: they multiply, divide and cancel exactly as numbers do. This is dimensional analysis, and it needs nothing but care.

• Line 1: 0.210 m × 0.297 m = 0.06237 m². Fine: metres times metres gives square metres.
• Line 2: 21 g ÷ 0.06237 m² = 337 g/m². Fine, and since 21 g has two significant figures, it is written 340 g/m².
• Line 3: 3 × 100 mm × 70 mm = 21,000 mm². Fine, but look at the unit: square millimetres.
• Line 4: here it breaks. Converting with a factor that equals one gives 21,000 mm² × (1 m ÷ 1,000 mm)². The factor has to be squared, because the unit being cancelled, mm², is squared: (1 m ÷ 1,000 mm)² = 1 m² ÷ 1,000,000 mm². So the area is 21,000 ÷ 1,000,000 = 0.021 m².
• Line 5, redone: 0.021 m² × 340 g/m² = 7.1 g. The square metres cancel and leave grams, the unit the question asked for.

The arm weighs 7.1 g, not 7.1 kg: a thousand times less, and under 2 percent of the 400 g allowance. If line 4 had been written with its units, the student would have had to write 21,000 mm² × (1 m ÷ 1,000 mm) = 21 mm·m, a unit that is not square metres. The units would have refused to cancel, and that refusal is the whole reason for writing them.

Why does the slip feel right? Because 1 m = 1,000 mm is true, for lengths. A square metre is a square 1,000 mm on each side, so it holds 1,000 × 1,000 = 1,000,000 squares of 1 mm. A cubic metre holds 1,000 × 1,000 × 1,000 = 1,000,000,000 cubes of 1 mm, which is 1,000 litres. Square the factor for areas; cube it for volumes.

The estimate that catches it in five seconds

Units find the error by its shape. An estimate finds it by its size, and needs no algebra at all. Before line 1, look at the card. The three strips together are 21,000 mm²; the A4 piece is about 62,000 mm². So the arm uses about a third of the piece, and the piece weighs 21 g: the arm should weigh about 7 g. An answer of 7,140 g is a thousand times that. Even without the estimate, 7 kg is the mass of seven 1-litre bottles of water, claimed for three tubes of cereal-box card you could balance on a fingertip. Any answer you can check against your own hands, check.

So what?: in a units error the arithmetic is usually perfect, which is exactly why the error is hard to see. Write the units to catch it by its shape, and estimate first to catch it by its size.

The method: conversion factors that equal one

OpenStax's College Physics defines a conversion factor as a ratio saying how many of one unit equal another, and converts 80 m to kilometres by multiplying by 1 km ÷ 1,000 m: the metres cancel and leave 0.080 km. The top and bottom of the factor are the same length, so the factor equals one, and multiplying by one changes the unit without changing the amount. Four rules make the method safe:

1. Write the unit beside every number on every line, including the lines that seem too obvious to need it.
2. Turn the factor so that the unit you do not want cancels. To go from millimetres to metres, millimetres go on the bottom: 1 m ÷ 1,000 mm. Put it the wrong way up and the units come out as mm² per m, which you can see at once.
3. Square the factor for areas and cube it for volumes.
4. Check the final unit against the question. A mass must come out in grams or kilograms, a force in newtons, a time in seconds or hours. An answer in g·mm²/m² contains an unconverted unit, whatever its number says.

A fifth rule hides inside lesson 2's light calculation. A ratio of two lengths is a pure number only when both lengths are in the same unit. 0.40 m ÷ 2.0 m = 0.20, with no unit left over. But 40 cm ÷ 2.0 m = 20 cm/m, which is not a pure number until the factor 1 m ÷ 100 cm turns it into 0.20.

NIST's guide to the SI lists conversion factors for hundreds of units. The handful an engineer in the United States meets every week:

Look at the last two rows. The pound is a mass and the pound-force is a force, and in everyday speech both are called a pound. In the International System of Units (SI) they cannot be mixed up: mass is in kilograms and force in newtons, and the two are linked by gravity, 9.81 newtons for every kilogram at Earth's surface.

Wrong answer 2: a pillow brighter than the noon sun

Lesson 2 worked out how much light from a bare lamp reaches the sleeper's pillow: 300 lux × (0.40 m ÷ 2.0 m)² = 12 lux. Suppose someone measures the lamp's height above the page with a ruler in centimetres and the room with a tape in metres, and writes:

E = 300 × (40 ÷ 2.0)² = 300 × 20² = 300 × 400 = 120,000 lux

Two quick estimates reject this before any unit check. The first is about direction: the pillow is farther from the lamp than the page is, so it must get less light than the page's 300 lux, not more. The second is about size: direct sunlight gives 32,000 to 100,000 lux, so the answer claims that a battery lamp lights a pillow 2 m away more brightly than the noon sun does. The trace is short. The ratio 40 cm ÷ 2.0 m is 20 cm/m, not a pure number; multiplied by 1 m ÷ 100 cm it becomes 0.20. The ratio was 100 times too big, and squaring it made the answer 10,000 times too big: 120,000 ÷ 12 = 10,000.

A direction check, asking only whether the answer should be bigger or smaller than something you already know, is the cheapest estimate there is. It takes no arithmetic, and here it is enough to throw the answer out.

Wrong answer 3: a full tank that was less than half full

On 23 July 1983, Air Canada Flight 143, a new Boeing 767, was to fly from Montreal to Edmonton by way of Ottawa. Canada was converting from imperial to metric units, and Air Canada's new 767s were the first of its aircraft calibrated in metric units. The aircraft's fuel gauges were not working, so the crew measured the fuel already in the tanks with a dripstick, a rod lowered into each tank, and worked out how much to add. The flight plan called for 22,300 kg of fuel. The dripstick showed 7,682 litres on board, and the density of the fuel was given as 1.77. Here is the working, beside the same working with its units:

The aircraft took on 4,917 L instead of 20,088 L. With 7,682 + 4,917 = 12,599 L on board, it carried 12,599 L × 0.803 kg/L = 10,117 kg, which is 45 percent of the 22,300 kg the plan required. At Ottawa the dripstick check was repeated with the same factor, the tanks seemed to hold plenty, and no fuel was added. Over Red Lake, Ontario, at 41,000 feet, the first fuel-pressure warning sounded, and soon both engines had stopped. The crew glided the aircraft to a former Royal Canadian Air Force base at Gimli, Manitoba, where a sports-car race was being run on part of the old runway, and landed it with no engines. All 69 people on board survived; the injuries were minor ones, suffered by passengers leaving down the rear evacuation slides. The aircraft has been called the Gimli Glider ever since.

Trace it. The number 1.77 was correct: it is the density of the fuel in pounds per litre, and 1.77 lb/L × 0.4536 kg/lb = 0.803 kg/L, the same density in metric units. What went wrong is that the factor travelled without its unit, exactly as the orbiter's impulses would sixteen years later. Written with units, the second line cannot even be done: kilograms minus pounds is not a subtraction anyone can finish without converting first.

An estimate would have caught it as well. Jet fuel floats on water, so a litre of it has a mass of less than 1 kg. Then 12,599 litres must weigh less than 12,599 kg, and can never reach 22,300 kg. One sentence of reasoning, with no calculator, shows the tanks were nowhere near full enough. The board of inquiry later recommended converting every Air Canada aircraft from imperial units to SI units at once. That is an engineering answer in its own right: the strongest protection against a unit error is a system in which the wrong unit has nowhere to appear.

Estimate first: one significant figure, before the calculator

The physicist Enrico Fermi was known for making good approximate calculations from little or no data. At the first nuclear test, Trinity, in July 1945, he dropped scraps of paper from his hand as the blast wave passed and estimated the energy released from how far they were blown. His figure, 10 kilotons of TNT, was about half the accepted value of 21 kilotons. That is the standard an estimate aims for: not the right answer, but the right size. A quick estimate of this kind is called a Fermi problem, or an order-of-magnitude estimate, because its goal is to get the power of ten right.

OpenStax's chapter on approximation gives the use in one line: an estimate lets you rule out unrealistic numbers. The habit has four steps:

1. Before calculating, write an estimate to one significant figure, with round numbers.
2. Build it from anchors you trust: values you have measured, or can feel in your hands.
3. Calculate properly, with units on every line.
4. Compare. Agreement within a factor of about 3 is normal for an estimate. A factor of 10, 100 or 1,000 means an error somewhere, and you find it before going on.

A worked estimate: what does 20 lb bond weigh?

In the United States, printer paper is sold with a label like "20 lb bond". Estimate first: a ream is 500 sheets of about 5 g each, so 500 × 5 g = 2,500 g, about 2.5 kg, or about 5.5 pounds. The ream in your hands does not weigh 20 pounds. So where does 20 come from? It is a basis weight: the weight of a ream of 500 sheets cut to a standard basis size, which for this paper is 17 by 22 inches. That basis sheet is 17 × 22 = 374 square inches, four times a Letter sheet of 8.5 × 11 = 93.5 square inches. So a ream of Letter sheets weighs 20 ÷ 4 = 5 lb, which matches the estimate. Now convert, with units on every line:

Basis area: 374 in² × (0.0254 m ÷ 1 in)² = 0.2413 m² per sheet, so 500 sheets cover 120.6 m².
Mass: 20 lb × 453.6 g/lb = 9,072 g.
Grammage: 9,072 g ÷ 120.6 m² = 75.2 g/m².

So 20 lb bond is about 75 g/m², a little lighter than the 80 g/m² that is typical elsewhere, and a Letter sheet of 0.0603 m² weighs 0.0603 m² × 75.2 g/m² = 4.5 g. The estimate got the size right in one line, the calculation got the decimals right, and together they explained a label whose number describes a sheet you have never seen.

Where units go missing: at the interfaces

None of the three errors happened in the middle of a calculation. Each happened where a number passed from one person, team or program to another: from SM_FORCES to the navigation software, from the fuel figures to the flight crew, from a ruler in centimetres to a tape in metres. The place where two parts of a system meet is an interface, and interfaces are where units go missing. The orbiter's board found that the interface specification existed but was not properly used, and that end-to-end testing of the small forces software against it did not appear to have been done. It also urged that every concern be written up formally rather than mentioned in passing.

The same habits work on the reading light today:

• Put the unit in every column heading and every file name or note: "Pillow rise (lux)", "Mass (g)". Lesson 4's test log already does this.
• Choose one system for the project and keep to it. The brief is in SI throughout: lux, grams, centimetres, degrees Celsius, hours.
• Test each interface with a case whose answer you already know. Before weighing the lamp, put a litre of water in a container you have already weighed on the kitchen scale: it should read close to 1,000 g more.
• When a number looks odd, write it down where the team will see it, with the readings, and do not let it go until it is explained.

Common misconceptions

• "If every step of the arithmetic is right, the answer is right." The orbiter's software did every multiplication correctly and still modelled each thruster firing at 22 percent of its real strength. Correct arithmetic on a number with the wrong unit gives a wrong answer with great precision.
• "A metre is 1,000 millimetres, so a square metre is 1,000 square millimetres." A square metre is 1,000 mm by 1,000 mm, which is 1,000,000 mm². The factor is squared for areas and cubed for volumes, and forgetting it made a 7.1 g arm weigh 7.1 kg.
• "Units can be added to the answer at the end." Units do their checking in the middle. Written with units, the Gimli fuel sum becomes 22,300 kg minus 13,597 lb, which visibly cannot be done; written as bare numbers it looked fine.
• "An estimate is only a guess, so a calculation always beats it." An estimate is an independent check made by a different route. Fermi's 10 kilotons was half the true value, which was good enough to tell the right size from a wrong one, and that is all an estimate is for.

Where this leaves us

The Mars Climate Orbiter was lost on 23 September 1999 because ground software reported thruster impulses in pound-force seconds where the interface specification required newton-seconds; every firing was modelled at 1 ÷ 4.45 of its strength, and the spacecraft reached Mars at about 57 km instead of 226 km, below the 80 km it could survive. Flight 143 left Montreal in 1983 with 10,117 kg of fuel instead of 22,300 kg because a density of 1.77 lb/L was used as if it were in kilograms per litre. On the reading light, dividing square millimetres by 1,000 instead of 1,000,000 made a 7.1 g arm weigh 7.1 kg, and mixing centimetres with metres put 120,000 lux on a pillow. In each case the arithmetic was right and the units were lost, usually at an interface. Writing units on every line, converting with factors equal to one, squared or cubed where needed, and checking the final unit catch these errors by their shape; an estimate to one significant figure, built from anchors such as a 5 g sheet of paper or a 1 kg litre of water, catches them by their size.

So what?: treat a number without a unit as unfinished, and never accept an answer you have not first estimated. A rough estimate and a column of units cost a minute; the orbiter cost a spacecraft.

Sources

1. Mars Climate Orbiter Mishap Investigation Board. (1999). Phase I report (10 November 1999). NASA. NASA
2. Thompson, A., & Taylor, B. N. (2008). Appendix B.8: Factors for units listed alphabetically. In Guide for the use of the International System of Units (SI) (NIST Special Publication 811). National Institute of Standards and Technology. NIST
3. Urone, P. P., & Hinrichs, R. (2022). 1.2 Physical quantities and units. In College physics 2e. OpenStax. OpenStax
4. Urone, P. P., & Hinrichs, R. (2022). 1.4 Approximation. In College physics 2e. OpenStax. OpenStax
5. Wikipedia contributors. (2026). Gimli Glider. In Wikipedia, The Free Encyclopedia. Wikipedia

1676: fourteen letters at the end of a book

In 1676 Robert Hooke, curator of experiments at the Royal Society in London, ended a book describing instruments for looking at the Sun with fourteen scrambled letters: ceiiinosssttuu. Scientists of the time used anagrams to put a dated claim on an idea without giving it away. In 1678 Hooke published the answer: ut tensio, sic vis, "as the extension, so the force." Pull a spring twice as hard and it stretches twice as far. Hooke wrote that he had known it since 1660.

What is now called Hooke's law is a law about stiffness: how much a thing gives under a load. Read it again for what it leaves out. It says nothing about the load at which the spring stops springing back, and nothing about the load at which it snaps. Those are separate properties of the material, found by separate tests and printed as separate numbers on a data sheet. Mixing them up is one of the commonest mistakes in choosing a material, and the mistake has a short name: strong is not stiff.

This lesson puts eight materials side by side in one table, reads the table column by column, and then uses it for a decision lesson 1 promised: what to make the reading light's arm from.

Five numbers on a data sheet

Every number in the table is a property of the material, not of one object: a thick steel bar and a thin steel wire share the same values. That works because the numbers are written per unit of size. The key idea is stress, which OpenStax's University Physics defines as force per unit area. Its unit, the pascal, is one newton per square metre, which is tiny, so data sheets use the megapascal (MPa), a million pascals. One megapascal is exactly one newton per square millimetre, which makes it easy to picture: hang a 10 kg mass from a strip 2 mm by 5 mm, an area of 10 mm², and the stress is 10 kg × 9.81 N/kg ÷ 10 mm² = 9.81 N/mm², or 9.81 MPa. Lesson 9 works with stress in detail. Here you need only the five numbers it makes possible.

1. Density, ρ: mass per volume, in grams per cubic centimetre. (The data book's unit, Mg/m³, gives the same number.) Water is 1.0.
2. Young's modulus, E: the material's stiffness, in gigapascals (1 GPa = 1,000 MPa). It is the stress divided by the fractional stretch the stress causes, so a large E means a small stretch. Hooke found the proportion; the modulus puts a number on it for each material. It is named after the British scientist Thomas Young, although Leonhard Euler had worked with the idea in 1727.
3. Yield strength, σ_y: the stress at which the material stops springing back and begins to bend or stretch for good.
4. Tensile strength, σ_ts: the highest stress it carries in a pulling test before it breaks.
5. Fracture toughness, K_IC: how well it resists a crack growing. Its unit, MPa√m, looks odd; read it simply. A low number means a small scratch or notch can make the material snap under load. A high number means it can be cut, dented and scratched and still carry its load.

The table: eight materials side by side

These are typical ranges from the Cambridge University Engineering Department's Materials Data Book, compiled from a commercial materials database and used to teach engineering students. Each property is a range, not one number, because composition and treatment change it.

Glass has no yield strength in the table because it breaks before it yields. The data book notes that ceramics such as glass are roughly ten times stronger when squeezed than when pulled: soda-lime glass takes 360 to 420 MPa in compression but only 31 to 35 MPa in tension. And the book gives a warning that every engineer repeats: properties vary widely with composition and treatment, so any final design should use the maker's or supplier's data for the actual material. Aluminium alloys run from 30 to 500 MPa in yield strength depending on the alloy and its heat treatment, which is why "aluminium" on its own is not a specification.

Reading down the stiffness column

Take a middle value from each range and compare. Steel, at about 205 GPa, is about 2.7 times as stiff as aluminium alloy at about 75 GPa. Aluminium is about 6 times as stiff as wood along the grain (75 ÷ 13 = 5.8). Wood is about 6.5 times as stiff as ABS (13 ÷ 2.0). And steel is about 100,000 times as stiff as rubber (205 ÷ 0.002 = 102,500). Those factors are why a steel ruler barely bends under your thumb, a plastic one bows, and a rubber band stretches to twice its length.

One rule turns the column into a design tool. For parts of the same shape under the same load, the deflection is inversely proportional to E. Change the material and nothing else, and the bend changes by the ratio of the moduli: an aluminium bracket that replaces a steel one of identical shape bends 205 ÷ 75 = 2.7 times as far.

Strong is not stiff: three pairs that prove it

Steel and aluminium alloy: the same strength, a third of the stiffness. The strength ranges overlap. Mild steel yields at 250 to 395 MPa, and the stronger aluminium alloys reach 500 MPa. So a well-chosen aluminium alloy can carry as much stress as mild steel before it bends for good, yet under the same load it gives 2.7 times as much.

Glass and aluminium alloy: the same stiffness, a fraction of the toughness. Window glass has E of 68 to 72 GPa, inside aluminium's 68 to 82. A glass pane bends no more than an aluminium sheet of the same size. But glass breaks in tension at 31 to 35 MPa, and its toughness of 0.55 to 0.7 is about forty times below aluminium's 22 to 35 (22 ÷ 0.55 = 40). Glass is not floppy. It is brittle: one scratch and it snaps.

Rubber and acrylic: similar strength, wildly different stiffness. Rubber's tensile strength, 22 to 32 MPa, is about 40 percent of acrylic's 48 to 80 (27 ÷ 64 = 0.42), not so very different. But rubber's E is about 1,500 times lower: 0.002 GPa against 3.0. Rubber is not weak; it is floppy. It reaches its breaking stress only after stretching to several times its length, while acrylic hardly stretches at all before it cracks.

Worth holding on to: stiffness says how much a part gives; strength says how much it can take before it is damaged. A material can have plenty of one and little of the other, so every requirement must say which one it needs.

Toughness: the column people skip

Read the last column. Mild steel, at 41 to 82, and aluminium alloys, at 22 to 35, are tough: they can be scratched, drilled and dented and still carry load, because the metal at the tip of a crack yields and blunts it. Wood along the grain manages 5 to 9. The plastics vary: ABS reaches about 4, acrylic only 0.7 to 1.6. Glass is lowest at 0.55 to 0.7. A material with low toughness is brittle: it fails suddenly from a small flaw, with no warning stretch. One that yields and stretches a long way before it breaks is ductile.

You can see the column in everyday things. A steel paper clip bends back and forth many times before it breaks. A glazier cuts glass by scoring a line on it first: the score is a flaw put there on purpose, and the brittle sheet then snaps along it. For a part that will be dropped, knocked or drilled, the toughness column often matters more than the strength column.

Direction matters: wood along and across the grain

Wood takes two rows in the table, because its properties depend on direction. Along the grain it has E of 6 to 20 GPa and tensile strength of 60 to 100 MPa; across the grain only 0.5 to 3 GPa and 4 to 9 MPa. Pulled across the grain, it is roughly ten times weaker and about seven times less stiff. The reason is its structure: wood is a bundle of long hollow cells joined side by side. Pull along the bundle and you stretch the cell walls; pull across it and you peel the cells apart. A material whose properties change with direction is called anisotropic.

That gives the reading light two rules. A wooden arm must have its grain running along its length. And a hole drilled close to the end of a piece, such as the hole for lesson 6's pin, loads the wood across the grain, so keep it several diameters from the end or it can split out.

Per kilogram: why light materials win in bending

Engineers seldom want stiffness alone. They want stiffness for the least mass, so they divide E by density, a ratio called the specific modulus. Steel: 205 ÷ 7.85 = 26. Aluminium: 75 ÷ 2.7 = 28. Glass: 70 ÷ 2.47 = 28. Wood: 13 ÷ 0.7 = 19. ABS: 2.0 ÷ 1.1 = 1.8. Pulled straight, kilogram for kilogram, steel, aluminium and glass are almost equally stiff, wood is close behind, and the plastics are more than ten times worse.

Bending changes the picture, and the reason is geometry. For a strip of rectangular section, the resistance to bending grows with the cube of its thickness: mechanics texts write the strip's second moment of area as I = b h³ ÷ 12, where b is the width and h the thickness. Double the thickness and the strip is 2³ = 8 times as stiff in bending. Now compare a steel strip 1 mm thick with a wooden plank of the same width and the same mass. Wood is 7.85 ÷ 0.7 = 11.2 times less dense, so the plank can be 11.2 mm thick. Its bending stiffness, compared with the steel strip's, is the ratio of moduli times the ratio of thicknesses cubed:

(13 GPa ÷ 205 GPa) × (11.2 mm ÷ 1 mm)³ = 0.063 × 1,405 = 89

The plank is about 89 times as stiff. Even with wood's E at the bottom of its range, 6 GPa, it is about 40 times as stiff. The same reasoning gives aluminium sheet a ninefold advantage over steel sheet of the same mass: (75 ÷ 205) × (7.85 ÷ 2.7)³ = 0.37 × 24.6 = 9. A light material spent on thickness beats a stiff material spent thin. Mechanics texts make the same point about beams, which are often given wide flanges at the top and bottom to increase I; lessons 11 and 12 use the idea when they put material where it does the most work.

The reading light's arm, five ways

Now use the table. Lesson 6's arm was three card tubes, and as a prototype it did its job. With the hood on, it drooped about 4 mm, which was acceptable; but the first time it was knocked off the shelf, one tube folded flat and stayed folded (illustrative results). In this shape card is stiff enough and not tough enough. Set the requirements first, as lesson 3 taught, then compare:

• The hood end shall droop no more than 5 mm under the 60 g hood. A pool of light moved 5 mm goes unnoticed; a hood that visibly sags looks broken.
• The arm shall survive a fall from the shelf to the floor.
• The arm shall be as light as possible, because R5 limits the whole lamp to 400 g.
• The arm shall be workable at home with hand tools.

To compare materials fairly, give them all the same shape: a solid rod 8 mm across and 300 mm long, a volume of π × (4 mm)² × 300 mm = 15,080 mm³, or 15.1 cm³. A pine dowel of that size, clamped at one end with the 60 g hood on the other, drooped 3 mm (an illustrative measurement). Taking 10 GPa as this dowel's modulus, a typical value inside wood's range, the other droops follow from the ratio of moduli, and the masses from density × volume. The arm's own weight is ignored, which matters only for steel.

Read it the way lesson 3 read its matrix. Steel and aluminium are the stiffest and toughest, and steel's mass rules it out. ABS and acrylic fail the stiffness requirement at this size; they could pass in a deeper section, since bending stiffness grows so fast with thickness, but they would gain mass doing it. The dowel passes every requirement, weighs about a quarter of the aluminium rod, and can be cut with a small saw and drilled for lesson 6's pin. It goes forward. If a later test showed that 3 mm of droop was too much, aluminium would be next: the table says where to go, not just what to choose.

One caution from the data book applies here too. The dowel's modulus was assumed, not measured, and wood varies more from piece to piece than anything else in the table. Measure the actual droop of the actual dowel before trusting any prediction, which is what the activity below asks you to do with three materials.

Measuring E with a ruler and coins

You can measure Young's modulus at home by bending a strip. Clamp a strip of width b and thickness t at a table edge so that a length L sticks out (a cantilever), hang a weight F at the tip, and measure how far the tip drops, δ. Beam theory, in Louis Bucciarelli's MIT text, gives the tip deflection of an end-loaded cantilever as F L³ ÷ (3 E I). With I = b t³ ÷ 12 for a rectangular strip, that rearranges to:

E = 4 F L³ ÷ (δ b t³)

Try it with numbers. A wooden tongue depressor 18 mm wide sticks out 120 mm. Ten of them stacked measure 16 mm, so t = 1.60 mm. Ten coins of 5.0 g each, 50 g in all, weigh 0.050 kg × 9.81 N/kg = 0.49 N, and the tip drops 4.5 mm. In metres:

E = 4 × 0.49 N × (0.120 m)³ ÷ (0.0045 m × 0.018 m × (0.0016 m)³) = 1.0 × 10¹⁰ Pa = 10 GPa

That sits inside wood's 6 to 20 GPa along the grain. Two cautions. The formula assumes a small bend, so choose a load that drops the tip by no more than about a tenth of L. And the thickness is cubed, so it controls the accuracy: stacking ten sticks gives t to ±0.05 mm, about 3 percent, which becomes about 9 percent in E. A single reading of a 2.5 mm ruler to ±0.5 mm is 20 percent, which becomes a range from about half to almost twice the true E.

Common misconceptions

• "Stiff and strong mean the same thing." Glass is as stiff as aluminium and breaks in tension at 31 to 35 MPa; rubber reaches 22 to 32 MPa yet is about 1,500 times less stiff than acrylic. They are separate properties with separate numbers.
• "Steel is always stronger than aluminium." Aluminium alloys span 30 to 500 MPa in yield strength, and the strongest exceed mild steel's 250 to 395. What steel always has is about 2.7 times the stiffness and about 2.9 times the density.
• "A denser, heavier material makes a stiffer part." For the same mass, a lighter material can be made thicker, and in bending thickness counts cubed. A wooden plank is about 89 times as stiff as a steel strip of the same width and mass.
• "Plastics are weak." Acrylic's tensile strength, 48 to 80 MPa, overlaps wood's 60 to 100 along the grain. What plastics lack is stiffness, at 1 to 4 GPa, which is why plastic parts are made thicker or given ribs.

Summing up

Hooke's law of 1678, as the extension, so the force, describes stiffness and says nothing about strength. A data sheet separates five properties: density; Young's modulus, the stiffness; yield strength, where permanent bending starts; tensile strength, where the material breaks; and fracture toughness, its resistance to a crack. In the Cambridge data book, steel at about 205 GPa is 2.7 times as stiff as aluminium alloy, which matches it in strength; glass matches aluminium in stiffness and has about a fortieth of its toughness; rubber has about 40 percent of acrylic's strength and a 1,500th of its stiffness. Wood is about ten times weaker across the grain than along it. For parts of the same shape, deflection goes as 1 ÷ E and mass as density; per kilogram, light materials win in bending because thickness counts cubed, which makes a wooden plank about 89 times as stiff as a steel strip of equal mass. For the reading light's arm, an 8 mm pine dowel drooped 3 mm, passed every requirement at 10.6 g, and goes forward, with aluminium as the fallback.

Worth holding on to: decide which property the part really needs, then compare materials by ratios for the same shape or the same mass, and check the supplier's figures and your own measurement before you trust a range from a table.

Sources

1. Cambridge University Engineering Department. (2003). Materials data book (2003 ed.). University of Cambridge. University of Cambridge
2. Moebs, W., Ling, S. J., & Sanny, J. (2016). 12.3 Stress, strain, and elastic modulus. In University physics volume 1. OpenStax. OpenStax
3. Bucciarelli, L. L. (2002). Chapter 10: Deflections due to bending. In Engineering mechanics for structures. MIT OpenCourseWare, 1.050 Solid Mechanics. MIT OpenCourseWare
4. Roylance, D. (n.d.). 4.2 Stresses in beams. In Mechanics of materials. Engineering LibreTexts, from MIT OpenCourseWare. LibreTexts
5. Cooper, K. (2025, December 17). Introducing laws of physics. Ex Libris Universum. American Institute of Physics. AIP

99 Southwark Street: Facts not Opinions

In September 1865 a machine 47 feet 7 inches long, about 14.5 m, and weighing some 116 tons was delivered to London for David Kirkaldy, a Scottish engineer who had made his name by pulling bars of iron and steel apart and publishing the results. It was a testing machine: it gripped a sample at both ends, pulled with a force that could be read off, and kept pulling until the sample broke. In 1874 Kirkaldy moved his testing works to 99 Southwark Street, where the machine still stands in a museum, and put three words above the door: Facts not Opinions.

The motto earned its place. After the Tay Bridge in Scotland fell in a gale on 28 December 1879, taking a train with it, Kirkaldy tested its parts for the official inquiry. The tie bars that braced the piers, and the lugs cast onto the iron columns to hold them, failed at about 20 tons of pull, well below what had been expected; four of the fourteen lugs he tested had flaws in the casting and failed at even less. The bridge had been designed on opinions about those parts. The machine supplied the facts.

A testing machine records two numbers over and over: the force, and how far the sample has stretched. From those two columns engineers draw the most useful single picture of a material there is, the stress-strain curve. This lesson builds it, but it starts the way real mistakes start, from four plausible wrong answers, and traces each one to the step where the reasoning breaks.

Wrong answer 1: "It stretched only 1.8 mm, so it is safe"

OpenStax's University Physics works an example you can check. A steel rod 2.0 m long, with a cross-section of 3.0 × 10⁻⁵ m² (30 mm², a rod about 6 mm across), hangs from a support and holds a 550 kg platform at its lower end. How much does it stretch? The working runs in three steps.

1. Force. F = m g = 550 kg × 9.81 N/kg = 5,400 N.
2. Stress. Stress is force per unit area, written σ (the Greek letter sigma): σ = F ÷ A = 5,400 N ÷ (3.0 × 10⁻⁵ m²) = 1.8 × 10⁸ Pa, or 180 MPa.
3. Strain and stretch. Strain, written ε (epsilon), is the stretch divided by the original length, so it has no unit. Hooke's law in its engineering form says ε = σ ÷ E. With E = 2.0 × 10¹¹ Pa for steel, ε = 1.8 × 10⁸ ÷ 2.0 × 10¹¹ = 0.0009, or 0.09 percent. The stretch is ε × L = 0.0009 × 2.0 m = 0.0018 m = 1.8 mm.

Now the plausible wrong answer. A rod 2 m long that stretches 1.8 mm, less than a tenth of a percent, looks untroubled, so it must be safe. Trace it. The stretch is set by the stiffness, E, which lesson 8 showed is a different property from strength. Safety depends on how the stress compares with the stress the steel can take. The Cambridge data book gives mild steel a yield strength of 250 to 395 MPa. If the rod is at the bottom of that range, the ratio of what it can take to what it carries is 250 ÷ 180 = 1.4.

That ratio is the factor of safety, and 1.4 is thin for a platform people stand on, built from steel whose grade nobody checked. The stretch was tiny because steel is stiff, not because the rod was lightly loaded. Small deformation and low stress are different facts, and only the second one says anything about safety.

Wrong answer 2: "The thick line held more, so it is the stronger material"

Two nylon fishing lines are pulled until they break: a 0.20 mm line breaks at 17 N and a 0.40 mm line at 68 N (illustrative results). The thick line held four times as much, so the plausible conclusion is that it is made of a stronger nylon. Trace it with areas. A circle's area is π times the radius squared:

• 0.20 mm line: A = π × (0.10 mm)² = 0.0314 mm², so the breaking stress is 17 N ÷ 0.0314 mm² = 540 N/mm² = 540 MPa.
• 0.40 mm line: A = π × (0.20 mm)² = 0.1257 mm², so the breaking stress is 68 N ÷ 0.1257 mm² = 540 MPa.

The two breaking stresses are identical. Doubling the diameter quadrupled the area, so the thick line carried four times the force at the same stress: it is the same material, just more of it. That is the whole reason engineers talk about stress and not force. Force describes one particular object; stress describes the material, so a test on a thin sample predicts the behaviour of a thick part.

Wrong answer 3: "The longer wire stretched twice as much, so it is half as stiff"

Hang the same 550 kg from two rods of the same steel and the same 30 mm² section, one 1.0 m long and one 2.0 m long. The short one stretches 0.9 mm and the long one 1.8 mm. Is the long rod less stiff? Work out the strain for each: 0.9 mm ÷ 1,000 mm = 0.0009, and 1.8 mm ÷ 2,000 mm = 0.0009. Every millimetre of each rod carries the same stress and stretches by the same fraction, and a rod with twice as many millimetres stretches twice as far. Strain does for stretch what stress does for force: it removes the size of the sample, so that what is left belongs to the material.

The core of it: divide force by area to get stress, and stretch by original length to get strain. Those two numbers describe the material, and every comparison, limit and safety check is made with them.

The curve: what a material does from first load to fracture

Load a sample in steps, work out the stress and strain at each step, and plot stress up the side against strain along the bottom. OpenStax describes the shape for a metal, and the diagram shows it with two other common shapes.

Follow the blue curve, a ductile metal such as mild steel, from the origin.

• The elastic region. At first the curve is a straight line through the origin: stress is proportional to strain, and the slope is Young's modulus. OpenStax calls the end of the straight part the proportionality limit, and the stress beyond which the material no longer springs back when unloaded the elastic limit. Unload anywhere in this region and the sample returns to its original length.
• Yield. Past the elastic limit the material deforms plastically: unload it and it stays longer than it was. For mild steel this starts at the yield strength, 250 to 395 MPa, at a strain of only about 0.1 to 0.2 percent (250 MPa ÷ 200,000 MPa = 0.00125). At true scale the straight part would be too steep and short to see; the diagram stretches it sideways.
• Strain hardening. The curve keeps rising, more slowly. The metal is getting harder to stretch as it deforms, which is work hardening, the reason a bent paper clip is stiffer at the bend.
• Tensile strength. The highest point of the curve is the tensile strength, 345 to 580 MPa for mild steel.
• Necking and fracture. After the peak, one section of the sample begins to thin faster than the rest, a neck. The force the sample can carry falls, and because the curve is drawn with the original area, the plotted stress falls too, until the sample breaks at the neck. OpenStax calls the stress at the fracture point the breaking stress.

The red line is a brittle material such as glass or acrylic: straight almost to the end, then fracture, with no yield and no warning. The grey line is rubber-like: soft at first, stiffening as its long molecules straighten out. Engineers read three things from any curve: the slope at the start (stiffness), the stress where the curve stops being straight (where permanent damage begins), and the area under it, which is the energy the material absorbs before it breaks. A ductile metal's curve encloses a large area; a brittle material's encloses very little, which is lesson 8's toughness seen on a graph.

Wrong answer 4: "The stress was below the yield strength, so it could not break"

On 16 January 1943, the SS Schenectady, a new welded tanker 159 m long, lay moored at the fitting-out dock of the Kaiser shipyard at Swan Island in Portland, Oregon. She had been declared complete on 31 December. The weather was calm. Without warning, and with a noise heard at least a mile away, her hull cracked almost in half just behind the superstructure, the cracks running down both sides nearly to the keel; only the bottom plates held. She was not at sea and not overloaded.

The plausible reasoning says this cannot happen: if the stress in the steel is below its yield strength, measured on a tensile curve like the blue one, the steel is in its elastic region and must spring back. Trace where it fails. The yield strength comes from a test on a smooth bar, usually at room temperature. The ship's steel was neither smooth nor warm. The official Coast Guard report blamed faulty welding, and the Board of Investigation weighed locked-in stresses left by welding, sharp changes in temperature, and flaws in the design. Later research pointed to brittle fracture: the low-grade steel used became highly brittle in cold weather. A brittle material does not need to yield before it breaks. Where there is a crack or a sharp notch, the stress at its tip is far higher than the average stress across the plate, and in a material with low toughness the crack runs. The ductile blue curve, measured warm on a smooth bar, simply did not describe that steel, that cold, with those flaws.

So a curve is a fact about one material in one condition. Change the temperature, add a notch, or weld it, and you may be on a different curve. That is why later rules for ship steel, and for anything that must not break suddenly, test toughness as well as strength, and test it cold.

How big should the factor of safety be?

A factor of safety covers everything the calculation does not know: loads larger than expected, material weaker than the table, flaws, wear, corrosion, mistakes. So its size depends on how much is unknown. NASA's standard for spaceflight structures, NASA-STD-5001B, sets a minimum ultimate design factor of 1.4 for metal structures, meaning they must be designed to carry 1.4 times the highest load expected before they fail. That sounds low, and it is low on purpose: every gram launched is expensive, so the margin is kept small and paid for with detailed analysis and with strength tests on flight-like hardware. The standard calls its factors minimum acceptable values.

A home project knows far less. The dowel's wood was never tested, fishing line is weakened by its knots, glue joints vary, and nobody will analyse the shelf the lamp sits on. Where the material and the loads are uncertain, factors of 3 or 4 are common working choices. The point is not a magic number. It is that the factor is a decision, written down with its reasons, like every other number in the brief.

Common misconceptions

• "Stress and force are the same thing." Force is a push or pull in newtons; stress is force per unit area, in pascals. The 0.40 mm line carried four times the force of the 0.20 mm line at exactly the same 540 MPa.
• "Strain is measured in millimetres." Strain is stretch divided by original length, so it has no unit. Both steel rods, 1 m and 2 m long, had a strain of 0.0009, though one stretched 0.9 mm and the other 1.8 mm.
• "If a part hardly deforms, it is lightly stressed." Deformation depends on stiffness. The steel rod stretched only 0.09 percent while carrying 180 MPa, within a factor of 1.4 of mild steel's lowest yield strength.
• "The tensile strength is the stress at which a sample breaks." For a ductile metal the tensile strength is the peak of the curve. After it, the sample necks and the plotted stress falls, so the breaking stress is lower than the tensile strength.

What to remember

David Kirkaldy's machine, with Facts not Opinions over the door, showed that the Tay Bridge's ties and lugs failed at about 20 tons, well below what the design had expected. Stress is force divided by area, in pascals or N/mm²; strain is stretch divided by original length, with no unit; in the elastic region E = σ ÷ ε. OpenStax's 2.0 m steel rod holding 550 kg carries 180 MPa and stretches only 1.8 mm, yet its factor of safety against mild steel's lowest yield strength is 250 ÷ 180 = 1.4. A 0.20 mm and a 0.40 mm line that break at 17 N and 68 N both break at 540 MPa, and rods of 1 m and 2 m under the same stress share a strain of 0.0009. The stress-strain curve of a ductile metal runs straight, yields, hardens to its tensile strength, necks and breaks; a brittle material breaks while still straight. The SS Schenectady broke in calm water in January 1943 because a curve measured on smooth, warm steel did not describe brittle steel with welding flaws in the cold. NASA designs spaceflight metal structures to a minimum ultimate factor of 1.4, kept low by analysis and testing; less certain work needs more.

The core of it: compare materials by stress and strain, judge safety by stress against strength and never by how little something stretches, and choose a factor of safety that matches how much you do not know.

Sources

1. Moebs, W., Ling, S. J., & Sanny, J. (2016). 12.3 Stress, strain, and elastic modulus. In University physics volume 1. OpenStax. OpenStax
2. Moebs, W., Ling, S. J., & Sanny, J. (2016). 12.4 Elasticity and plasticity. In University physics volume 1. OpenStax. OpenStax
3. NASA Office of the Chief Engineer. (2022). Structural design and test factors of safety for spaceflight hardware (NASA-STD-5001B with Change 3). NASA. NASA
4. Cambridge University Engineering Department. (2003). Materials data book (2003 ed.). University of Cambridge. University of Cambridge
5. Wikipedia contributors. (2026). SS Schenectady. In Wikipedia, The Free Encyclopedia. Wikipedia

Day eight: the lamp that went over the edge

The first complete reading light stood on the shelf beside the desk for a week. Its base was the box from lesson 3's concept B, 150 mm from front to back and 120 mm wide, with the power bank inside. A wooden post 12 mm across rose 450 mm from the back of the base, 130 mm behind its front edge and centred across its width. The 300 mm dowel arm chosen in lesson 8 reached forward from the top of the post, and the 60 g hood hung from its end. On the eighth evening the reader swung the arm round to the side, toward the bed, to read a letter held in their lap. The lamp leaned, paused and fell off the shelf. (The events and masses in this lesson are illustrative.)

The kitchen scale gives the parts: base with power bank and switch, 190 g; post, 36 g; arm, 11 g; hood, 60 g; 297 g in all, comfortably inside R5's 400 g. Two questions follow, and they are the questions of this lesson. Why did the lamp stand for a week facing forward and fall the first time it faced sideways? And what is the smallest change that stops it falling without breaking R5 or the shelf-area limit, C3?

Step 1: draw every force on one object

Guessing at tipping problems goes round in circles, so engineers start with a free-body diagram: one object, cut free from everything around it, with every force that acts on it drawn as an arrow at the point where it acts. OpenStax's University Physics gives the rules. Draw the object, simply. Include every force that acts on it. Do not include forces the object exerts on its surroundings, and never include both forces of an action-reaction pair. If there are several objects, draw a separate diagram for each.

Choose the whole lamp as the object. Two kinds of force act on it from outside. Each part's weight, its mass times 9.81 N/kg, acts straight down through that part's centre of mass. And the shelf pushes up, the normal force. Forces between the parts, such as the post pressing on the base, do not appear: they act inside the object and cancel in pairs. Here is the diagram with the arm swung sideways, at the moment the lamp is about to tip, seen from in front.

Notice where the normal force is drawn. While the lamp stands firmly, the shelf pushes up all along the underside of the base. As the lamp starts to tip, the base lifts off everywhere except along one edge, so at the point of tipping the whole normal force acts at that edge. That single fact makes the problem easy to solve.

Step 2: forces alone cannot answer the question

OpenStax states two conditions for an object to stay at rest. The first is that the forces balance: the net force is zero. Up must equal down, so the normal force equals the total weight: N = 0.297 kg × 9.81 N/kg = 2.91 N. That is true whether the lamp faces forward or sideways, and it says nothing about tipping. A lamp can have perfectly balanced forces and still rotate. The second condition deals with rotation: the net moment, also called torque, must be zero as well.

Step 3: what a moment is

A moment is the turning effect of a force about a point. Its size is the force multiplied by its lever arm, which OpenStax's College Physics defines as the shortest distance from the pivot to the line along which the force acts. For a weight, which acts straight down, that is the horizontal distance from the pivot. The unit is the newton metre, N·m. The book's seesaw example shows the idea: a 26.0 kg child sits 1.60 m from the pivot, and a 32.0 kg child balances them by sitting closer, at 26.0 × 1.60 ÷ 32.0 = 1.30 m. Weight times distance on one side equals weight times distance on the other.

Turning directions need a sign. The usual convention counts anticlockwise moments as positive and clockwise as negative. In this lesson it is simpler to name the two directions by what they do: a restoring moment presses the base back down onto the shelf, and a tipping moment lifts it.

Step 4: take moments about the edge it would tip over

The second condition holds about any point you choose, and a good choice makes the work short. Take moments about the edge the lamp would tip over. The normal force acts exactly there, so its lever arm is zero and it drops out of the calculation, leaving only the weights. Here is the sideways case. Each distance is measured horizontally from the side edge of the 120 mm base; the post is centred across the width, 60 mm inside the edge, and the arm reaches 300 mm out from it.

The tipping moment beats the restoring moment by 0.151 − 0.133 = 0.018 N·m, so the lamp rotates about the edge and falls. Look at where the trouble comes from: the hood weighs less than a third of the base, but it acts four times as far from the edge, so its moment is larger than the base's.

Now face the arm forward and do the same about the front edge. The post stands 130 mm behind the front edge, the base's centre is 75 mm behind it, the middle of the arm is 150 − 130 = 20 mm in front of it, and the hood 300 − 130 = 170 mm in front. Restoring: 1.864 × 0.075 + 0.353 × 0.130 = 0.140 + 0.046 = 0.186 N·m. Tipping: 0.108 × 0.020 + 0.589 × 0.170 = 0.002 + 0.100 = 0.102 N·m. The restoring moment is 0.186 ÷ 0.102 = 1.8 times the tipping moment, so facing forward the lamp stood with room to spare. That is why it survived a week. The base is deeper than it is wide, and the post sits at the back, so swinging the arm to the side took away most of its margin and then some.

The same answer from the centre of mass

There is a second way to see it. The four weights together act as one weight at the lamp's centre of mass, whose horizontal position is each mass times its position, added up and divided by the total mass. Measuring inward from the side edge, with positions outside the edge counted as negative:

x = (190 × 60 + 36 × 60 + 11 × (−90) + 60 × (−240)) ÷ 297 = (11,400 + 2,160 − 990 − 14,400) ÷ 297 = −1,830 ÷ 297 = −6.2 mm

The centre of mass lies 6.2 mm outside the base, so nothing the shelf can do will hold the lamp up. Facing forward, the same sum about the front edge gives (190 × 75 + 36 × 130 + 11 × (−20) + 60 × (−170)) ÷ 297 = 8,510 ÷ 297 = 28.7 mm inside. OpenStax's chapter on stability puts the rule in one line: the critical point is reached when the centre of gravity is no longer above the base of support.

The most famous leaning building in the world obeys the same rule. The Leaning Tower of Pisa, begun in 1173, started to lean during construction because the ground under it was soft. By 1990 the tilt had reached 5.5 degrees, and the tower was closed to the public on 7 January that year. Between 1993 and 2001 engineers added 870 tonnes of lead counterweights and drew 38 cubic metres of soil out from under the raised side, and the tower settled back to 3.97 degrees, with its top 3.9 m out of line over a height of about 56 m. It stands because the vertical line through its centre of mass still falls inside its base; the work moved that line back toward the middle before the ground could let it creep any further out.

Bottom line: an object on a base tips when the vertical line through its centre of mass passes outside the base. Taking moments about the edge it would tip over tells you whether it will, and by how much.

Three fixes, each checked by moments

Fix 1: add ballast. A mass m placed in the middle of the base adds a restoring moment of m × 9.81 × 0.060 m. To balance exactly needs (0.151 − 0.133) ÷ (9.81 × 0.060) = 0.0305 kg, about 31 g. But balanced exactly means the slightest nudge tips it. For the restoring moment to be 1.5 times the tipping moment, the ballast must supply 1.5 × 0.151 − 0.133 = 0.093 N·m, which takes 0.093 ÷ (9.81 × 0.060) = 0.159 kg. The lamp would then weigh 297 + 159 = 456 g, breaking R5's 400 g. Rejected.

Fix 2: stop the arm swinging sideways. A peg that limits the swing would work, and the moments for the forward position already pass. But reading in bed was the very use that turned the arm, and the brief's need statement is about the reader. Kept as a fallback.

Fix 3: widen the base. C3 allows a shelf area of 30 cm by 20 cm, so the base can grow to 200 mm wide while staying 150 mm deep. With the post still centred across the width, it now stands 100 mm from each side edge. Sideways: restoring (1.864 + 0.353) × 0.100 = 0.222 N·m; tipping 0.108 × (0.150 − 0.100) + 0.589 × (0.300 − 0.100) = 0.005 + 0.118 = 0.123 N·m. The ratio is 0.222 ÷ 0.123 = 1.8, the same margin the lamp had facing forward, and the centre of mass is 33.8 mm inside the side edge. The extra card in the wider base adds a few grams, which only helps. This fix is chosen: it costs almost nothing in mass and keeps the swinging arm.

Moments also say how much margin 1.8 really is. The wider base can take an extra tipping moment of 0.222 − 0.123 = 0.099 N·m before it goes over. A finger pressing down on the hood, 200 mm outside the side edge, reaches that with a push of 0.099 ÷ 0.200 = 0.49 N, about the weight of 50 g. So the lamp is stable in use but not when someone leans on the hood to adjust it, and the instructions will say to steady the base with the other hand. Writing that down is part of the design.

One part on its own: the arm and its joint

A free-body diagram can be drawn for any part, and the arm is worth one. Cut it free at the joint on top of the post. Its own weight, 0.108 N, acts 150 mm from the joint, and the hood's 0.589 N acts at 300 mm. For the arm to stay still, the joint must push up with 0.108 + 0.589 = 0.70 N, and it must also supply a moment equal and opposite to the weights' moment about the joint: 0.589 × 0.300 + 0.108 × 0.150 = 0.177 + 0.016 = 0.193 N·m. If the joint cannot resist 0.193 N·m, the arm swings down until the hood rests on the desk. In lesson 6's design the wing nut squeezes the arm pieces together and friction supplies that moment; lesson 13 works out how hard it must squeeze.

Common misconceptions

• "A heavier base always makes a lamp harder to tip." Moments depend on distance as well as weight. The 60 g hood at 240 mm outside the edge out-turned the 190 g base at 60 mm inside it. Widening the base did more than adding 159 g of ballast would have.
• "The shelf pushes up at the middle of the base." The normal force spreads under the whole base when the lamp is secure, and shifts toward the edge as it starts to tip. At the point of tipping, all of it acts at the edge.
• "A free-body diagram shows all the forces involved." It shows only the forces acting on the chosen object. Forces the lamp exerts on the shelf, and forces between its own parts, are left out.
• "The moment arm is the distance to where the force is applied." It is the perpendicular distance from the pivot to the force's line of action. For a weight acting straight down, that is the horizontal distance, however high up the part is.

Pulling it together

The reading light fell because, with its arm swung sideways over a 120 mm base, the hood's 0.589 N at 240 mm outside the edge and the arm's 0.108 N at 90 mm made a tipping moment of 0.151 N·m, against a restoring moment of only 0.133 N·m from the base and post 60 mm inside it. Facing forward the margin was 1.8, which is why it stood for a week. A free-body diagram shows every external force on one object at the point where it acts; the forces balance, so the shelf pushes up with the full 2.91 N, but only moments can say whether the object rotates. Taking moments about the tipping edge removes the normal force from the sum. The centre of mass told the same story, 6.2 mm outside the base. Ballast enough for a margin of 1.5 would have taken the lamp to 456 g; widening the base to 200 mm, inside C3, restored a ratio of 1.8 at no cost in mass, though a push of about 0.49 N on the hood can still tip it. The arm's own diagram showed that its joint must resist 0.193 N·m.

Bottom line: draw the object alone with every force on it, take moments about the point where it would turn, and design so that the centre of mass stays well inside the base in every position the user can reach.

Sources

1. Moebs, W., Ling, S. J., & Sanny, J. (2016). 5.7 Drawing free-body diagrams. In University physics volume 1. OpenStax. OpenStax
2. Moebs, W., Ling, S. J., & Sanny, J. (2016). 12.1 Conditions for static equilibrium. In University physics volume 1. OpenStax. OpenStax
3. Urone, P. P., & Hinrichs, R. (2022). 9.2 The second condition for equilibrium. In College physics 2e. OpenStax. OpenStax
4. Urone, P. P., & Hinrichs, R. (2022). 9.3 Stability. In College physics 2e. OpenStax. OpenStax
5. Wikipedia contributors. (2026). Leaning Tower of Pisa. In Wikipedia, The Free Encyclopedia. Wikipedia

Four straws and a fingertip

Pin four drinking straws into a square with a paper fastener through the ends at each corner, lay it flat and push one corner sideways with a fingertip. It folds into a flat diamond with almost no effort. Every straw is still exactly the same length; only the angles at the corners have changed. Now pin three straws into a triangle and push the same way. It does not move. Three sides of fixed length fit together in only one shape, so a pin-jointed triangle cannot change shape unless one of its sides stretches, shortens or breaks. A square can rack; a triangle cannot.

That is the whole reason the bridges of the American railroad age were built of triangles. In 1840 William Howe, a contractor in Massachusetts, patented the Howe truss: wooden diagonal braces with iron vertical rods. In 1844 Thomas and Caleb Pratt patented the Pratt truss, with the diagonals sloping the other way, down toward the centre. The two look like mirror images. They are not: under the same load, every diagonal in one is pulled and every diagonal in the other is pushed, and that difference decided which materials each one suited.

This lesson solves a small Pratt truss step by step, the way engineers do, then reverses its diagonals into a Howe truss and runs the procedure again to see what flips. The truss is sized like the card bridge you will build in lesson 12.

What makes a truss a truss

A truss, in Louis Bucciarelli's MIT text, is built so that its members carry their loads in tension or compression. On paper it is made of straight members joined by frictionless pins, with every load applied at a joint. Each member then has forces at only its two ends, and a member with forces at only two points can be in equilibrium only if those forces pull straight along it or push straight along it. So every member is in pure tension, being stretched, or pure compression, being squashed, and never bends. That is efficient: the whole cross-section carries the same stress, the uniform F ÷ A of lesson 9, instead of only the outer layers working hard as they do in a bent beam.

Real joints are bolted, riveted or glued, not frictionless pins, and Bucciarelli admits that such pins do not exist. But a structure designed and built according to accepted practice behaves closely enough like the ideal one that the ideal analysis is the basis of its design. One counting rule tells you whether a flat truss has just enough members: for j joints, a stable truss that can be solved by equilibrium alone has m = 2j − 3 members. Fewer and it can fold like the square; more and the extra members share the load in ways equilibrium alone cannot work out.

The truss to solve

Here is a Pratt truss 400 mm long and 100 mm high in four panels of 100 mm, with eight joints, A to H, and a load of P = 20 N, the weight of about 2 kg, hanging from the middle of the bottom chord at C. It rests on a pin at A and a roller at E, which can push up but cannot push sideways. Count: 8 joints, so 2 × 8 − 3 = 13 members are needed, and the truss has 13: four in the bottom chord, two in the top chord, two sloping end posts, three verticals and two diagonals.

Step 1: find the support reactions

Treat the whole truss as one free body, as in lesson 10. Take moments about A: the roller at E pushes up with a force E_y at 0.400 m, and the load pushes down with 20 N at 0.200 m, so E_y × 0.400 = 20 × 0.200 and E_y = 10 N. Vertical forces balance, so the pin at A pushes up with 20 − 10 = 10 N. Nothing pushes sideways, so the pin's horizontal reaction is zero. With the load in the middle, each support takes half, as you would expect.

Step 2: start at a joint with only two unknowns

The method of joints treats each joint as a small free body held still by the members meeting there. Each joint gives two equations, horizontal forces balance and vertical forces balance, so a joint can be solved when no more than two member forces at it are unknown. Engineering statics texts use one convention that saves endless sign trouble: assume every unknown member is in tension, pulling away from the joint. A positive answer then means tension and a negative one compression.

Joint A. Three forces: the 10 N reaction upward, the end post AF pulling up and to the right at 45 degrees, and the bottom chord AB pulling to the right. Because the truss is as high as a panel is wide, the end post and both diagonals lie at 45 degrees, and the sine and cosine of 45 degrees are both 0.707.

• Vertical: F_AF × 0.707 + 10 = 0, so F_AF = −14.1 N. Negative: the end post is in compression, pushing on the joint with 14.1 N.
• Horizontal: F_AB + F_AF × 0.707 = 0, so F_AB = −(−14.1 × 0.707) = +10 N. The bottom chord is in tension.

Step 3: joint B, and a member that carries nothing

At B meet AB and BC, both horizontal, and the vertical BF, with no load. The only vertical force at B is the one in BF, so it must be zero: F_BF = 0. Horizontally, BC must balance AB: F_BC = 10 N, tension. A member that carries no force under a particular load is a zero-force member. It is not useless. Move the load to B and BF carries all of it, and in a real bridge such members also brace the long compression members against bending sideways.

Step 4: joint F, where the diagonal starts

At F meet the end post AF, now known to push with 14.1 N, the zero-force vertical BF, the top chord FG and the diagonal FC, which runs down and to the right at 45 degrees. Two unknowns, two equations.

• Vertical: the end post's push on F has an upward part of 14.1 × 0.707 = 10 N. The diagonal, if in tension, pulls F downward with F_FC × 0.707. So F_FC × 0.707 = 10, and F_FC = +14.1 N: the diagonal is in tension.
• Horizontal: the end post pushes F to the right with 10 N, and the diagonal pulls it to the right with 14.1 × 0.707 = 10 N, so the top chord must pull left with 20 N: F_FG = −20 N, compression.

Step 5: the rest by symmetry, and a check that must come out right

At G meet FG, GH and the vertical GC, with no load. Horizontally, GH must equal FG, so GH is also 20 N in compression; vertically, GC carries nothing. The right half mirrors the left. Then use joint C as a check, since every force there is now known: the two diagonals each pull C upward with 14.1 × 0.707 = 10 N, which together hold up the 20 N load exactly, and the bottom chords pull 10 N each way. Everything balances. A check that closes is the sign the arithmetic is right.

The pattern is worth noticing: under a downward load, the top of the truss is squeezed and the bottom is stretched, just as the top of a bent ruler shortens and the bottom lengthens.

A shortcut that checks the top chord: cut the truss in two

The method of sections cuts the truss through no more than three members and treats one piece as a free body. Cut through FG, FC and BC, between B and C, and keep the left piece. Take moments about joint C, where two of the three cut members meet: their forces pass through C and drop out, leaving the reaction at A and the top chord. The reaction, 10 N at 0.200 m from C, turns the piece one way with 10 × 0.200 = 2.0 N·m; the top chord, 0.100 m above C, must turn it back with the same moment, so F_FG = 2.0 ÷ 0.100 = 20 N, compression. Same answer, one equation. It also shows where the top chord's force comes from: the bending moment at the middle of the span, divided by the truss's height. A truss twice as high would have half the force in its chords.

Why this matters: the procedure, reactions first, then joints with two unknowns, tension assumed, a check at the end, turns a web of members into a list of forces you can design each member for. Without it, every member is a guess.

Second run: reverse the diagonals and the answer flips

Now change one thing. Take out FC and HC and put in BG and DG, so that the diagonals slope up toward the centre. That is a Howe truss. Same joints, same load, same supports, same 13 members. Run the procedure again: joint A gives the same 14.1 N compression in AF and 10 N tension in AB. At F there is now no diagonal, so the end post's 10 N upward push must be balanced by the vertical BF pulling down: F_BF = +10 N, tension, and the top chord takes the horizontal 10 N: F_FG = −10 N. At B, the vertical's 10 N pull upward must be balanced by the diagonal BG: F_BG × 0.707 = −10, so F_BG = −14.1 N, compression, and BC becomes 20 N in tension. At G the two diagonals push down on the joint with 10 N each, so the vertical GC must pull with 20 N.

Every diagonal has gone from pulling to pushing, and the verticals have gone from carrying nothing to pulling. That is what Wikipedia's descriptions of the two historical designs record: the Howe truss used iron vertical rods in tension with wooden diagonal braces in compression, and the Pratt truss used its diagonals in tension, and became common on railroads as bridges moved from wood to metal. Iron and steel rods are excellent in tension and slender enough to be poor in compression; thick timbers push well.

Why the direction matters: a pulled member is safe, a pushed one buckles

A member in tension fails when its stress reaches the material's strength, so a thin rod or a string can carry a large pull. A slender member in compression usually fails much earlier, by buckling: it bows sideways and collapses. Euler's formula for the buckling load of a slender column makes the load proportional to 1 ÷ L², where L is the length, so a member twice as long buckles at a quarter of the load. That is why the string test in the activity works: a string can replace a member in tension, never one in compression.

Use it to choose between the two layouts for members made of drinking straws, which are slender. The diagonals are 100 × 1.414 = 141 mm long, and a 141 mm straw buckles at (100 ÷ 141)² = 0.50 of the load a 100 mm straw can take. In the Howe truss, 14.1 N of compression in a 141 mm diagonal is as hard on it as 14.1 ÷ 0.50 = 28 N would be on a 100 mm member. In the Pratt truss, the worst compression is the top chord's 20 N, in members only 100 mm long. So with straws the Pratt should carry about 28 ÷ 20 = 1.4 times as much before something buckles. The changed input, the direction of the diagonals, changed which member fails and roughly how soon.

Common misconceptions

• "A member that carries no force can be left out." The verticals of the Pratt truss carry nothing only while the load hangs at C. Move it to B and BF carries all of it. Zero-force members also stop long compression members from bowing sideways.
• "A triangle is strong just because of its shape." A triangle keeps its shape only while its sides keep their lengths. It is only as strong as its members and its joints, and Bucciarelli notes that structures often fail at their joints, as you may find in lesson 12.
• "A load pushing down puts everything in compression." The bottom chord and the Pratt diagonals are in tension. Under a downward load the top of a truss is squeezed and the bottom stretched.
• "A member fails at the same force whether it is pulled or pushed." A slender member buckles in compression at a small fraction of the force it carries in tension, and the longer it is the worse it gets, as 1 ÷ L².

The takeaway

A pin-jointed square racks into a diamond without any side changing length; a triangle cannot change shape without a side stretching or shortening, so trusses are built of triangles, and with loads at the joints every member carries pure tension or compression. A flat truss with j joints needs 2j − 3 members. To solve one, find the support reactions, then work joint by joint where no more than two forces are unknown, assuming tension so that negative answers mean compression, and finish with a check. For the 400 mm Pratt truss with 20 N at the middle: the end posts push with 14.1 N, the bottom chord pulls with 10 N, the top chord pushes with 20 N, the diagonals pull with 14.1 N and the verticals carry nothing, and a cut through the middle panel gives the top chord's 20 N in one moment equation, 2.0 N·m divided by the 0.100 m height. Reversing the diagonals into a Howe truss turns the diagonals into 14.1 N struts and the verticals into 10 N and 20 N ties. Because slender members buckle in proportion to 1 ÷ L², a 141 mm diagonal in compression is the weakest link, and the Pratt layout suits slender members such as straws, rods and steel.

Why this matters: solving a truss tells you which members pull and which push, and by how much. Put the long members in tension, keep the compressed ones short or thick, and you get far more bridge from the same material.

Sources

1. Bucciarelli, L. L. (2002). Chapter 3: Internal forces and moments. In Engineering mechanics for structures. MIT OpenCourseWare, 1.050 Solid Mechanics. MIT OpenCourseWare
2. Baker, D. W., & Haynes, W. (n.d.). 6.4 Method of joints. In Engineering statics: Open and interactive. Engineering Statics
3. Baker, D. W., & Haynes, W. (n.d.). 6.5 Method of sections. In Engineering statics: Open and interactive. Engineering Statics
4. Wikipedia contributors. (2026). Pratt truss. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Wikipedia contributors. (2026). Howe truss. In Wikipedia, The Free Encyclopedia. Wikipedia

Millwall: a 75-foot model of a bridge that did not yet exist

Robert Stephenson needed to carry the Chester and Holyhead Railway across the Menai Strait in North Wales, on two main spans of 460 feet (140 m). His idea was a tube of wrought iron plates big enough for trains to run inside. Nobody knew how such a tube would fail, so the engineer and shipbuilder William Fairbairn ran a series of experiments on tubes of different shapes, enlisting Eaton Hodgkinson, then the leading authority on the strength of iron beams. Stephenson reported their findings to the railway's directors in February 1846, and a model tube with a 75-foot (23 m) span was built and tested at Fairbairn's shipyard at Millwall on the Thames and used as the basis for the final design. The experiments showed that, without special precautions, the tube would fail by buckling of its top plate in compression. Fairbairn's answer was to build the top of the tubes as a row of cells, small tubes inside the big one. The Britannia Bridge opened to trains in 1850.

The engineers did not load their model to failure for the drama. They did it because the first thing to fail in a structure is the one fact a designer most needs and can least easily calculate, and a model costs far less to lose than a bridge. This lesson follows one student team, with illustrative numbers, through the same process at kitchen-table scale: a card bridge designed on paper, its material tested, its failure load predicted, then built and loaded until it broke, and built again.

The brief: a span, a limit and a figure of merit

The team wrote the brief first, as lesson 2 taught.

• Constraints. Clear span 400 mm between two tables. Mass no more than 60 g. Materials: cereal-box card and white PVA glue only. The load hangs from a cross-bar at the middle of the span.
• Figure of merit. Failure load divided by the bridge's own mass. A heavy bridge that holds a lot is not impressive; a light one that holds a lot is. Real structures are judged the same way, since every kilogram of bridge is a kilogram that must itself be carried.
• Verification. Load in measured steps until failure, filming the test, with the mass checked on a kitchen scale before loading.

The structure: two Pratt trusses, with tubes where they push

The team chose two Pratt trusses of the lesson 11 pattern, 400 mm long and 100 mm high in four panels, standing side by side and joined by card cross-pieces at the joints. Lesson 11's analysis says which members pull and which push. Under a load P at the middle, each truss carries P ÷ 2, and for each truss the top chord pushes with the full P ÷ 2, the end posts push with 0.707 × P ÷ 2, the diagonals pull with 0.707 × P ÷ 2 and the bottom chord pulls with 0.5 × P ÷ 2.

The members in tension could be flat strips of card, because a strip pulled along its length has nothing to buckle. The members in compression could not. A flat strip 10 mm wide and 0.43 mm thick bends sideways under a few grams. Bending resistance depends on the second moment of area, which for a rectangle is width times thickness cubed divided by 12, as lesson 8 showed:

Flat strip: I = 10 × 0.43³ ÷ 12 = 0.066 mm⁴
Square tube folded from the same card, 10 mm across: the outer square minus the inner, I = (10⁴ − 9.14⁴) ÷ 12 = (10,000 − 6,979) ÷ 12 = 252 mm⁴

The tube uses four times as much card as the strip, since its perimeter is 40 mm against the strip's 10 mm, and resists bending about 3,800 times as well. Euler's buckling load is proportional to I, so the tube is the same trick as Fairbairn's cells: put the material far from the middle of the section, where it does the most work.

Testing the card before trusting it

No table lists cereal-box card, so the team measured it, as Kirkaldy would have.

• Tension. Three strips 10 mm wide were hung with a bucket and loaded with water until they tore: 118, 126 and 122 N. The middle value, 122 N, over an area of 10 mm × 0.43 mm = 4.3 mm², gives a tensile strength of 122 ÷ 4.3 = 28 MPa.
• Compression. Square tubes 10 mm across were stood on end on a kitchen scale and pressed down by hand, slowly, until they buckled; the scale's peak reading gives the force. Tubes 100 mm long buckled at 4.1 kg, or 4.1 × 9.81 = 40 N. Tubes 141 mm long buckled at 2.0 kg, 20 N. The ratio, 20 ÷ 40 = 0.49, matches the 1 ÷ L² rule from lesson 11, which predicts (100 ÷ 141)² = 0.50.
• A glued joint. Two strips 10 mm wide, glued with a 10 mm overlap and left overnight, were pulled straight apart. The joint held 55 N.

Predicting the failure before building anything

Now combine the member forces with the test results. For each member, the largest load per truss it allows is its tested capacity divided by its force coefficient.

The weakest link is the end posts: at 28 N per truss, a total load of 2 × 28 = 57 N, or 5.8 kg. The long compression members, as lesson 11 warned, are the problem. So the team changed the design on paper, before cutting any card: the end posts became 15 mm square tubes. For the same card, I = (15⁴ − 14.14⁴) ÷ 12 = 887 mm⁴, 3.5 times the 10 mm tube's, so the 141 mm end posts should now buckle at about 20 × 3.5 = 70 N, allowing 70 ÷ 0.707 = 100 N per truss. Now the top chord governs at 40 N per truss: a predicted failure load of 80 N, or 80 ÷ 9.81 = 8.2 kg, by buckling of the top chord.

Key idea: the analysis tells you what each member carries, the tests tell you what each member can take, and dividing one by the other finds the weakest link before you build. Fix the weakest link, and the next one becomes the limit.

Building it

Building took two evenings. The team drew the truss full size on paper and built both trusses directly on the drawing, so that every joint landed in the same place; lesson 6's tolerances apply here, because a member cut 2 mm short pulls the truss out of shape and starts it off stressed before any load goes on. They folded the tubes round a 10 mm square wooden strip and a 15 mm one, glued them along a lapped seam, and cut every craft-knife line on a cutting board with the blade moving away from their fingers. PVA glue needs time to reach its strength, so the finished bridge sat overnight before testing. On the kitchen scale it weighed 52 g, inside the 60 g limit.

The test: 6.8 kg, then a sound like a zip

The loading rig was a bucket hung by string from the cross-bar at mid-span, filled with sand from a jug in 0.5 kg steps, each step weighed. A ruler taped upright beside the middle of the bridge measured the sag, and a phone filmed the joints at C and G in slow motion. For safety the bucket hung only 5 cm above a cardboard box lined with a towel, nobody's feet were under it, and the team wore safety glasses, because card under load can snap and flick fragments.

The sag grew steadily: 1.0 mm at 2 kg, 2.2 mm at 4 kg, 3.5 mm at 6 kg. At 6.5 kg there was a faint crackle. At 6.8 kg, eleven seconds after the last sand went in, the bridge folded and the bucket dropped onto the towel. Total load: 6.8 × 9.81 = 66.7 N, or 33.4 N per truss. That was 6.8 ÷ 8.2 = 83 percent of the prediction, and it was not the top chord.

Reading the wreck

After a collapse, almost everything is damaged, and most of the damage is a result, not a cause. The job is to find the first failure. The slow-motion video settled it: the joint at C, where the two diagonals meet the bottom chord, opened first. A tenth of a second later the bottom chord dropped, the trusses lurched, and only then did the top chord buckle. The broken joint confirmed it. The glue tab had come away carrying a layer of card fibres with it: the glue had held, and the card had split within its own thickness.

Why did a joint tested at 55 N fail when each diagonal pulled only 0.707 × 33.4 = 23.6 N? Because the test was not the use. In the test the strips were pulled straight along the glued overlap, so the glue was loaded in shear across its whole area. At C the diagonal pulled at 45 degrees to the bottom chord and lifted the end of its tab, so the force was concentrated along one edge and peeled it open, the way you peel a sticker by lifting a corner. A test predicts only what is loaded the way it was tested, which is Kirkaldy's lesson again. Bucciarelli's MIT text puts it bluntly: structures often fail at their joints.

Version 2: one change, and the predicted failure

As lesson 4 taught, the team changed one thing. They glued gusset plates, triangles of card about 30 mm across, over both faces of every joint, so that each diagonal's pull spread across a large glued area in shear instead of peeling one edge. The gussets added 3 g, making 55 g. Same rig, same steps. Version 2 carried 8.6 kg and failed when the top chord buckled sideways next to G, the mode and place the prediction named, at 8.6 ÷ 8.2 = 105 percent of the predicted load. Figure of merit: version 1, 6.8 kg ÷ 0.052 kg = 131; version 2, 8.6 ÷ 0.055 = 156. Three grams of gusset raised the load the bridge carried for each gram of its own mass by about a fifth.

The next change would be the top chord itself, perhaps a 12 mm tube, and the next limit would then appear somewhere else. That is how the process ends, not with a perfect bridge, but with a record of where each version failed and why. Fairbairn's cells and the team's gussets are the same move: find the first thing that fails, and put material exactly there.

Common misconceptions

• "The bridge broke where the damage is worst." A collapse damages everything at once. The first failure was a small peeled joint at C; the buckled top chord was a consequence. Video and the broken surfaces find the origin.
• "If the calculation says 8.2 kg, the bridge will hold 8.2 kg." The calculation assumed ideal joints. Version 1 failed at 83 percent of the prediction, by a mode the calculation did not include.
• "A flat strip and a tube of the same card are about equally strong in compression." The tube uses four times the card and resists bending about 3,800 times as well, because its material sits far from the middle of the section.
• "A test result is a property of the part." It is a property of the part loaded in a particular way. The joint held 55 N pulled straight and peeled open at 23.6 N pulled at an angle.

What to carry forward

Fairbairn's experiments for the Britannia Bridge, which went as far as a 75-foot model tube, showed that the top plate would buckle in compression, and the finished tubes were given cellular tops. The card bridge followed the same path. The brief set a 400 mm span, a 60 g limit and a figure of merit of load divided by mass. Lesson 11's member forces, combined with tested capacities (strips at 28 MPa, 10 mm tubes buckling at 40 N when 100 mm long and 20 N when 141 mm long, a straight-pulled glue joint at 55 N), predicted failure of the end posts at 5.8 kg; 15 mm end-post tubes, with 3.5 times the second moment of area, moved the prediction to 8.2 kg by top-chord buckling. The 52 g bridge failed at 6.8 kg, 83 percent of the prediction, when the joint at C peeled; gusset plates, adding 3 g, raised the failure load to 8.6 kg, 105 percent of the prediction, by the predicted top-chord buckling, and the figure of merit rose from 131 to 156.

Key idea: predict before you build, load to failure, find the first thing that broke, and change one thing. A failure you can explain is worth more than a success you cannot.

Sources

1. Bucciarelli, L. L. (2002). Chapter 3: Internal forces and moments. In Engineering mechanics for structures. MIT OpenCourseWare, 1.050 Solid Mechanics. MIT OpenCourseWare
2. Roylance, D. (n.d.). 4.2 Stresses in beams. In Mechanics of materials. Engineering LibreTexts, from MIT OpenCourseWare. LibreTexts
3. Wikipedia contributors. (2026). Britannia Bridge. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Euler's critical load. In Wikipedia, The Free Encyclopedia. Wikipedia

Two centimetres a night

Once its base was widened, the reading light stopped falling over. Its arm, though, would not stay where it was put. Each evening the reader set the hood at the right height; each morning it hung about 2 cm lower. (The events and measurements in this lesson are illustrative.) The arm's joint works like this: two plywood cheeks at the top of the post sandwich a plywood tab on the end of the arm. Lesson 6's 5 mm pin passes through all three and locates them, so the arm swings about the pin. A second bolt, an M5 with a wing nut, passes through a curved slot in the cheeks 20 mm from the pin; tightening the wing nut squeezes the cheeks onto the tab, and friction between them holds the arm's angle. Lesson 10's free-body diagram of the arm showed that, with the arm level, the joint must resist a moment of 0.193 N·m.

The reader had tightened the wing nut until it felt snug. Nobody knew what clamping force that was, or what clamping force was needed. Three questions follow. How hard must the wing nut squeeze? Can a finger and thumb supply that much? And where does the effort of tightening go? Answering them takes friction, two simple machines and the idea of work.

Friction depends on the squeeze, not on the area

When two surfaces are pressed together, friction resists their sliding over each other. OpenStax's College Physics gives the two working laws. Static friction, before anything moves, can be anything up to a limit: f_s ≤ μ_sN, where N is the normal force pressing the surfaces together and μ_s is the coefficient of static friction. Once sliding starts, kinetic friction takes over: f_k = μ_kN, with μ_k usually smaller. The book's table gives some typical pairs:

Two features of the laws surprise people. Friction is proportional to the normal force but not to the area in contact, which OpenStax calls a somewhat counterintuitive notion: a brick slides as easily on its narrow side as on its wide one. And the table's values are stated to only one or two digits, because real surfaces vary with finish, dirt and moisture. For a design that depends on friction, measure your own surfaces.

Measuring μ for the actual plywood

Lay one piece of the plywood on a longer board of the same plywood and lift one end of the board slowly until the piece starts to slide. At that moment the weight's component down the slope, mg sin θ, just equals the largest static friction, μ_s × mg cos θ, so μ_s = sin θ ÷ cos θ = tan θ. The mass cancels, which is why the test works with any size of piece.

On a board 400 mm long, the piece slid when the raised end was at heights of 132, 140, 128, 136 and 134 mm. The middle value, 134 mm, gives sin θ = 134 ÷ 400 = 0.335, so θ = 19.6 degrees and μ_s = tan 19.6° = 0.36. That is lower than the table's 0.5 for wood on wood, because this plywood is sanded smooth. The design will use the measured 0.36.

From a moment to a clamping force

Each cheek grips one face of the tab, so there are two friction faces. The friction acts around the clamping bolt, 20 mm from the pin about which the arm turns, so the largest moment the joint can resist is the friction force on both faces times that distance:

Moment capacity = 2 × μ_s × N × r = 2 × 0.36 × N × 0.020 m

Set that equal to the 0.193 N·m the arm needs and N = 0.193 ÷ (2 × 0.36 × 0.020) = 13.4 N. But the coefficient was measured on one board on one day, the joint will be bumped, and the wood will settle under the washer. Use a factor of safety of 2, as lesson 9 discussed: N = 2 × 13.4 = 26.8 N, about the weight of a 2.7 kg mass, pressing the cheeks onto the tab.

Check that the squeeze will not crush the plywood. An M5 washer has an outer diameter of 10 mm and a hole of 5.3 mm, an area of π ÷ 4 × (10² − 5.3²) = 56.5 mm². The pressure under it is 26.8 N ÷ 56.5 mm² = 0.47 MPa. The Cambridge data book gives wood across the grain a yield strength of 2 to 6 MPa, at least four times that, so the plywood is safe.

The wing nut is two machines at once

Can fingers supply 26.8 N of clamping force? The wing nut combines two simple machines. Its wings are a lever: your fingers push about 12 mm from the bolt's centre. And the thread is a screw, an inclined plane wrapped round a cylinder: each full turn moves the nut along by one pitch, which for an M5 coarse thread is 0.8 mm.

Ignore friction for a moment and use work. In one turn the fingers push through a circle of 2π × 12 mm = 75.4 mm, while the nut advances 0.8 mm. If no energy were lost, force times distance in would equal force times distance out, so the clamping force would be 75.4 ÷ 0.8 = 94 times the finger force. That ratio, distance moved by the effort divided by distance moved by the load, is the ideal mechanical advantage, and the article on the screw as a simple machine writes it as 2πr ÷ l, where l is the distance the screw advances per turn.

Real screws lose much of the effort to friction in the threads and under the nut. The same article notes that even well-lubricated jack screws manage only 15 to 20 percent efficiency. Assume a dry wing nut does worse, 10 percent. Then the actual mechanical advantage is 94 × 0.10 = 9.4, and the finger force needed is 26.8 N ÷ 9.4 = 2.8 N, the weight of about 290 g: an easy pinch. So the joint slipped not because fingers are too weak, but because snug was well short of 26.8 N. The fix is to tighten firmly, and to check after a week that the plywood has not settled and loosened the nut.

The inefficiency has an upside. A screw is self-locking, unable to be turned backwards by the load on it, if and only if its efficiency is below 50 percent. The friction that wastes 90 percent of your effort is also what stops the clamping force from simply unwinding the nut. The friction that takes energy is the friction that holds the arm up.

The point: work out the force a friction joint needs from the moment it must resist, a measured μ and a factor of safety; then check that the machine doing the squeezing can supply it, allowing for the energy its own friction takes.

Levers: trading force for distance

A lever is lesson 10's moments put to work. A force on one side of a pivot balances a force on the other when force times lever arm is equal on both sides. OpenStax defines the mechanical advantage (MA) of any simple machine as output force divided by input force, and shows that for a lever it equals the input lever arm divided by the output lever arm. Its wheelbarrow example: a load of 45.0 kg sits 7.50 cm from the wheel's axle, and the hands lift 1.02 m from it, so MA = 1.02 ÷ 0.0750 = 13.6, and the hands lift with 45.0 kg × 9.80 N/kg ÷ 13.6 = 32.4 N instead of 441 N. The price: the hands rise 13.6 times as far as the load.

Levers are sorted by where the pivot, or fulcrum, sits. In a first-class lever, such as a seesaw or a crowbar, the fulcrum is between effort and load. In a second-class lever, such as the wheelbarrow, the load is between the fulcrum and the effort, so MA is always above 1. In a third-class lever, such as tweezers or your forearm lifting a cup, the effort is between the fulcrum and the load, so MA is below 1: you push harder than the load, and in exchange the load moves further and faster than your effort.

Pulleys: counting the ropes

A single fixed pulley only changes the direction of a pull: MA = 1. Hang the load from a movable pulley, so that two lengths of rope hold it up, and each length carries half the load: MA = 2, but you must pull 2 m of rope to raise the load 1 m. OpenStax gives the rule for any arrangement, including a block and tackle: the MA is approximately the number of rope lengths pulling directly up on the load.

Measure one and the word approximately earns its place. A 2.0 kg bag of rice, weighing 2.0 × 9.81 = 19.6 N, hangs from a single movable pulley. Ideally it needs 19.6 ÷ 2 = 9.8 N of pull. A luggage scale on the rope reads 11.5 N while the bag rises steadily. The actual mechanical advantage is 19.6 ÷ 11.5 = 1.70, not 2.

Where the missing energy goes

The pulley's shortfall is easiest to see with work, force times the distance moved in the direction of the force, measured in joules. Raise the bag 0.30 m: the work done on it is 19.6 N × 0.30 m = 5.9 J. To raise it that far you pull 0.60 m of rope at 11.5 N: the work you do is 11.5 N × 0.60 m = 6.9 J. OpenStax defines efficiency as useful work out divided by energy in: 5.9 ÷ 6.9 = 0.85, or 85 percent. The same number comes from the force ratios: actual MA divided by ideal MA, 1.70 ÷ 2 = 0.85.

The missing 6.9 − 5.9 = 1.0 J has not vanished. Friction in the pulley's axle and in the rope bending round the wheel turned it into thermal energy; OpenStax describes friction as a nonconservative force that creates thermal energy which dissipates. No machine can do more work than the energy put into it. A machine multiplies force or distance, never energy, and every real machine returns a little less than it is given. For comparison, OpenStax's table of efficiencies lists an electric motor at 98 percent and a gasoline engine at 30 percent; lesson 17 returns to efficiency when the energy comes from a battery.

Common misconceptions

• "A machine with a big mechanical advantage gives you extra energy." It multiplies force and divides distance. The wing nut's 94 times the force comes with 1 ÷ 94 of the distance, and friction then takes most of the energy anyway.
• "More contact area means more friction." Friction is proportional to the normal force, not the area. Widening the cheeks would not help the arm joint; squeezing harder does.
• "Friction is always a loss to be eliminated." The arm joint, a screw that stays tight, shoes and brakes all depend on it. Friction is a loss in the pulley and a requirement in the clamp.
• "The table value of μ is the value for my materials." The table gives wood on wood as 0.5; the smooth plywood measured 0.36. Designing with 0.5 would have left the clamp a third too weak.

Recap

The reading light's arm sank about 2 cm a night because its friction joint, two plywood cheeks clamped onto a tab by a wing nut 20 mm from the pivot pin, was only snug. Friction is at most μ_sN, proportional to the squeeze and not the area. A tilt test gave the plywood μ_s = tan 19.6° = 0.36, below the table's 0.5. Two friction faces 20 mm from the pin must resist 0.193 N·m, so N = 0.193 ÷ (2 × 0.36 × 0.020) = 13.4 N, doubled to 26.8 N for safety, a pressure of 0.47 MPa under the washer, far below the plywood's crushing strength. The wing nut is a lever and a screw with an ideal mechanical advantage of 2π × 12 ÷ 0.8 = 94; at an assumed 10 percent efficiency, a finger force of 2.8 N supplies the clamp, and an efficiency below 50 percent keeps the screw from unwinding. A lever's MA is input lever arm over output lever arm, 13.6 for OpenStax's wheelbarrow; a pulley system's is roughly the number of supporting ropes. A movable pulley lifting 19.6 N needed 11.5 N, an actual MA of 1.70 and an efficiency of 5.9 J ÷ 6.9 J = 85 percent, the missing 1.0 J turned into heat by friction.

The point: machines trade force for distance and never create energy; friction takes a share of every exchange, and the same friction, measured and sized, is what holds a joint still.

Sources

1. Urone, P. P., & Hinrichs, R. (2022). 5.1 Friction. In College physics 2e. OpenStax. OpenStax
2. Urone, P. P., & Hinrichs, R. (2022). 9.5 Simple machines. In College physics 2e. OpenStax. OpenStax
3. Urone, P. P., & Hinrichs, R. (2022). 7.5 Nonconservative forces, and 7.6 Conservation of energy. In College physics 2e. OpenStax. OpenStax
4. Wikipedia contributors. (2026). Screw (simple machine). In Wikipedia, The Free Encyclopedia. Wikipedia
5. Wikipedia contributors. (2026). ISO metric screw thread. In Wikipedia, The Free Encyclopedia. Wikipedia

A wheel of 223 teeth, under the sea for 2,000 years

In 1901, probably in July, sponge divers working a Roman-era shipwreck off the Greek island of Antikythera brought up lumps of corroded bronze. The Antikythera mechanism, made in the second century BC, turned out to be a geared calculator of the sky. In 2005 a team from Cardiff University imaged the fragments with X-ray tomography, and the scans suggest the mechanism had 35 meshing bronze gears, 30 of which survive. The largest, about 13 cm across, originally had 223 teeth.

The researchers counted teeth because in a gear train the tooth counts are the calculation. One of the cycles the mechanism displayed is the Metonic cycle: 235 months of the Moon come very close to 19 years of the Sun, and the mechanism's Metonic dial covers 235 months in five turns. A train of gears whose teeth are in the right whole-number ratios turns one pointer at exactly the right speed relative to another, for ever. The bicycle in your shed does the same arithmetic with a chain. This lesson works out the arithmetic, step by step, for a bicycle on the flat, then changes one thing, a hill, and finds that the best answer turns upside down.

The one rule: teeth pass teeth

Where two gears mesh, or where a chain runs round two sprockets, the same number of teeth must pass the meeting point on each side, or the teeth would collide or slip. So the number of turns times the number of teeth is the same for both:

turns of the driver × teeth on the driver = turns of the driven × teeth on the driven

A 42-tooth gear driving a 21-tooth gear: one turn of the 42 pushes 42 teeth past, which is two full turns of the 21. The speed ratio, output turns per input turn, is driver teeth ÷ driven teeth = 42 ÷ 21 = 2. The small gear turns twice as fast.

Torque goes the other way. OpenStax gives the power of a turning shaft as torque times angular speed, P = τω. If no energy is lost, the power out equals the power in, so when the speed doubles the torque must halve. The torque ratio is the inverse of the speed ratio: driven teeth ÷ driver teeth. Gears trade speed for torque exactly as levers trade distance for force in lesson 13.

One warning before any numbers. Engineers usually quote a gear ratio as driven teeth over driver teeth, so a "3:1 reduction" means the output turns once for every three input turns. Cyclists quote theirs the other way up, chainring teeth over sprocket teeth, as wheel turns per pedal turn. Both are correct; mixing them is how a design ends up three times too fast or three times too slow. Always say which way round you mean.

Worked example 1: a bicycle on the flat, step by step

Step 1: count. The chain runs on a 42-tooth chainring at the pedals and a 21-tooth sprocket on the rear wheel.

Step 2: speed ratio. The chainring drives, so the wheel turns 42 ÷ 21 = 2.0 times per pedal turn. Wikipedia's article on bicycle gearing defines the bicycle's gear ratio this way, chainring teeth over sprocket teeth.

Step 3: wheel size. A 700C wheel has a rim 622 mm across, and the article gives the outside diameter as roughly the rim plus twice the tyre width: with 25 mm tyres, 622 + 2 × 25 = 672 mm. Its circumference is π × 0.672 m = 2.11 m, and its radius 0.336 m.

Step 4: development. The distance the bicycle travels for one turn of the pedals is called its development: speed ratio × circumference = 2.0 × 2.11 m = 4.22 m.

Step 5: road speed. At a cadence of 80 pedal turns a minute, the bicycle covers 80 × 4.22 m = 338 m each minute, which is 338 ÷ 60 = 5.63 m/s, or 5.63 × 3.6 = 20.3 km/h.

Step 6: torque and power at the pedals. Suppose the rider's push, averaged round the circle, is 70 N on cranks 170 mm long. The crank torque is 70 N × 0.170 m = 11.9 N·m. The pedals turn at 80 × 2π ÷ 60 = 8.38 rad/s, so the power is P = τω = 11.9 × 8.38 = 100 W.

Step 7: chain tension. A bicycle chain has a pitch of 12.7 mm, half an inch, from one pin to the next. A 42-tooth chainring therefore has a circumference along its pitch line of about 42 × 12.7 = 533 mm and a radius of about 533 ÷ 2π = 84.9 mm. The chain pulls with 11.9 N·m ÷ 0.0849 m = 140 N, about the weight of a 14 kg mass, through links a few millimetres across.

Step 8: torque and force at the wheel. Torque scales by driven teeth over driver teeth: 11.9 × 21 ÷ 42 = 5.95 N·m at the rear wheel. At the tyre, 0.336 m from the axle, that pushes the road with 5.95 ÷ 0.336 = 17.7 N.

Step 9: check the power. At the road: force × speed = 17.7 N × 5.63 m/s = 99.7 W, the same 100 W the rider put in, apart from rounding. The gear doubled the speed and halved the torque; it did not create or destroy power. A real drivetrain loses a little to friction: the article gives a well-maintained bicycle transmission an efficiency of 86 to 99 percent.

Worked example 2: the same rider, the same 100 W, up a 10 percent hill

Change one thing: the road now rises 10 m for every 100 m, a slope of 5.71 degrees. Rider and bicycle together have a mass of 80 kg. Ignoring air and rolling resistance, which are small at walking pace, the force needed to climb is the component of weight along the slope: 80 kg × 9.81 N/kg × sin 5.71° = 78.1 N. At 100 W the speed can be only 100 W ÷ 78.1 N = 1.28 m/s, 4.6 km/h. The wheel turns 1.28 ÷ 2.11 × 60 = 36.4 times a minute, and the torque it needs is 78.1 N × 0.336 m = 26.2 N·m. Now compare two gears.

Both gears deliver the same 100 W at the same 4.6 km/h. In the 2.0 gear the rider must grind the pedals round 18 times a minute, pushing with an average of 309 N, the weight of about 31 kg. In the 0.85 gear they spin 43 times a minute, pushing with 131 N. The result has flipped. On the flat, the high gear turned a comfortable cadence into speed. On the hill, the gear below 1 makes the wheel turn slower than the pedals and multiplies the torque instead. People pedal comfortably only within a fairly narrow band of cadence and force, so a bicycle carries many gears to keep the rider in that band whatever the road does.

What matters here: the right gear depends on the load. A gear ratio is a trade between speed and torque, and the best trade on the flat is the wrong one on a hill.

Gear trains: ratios multiply

Real machines join several pairs of gears in a gear train, and a few rules cover them.

• Ratios multiply. In a compound train, two gears of different sizes are fixed on one shaft, so the output of one pair is the input of the next. Build one from toy construction gears: an 8-tooth gear driving a 40-tooth gear is a 5:1 reduction; on the 40's shaft, another 8-tooth gear drives a 24-tooth gear, a 3:1 reduction. The output turns once for every 5 × 3 = 15 turns of the input, with 15 times the torque before friction. Ratios add only in a wrong calculation.
• Meshing gears turn opposite ways. Two gears in mesh rotate in opposite directions; two sprockets joined by a chain rotate the same way.
• An idler changes direction, not ratio. A gear placed between two others, touching both, reverses the output's direction, but its own tooth count cancels out of the calculation: whatever it gains from the driver it gives to the driven.
• Teeth must match. Gears mesh only if their teeth are the same size and shape, which is why gear makers specify tooth size, not just tooth count.
• A worm drive gives a big ratio in one step. A worm is a screw meshing with a gear wheel. With a single-start worm, each full turn advances the wheel by just one tooth, so a 20-tooth wheel gives 20:1. Some worm drives are self-locking, meaning the wheel cannot drive the worm backwards, the same friction effect that kept lesson 13's wing nut tight; the price is that worm drives are less efficient than ordinary gears.

A hand drill of the egg-beater kind is the simplest train to inspect: a big crank wheel, say 48 teeth, drives a small pinion of 12 teeth on the chuck, a speed increase of 48 ÷ 12 = 4. Every turn of the handle turns the drill bit four times, with a quarter of the handle's torque at the bit.

Common misconceptions

• "The bigger gear turns faster." In mesh, the gear with more teeth turns fewer times. The 42-tooth chainring turns once while the 21-tooth sprocket turns twice.
• "A high gear gives you more power." Gears trade speed for torque at the same power, apart from friction. On the hill the rider made 100 W in both gears; the low gear only changed how that 100 W was delivered.
• "An idler gear changes the ratio." It reverses the direction of rotation, and its own tooth count cancels out.
• "The ratios of a gear train add up." They multiply: 5:1 followed by 3:1 is 15:1, not 8:1.

The short version

The Antikythera mechanism, raised from a wreck in 1901, computed astronomical cycles with tooth counts, its largest gear having 223 teeth, because meshing gears pass equal numbers of teeth: turns times teeth is the same on both sides. The speed ratio is driver teeth over driven teeth and the torque ratio its inverse, so power, P = τω, is conserved apart from friction. A bicycle in 42 × 21 turns its 2.11 m wheel twice per pedal turn, a development of 4.22 m and 20.3 km/h at 80 turns a minute; a 70 N average push on 170 mm cranks makes 11.9 N·m and 100 W, tensions the chain to 140 N, and gives 5.95 N·m and 17.7 N at the tyre, 99.7 W, with real drivetrains 86 to 99 percent efficient. On a 10 percent hill at the same 100 W, 42 × 21 needs an average pedal push of 309 N at 18 turns a minute, while 34 × 40 needs 131 N at 43. In a gear train ratios multiply, meshing gears reverse direction, idlers change direction but not ratio, and a single-start worm gives a ratio equal to its wheel's tooth count.

What matters here: count the teeth, say which way round the ratio runs, and remember that every gain in speed is paid for in torque, and every gain in torque in speed.

Sources

1. Moebs, W., Ling, S. J., & Sanny, J. (2016). 10.8 Work and power for rotational motion. In University physics volume 1. OpenStax. OpenStax
2. Wikipedia contributors. (2026). Bicycle gearing. In Wikipedia, The Free Encyclopedia. Wikipedia
3. Wikipedia contributors. (2026). Bicycle chain. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Worm drive. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Wikipedia contributors. (2026). Antikythera mechanism. In Wikipedia, The Free Encyclopedia. Wikipedia

Berlin, 1827: a pamphlet nobody wanted

In 1827 Georg Ohm, a schoolteacher, published a pamphlet in Berlin called Die galvanische Kette, mathematisch bearbeitet, The Galvanic Circuit Investigated Mathematically. Working with equipment he made himself, he had found that the current through a conductor is directly proportional to the voltage across it. The work was coldly received; his college did not appreciate it, and Ohm resigned. Fourteen years later, in 1841, the Royal Society in London awarded him its Copley Medal, and today the unit of resistance, the ohm, written Ω, carries his name.

Ohm's law is now so familiar that it is easy to recite and hard to use. It connects three quantities that sound alike and behave very differently, and each is measured with the same meter connected in a different way. Connect the meter the wrong way for the quantity and you get a meaningless number, a blown fuse or a hot wire. This lesson sets the three side by side in one table, reads the table row by row, and then checks Ohm's law against real readings from a battery and a resistor, the first step toward the reading light's circuit.

Three quantities, side by side

Reading the voltage column

Voltage is a difference. The black probe goes in the meter's COM socket, the red in the socket marked V, and the two probes touch the two points you are comparing; SparkFun's multimeter guide describes it as poking at the two points in parallel. OpenStax explains why that works without disturbing anything: a voltmeter connected in parallel has a resistance a few orders of magnitude greater than the device, so very little current flows through it and the circuit is not appreciably affected. Set the dial to DC volts, the V with a straight line. The V with a wavy line is for alternating voltage, like that from a wall socket, and nothing in this course is ever measured at a wall socket.

Reading the current column

Current is a flow, so to count it the meter has to be in the flow. You break the circuit at one point and connect the meter into the gap, in series, so that every bit of charge passes through it. OpenStax: ammeters are connected in series and have a resistance very small compared with the devices in the circuit. That low resistance is exactly what makes the commonest mistake so destructive. Touch an ammeter's probes across a battery and you have connected a near-zero resistance straight across the supply. SparkFun's guide marks this as bad: the meter's 200 mA fuse blows. Two habits prevent it: move the red lead to the current socket only when you have opened the circuit to put the meter in series, and put the lead back in the voltage socket as soon as you finish, as the guide advises, so the next voltage reading is not a short circuit.

Reading the resistance column

An ohmmeter works by sending its own small current through the component and measuring the voltage that results, so the circuit must be off. In a working circuit the circuit's own voltage corrupts the reading; with other components connected in parallel, the meter measures them too. Take the resistor out, or lift one leg, before measuring it. A resistor's value is also printed on it in coloured bands: brown, black, brown is 1, 0 and one zero, 100 Ω; red, red, brown is 220 Ω; brown, black, red is 1,000 Ω, or 1 kΩ. A fourth band gives the tolerance, often gold for plus or minus 5 percent, which lesson 6 would recognise.

Remember: volts across, amps through, ohms with the power off. Most multimeter accidents are one of those three rules broken.

Ohm's law, checked against a real circuit

OpenStax states Ohm's law as I = V ÷ R, with the ohm defined as one volt per ampere. Its example: a car headlight that draws 2.50 A from 12.0 V has a resistance of R = V ÷ I = 12.0 ÷ 2.50 = 4.80 Ω. Now try it on the desk. The readings below are illustrative of what a cheap meter gives.

1. Measure the source. Three fresh AA cells in a holder read 4.71 V, about 1.57 V each.
2. Measure the resistor, unpowered. A resistor banded red, red, brown, marked 220 Ω, reads 218 Ω, within its 5 percent tolerance.
3. Predict. I = V ÷ R = 4.71 V ÷ 218 Ω = 0.0216 A, or 21.6 mA.
4. Measure. Break the circuit, put the meter in series on its 200 mA range, and close the circuit: 21.4 mA.
5. Compare. The reading is 0.9 percent below the prediction. Two effects explain it. The meter adds a little resistance of its own in series. And a battery has internal resistance, so its voltage drops slightly when current flows; the 4.71 V was read with almost no current flowing. Measure the voltage again with the resistor connected and it reads a few millivolts lower.

Swap in a 1 kΩ resistor that measures 992 Ω, and the prediction is 4.71 ÷ 992 = 4.75 mA; the meter reads 4.73 mA. Ohm's law holds for both, as it does for any ohmic component, one whose resistance does not depend on the voltage across it. OpenStax adds that Ohm's law, like Hooke's law, is not universally valid. Lesson 16 meets a component it fails for completely: the LED.

Power: what the resistor turns into heat

Electrical power, the rate at which energy is delivered, is P = IV. For a resistor, substituting Ohm's law gives two more forms, P = I²R and P = V² ÷ R; OpenStax notes that P = IV is always valid, and the other two only for resistors. For the 220 Ω circuit: P = IV = 0.0214 A × 4.71 V = 0.101 W, and P = I²R = 0.0214² × 218 = 0.100 W, the same within rounding. The headlight: 2.50 A × 12.0 V = 30 W.

That power becomes heat in the resistor, and every resistor has a power rating. The common small through-hole resistor is rated at 0.25 W, so the 220 Ω resistor, at 0.10 W, runs at 40 percent of its rating and barely warms. Now suppose someone grabs a 47 Ω resistor by mistake: I = 4.71 ÷ 47 = 0.100 A, and P = V² ÷ R = 4.71² ÷ 47 = 0.47 W, nearly twice the rating. It will get hot enough to burn a finger and to scorch its own coating. A component that is too hot to touch has been asked for more power than it can shed.

Batteries are safe low voltage, not safe energy

Everything in this course runs at 5 V or less, which cannot push a dangerous current through your skin. But a battery stores energy, and a short circuit releases it fast. A bare wire across a battery's terminals has almost no resistance, so the current is limited only by the battery itself, and the wire and battery heat quickly. Never let a wire, a key or a coin bridge a battery's terminals, and never put loose batteries in a pocket with metal objects. Lithium-ion power banks store far more energy than AA cells. The U.S. Environmental Protection Agency warns that a damaged or crushed lithium-ion battery is a fire hazard, that even used batteries can hold enough energy to injure or start fires, and that they must not go in household trash or recycling bins: tape their terminals and take them to a battery collection point. If a power bank swells, gets hot or smells odd, stop using it.

Common misconceptions

• "Current gets used up as it goes round the circuit." In a single loop the current is the same at every point: 21.4 mA leaves the battery and 21.4 mA returns. What the resistor uses up is energy, the volts each unit of charge loses crossing it.
• "A voltmeter reads the voltage at a point." It reads the difference between the two points its probes touch. "The voltage here" only means something once you say "compared with where".
• "A meter on its current setting is just another way of looking at a circuit." On current, the meter is nearly a plain wire. Across a supply it is a short circuit; in series it is harmless.
• "Low voltage means no danger." Five volts cannot shock you, but a shorted battery or a damaged lithium-ion cell can heat, burn and start a fire, because the danger is the stored energy released quickly.

Pulling it together

Georg Ohm's 1827 pamphlet, coldly received, gave the law that the current through a conductor is proportional to the voltage across it, and the ohm, one volt per ampere, now bears his name. Voltage is the push between two points and is measured across a component with the circuit on; current is the flow and is measured in series by breaking the circuit; resistance is measured with the power off and the component isolated. A voltmeter has a very high resistance and an ammeter a very low one, which is why an ammeter across a battery blows its fuse. With three AA cells at 4.71 V, Ohm's law predicted 21.6 mA through a 218 Ω resistor and the meter read 21.4 mA, the small gap due to the meter and the battery's internal resistance; a 992 Ω resistor gave 4.75 mA predicted and 4.73 mA measured. Power is P = IV, or I²R and V² ÷ R for resistors: 0.10 W in the 220 Ω resistor, 40 percent of a 0.25 W rating, but 0.47 W in a 47 Ω one, nearly twice the rating. Batteries are low voltage but stored energy, so short circuits and damaged lithium-ion cells are fire hazards.

Remember: volts across, amps through in series, ohms with the power off, and check every resistor's power against its rating before you switch on.

Sources

1. Urone, P. P., & Hinrichs, R. (2022). 20.2 Ohm's law: Resistance and simple circuits. In College physics 2e. OpenStax. OpenStax
2. Urone, P. P., & Hinrichs, R. (2022). 20.4 Electric power and energy. In College physics 2e. OpenStax. OpenStax
3. Urone, P. P., & Hinrichs, R. (2022). 21.4 DC voltmeters and ammeters. In College physics 2e. OpenStax. OpenStax
4. SparkFun Electronics. (n.d.). How to use a multimeter. SparkFun Learn. SparkFun
5. U.S. Environmental Protection Agency. (n.d.). Used lithium-ion batteries. EPA. EPA

Plug in, and nothing

The first version of the reading light's circuit took five minutes to build. Two white LEDs and one 100 Ω resistor were pushed into a breadboard in a neat row, positive lead of the USB power bank to the resistor, resistor to the first LED, first LED to the second, second LED back to the negative lead. The power bank was plugged in. Nothing lit. Not a glow, not a flicker. (The circuits and readings in this lesson are illustrative.)

This lesson debugs that circuit and three more that fail, each in a different way, and traces every failure to the exact assumption behind it. By the end you will have the circuit the reading light actually uses, with every resistor calculated and checked, and you will know why each of the wrong versions failed.

What an LED needs

A light-emitting diode, an LED, lets current through in one direction only. SparkFun's guide puts it plainly: the positive side, the anode, is marked by the longer lead, and current flows from the anode to the cathode and never the opposite way. Two numbers on its data sheet matter most. Adafruit's super-bright 5 mm white LED lists a typical forward voltage of 3.0 V at 20 mA, and a maximum continuous current of 20 mA. SparkFun's similar white LED gives the forward voltage as anything from 3.0 to 3.4 V, which matters later.

An LED is not ohmic. OpenStax's chapter on semiconductor devices describes a diode in forward bias: small changes in the voltage lead to large changes in the current. Below about its forward voltage, a white LED passes almost nothing and stays dark. A few tenths of a volt above it, the current is already far past anything the LED can survive. There is no fixed resistance to plug into Ohm's law; there is a knee, and the current goes almost straight up beyond it.

Failure 1: two LEDs in series on 5 V

Here is the calculation that should have been done before building. In a series circuit, OpenStax's rules say the same current flows through each part and the voltages across the parts add up to the supply. So the resistor must take whatever the LEDs leave over, and for 20 mA:

R = (V_supply − V_f1 − V_f2) ÷ I = (5.0 V − 3.0 V − 3.0 V) ÷ 0.020 A = −50 Ω

A negative resistance does not exist. The arithmetic is announcing that the design is impossible: the two LEDs together need 6.0 V before they conduct, and the supply offers 5.0 V. SparkFun says it in one line: the forward voltages of all the LEDs added together cannot exceed the supply voltage. Each LED got roughly 2.5 V, below its knee, so each passed a current too small to see. That is why there was not even a dim glow: an LED is not a lamp that gets dimmer as the voltage falls, but a valve that shuts.

Failure 2: one LED, the wrong way round

Down to one LED and the 100 Ω resistor: still nothing. The multimeter, set to DC volts and held across the LED, read 5.0 V, the whole supply. With no current flowing, the resistor dropped nothing and the LED had the full supply across it, the wrong way round: its short lead was on the positive side. Reversed, a diode passes almost no current. Turned round, the LED lit at once. A reversed LED on a low supply like this usually comes to no harm, but LED data sheets also give a maximum reverse voltage, and it is low, so it is not something to rely on. Lesson learned: check polarity first, because it is the cheapest fault to find, and a voltmeter across a dark LED that reads the full supply is the sign of an open path, a backwards LED or a broken connection.

Failure 3: no resistor at all

Someone then asked why the resistor was needed, pulled it out and connected the LED straight across the 5 V supply. It lit, very brightly, and the meter, in series on its 10 A range, read 0.31 A. Within seconds the LED dimmed and died. Trace it with the knee. With no resistor, nothing in the circuit holds the current near 20 mA; it rises until the supply's own small internal resistance and the LED's heating stop it, at fifteen times the LED's 20 mA maximum. SparkFun's guide notes that an LED connected directly to a source will try to take as much power as it can draw, and destroy itself. The resistor is not decoration. Because an LED's current shoots up past its knee, something with a real, fixed resistance must set the current, and that is the series resistor's whole job.

The resistor, calculated properly

For one LED in series with one resistor, the resistor takes the voltage the LED does not, and Ohm's law applies to the resistor alone:

R = (V_supply − V_f) ÷ I = (5.0 V − 3.0 V) ÷ 0.020 A = 100 Ω

The resistor's power is the voltage across it times the current: 2.0 V × 0.020 A = 0.040 W, 16 percent of a 0.25 W rating. That is the typical case, and typical is not enough for a design, because both numbers in the bracket vary. USB's supply voltage was originally specified to lie anywhere from 4.75 V to 5.25 V, and SparkFun's white LED gives V_f from 3.0 to 3.4 V. Check the worst cases with 100 Ω:

• Highest current: high supply, low forward voltage: (5.25 − 3.0) ÷ 100 = 0.0225 A = 22.5 mA, over the 20 mA maximum.
• Lowest current: low supply, high forward voltage: (4.75 − 3.4) ÷ 100 = 0.0135 A = 13.5 mA.

To keep the worst case at or below 20 mA, R must be at least (5.25 − 3.0) ÷ 0.020 = 112.5 Ω. Resistors come in standard values, and the next standard value up is 120 Ω. With 120 Ω the current stays between (4.75 − 3.4) ÷ 120 = 11.3 mA and (5.25 − 3.0) ÷ 120 = 18.8 mA, and is typically (5.0 − 3.0) ÷ 120 = 16.7 mA. The LED runs a little below its brightest, and it is never overdriven. The reading light uses 120 Ω.

In short: an LED's current is set by the resistor in series with it, R = (V_supply − V_f) ÷ I, and the resistor must be chosen for the worst combination of supply and LED, not the typical one.

Failure 4: four LEDs sharing one resistor

The reading light needs four LEDs. Since two in series cannot work on 5 V, they must go in parallel, where OpenStax's rules say each branch gets the full supply voltage and the branch currents add. A tempting shortcut is one resistor for all four: 80 mA in total, so R = (5.0 − 3.0) ÷ 0.080 = 25 Ω. Built that way, the four LEDs glowed unevenly, and the meter, moved into each branch in turn, read 34, 22, 15 and 9 mA. The total is 80 mA, as designed, but one LED is carrying 34 mA, far over its maximum.

Trace it. In parallel, all four LEDs share one voltage, the voltage left after the shared resistor. But no two LEDs have quite the same forward voltage; SparkFun's range runs over 0.4 V, and on the steep part of the curve a hundredth of a volt changes the current a lot. The LED with the lowest forward voltage takes the most current, which warms it, which lowers its forward voltage further, so it takes more still. Wikipedia's article on LED circuits warns that parallel LEDs must have closely matched forward voltages to have similar branch currents. The cure is a resistor in every branch. Then each branch's current is set by its own resistor, and a difference in forward voltage matters far less: with 120 Ω, an LED at 2.9 V takes (5.0 − 2.9) ÷ 120 = 17.5 mA and one at 3.1 V takes (5.0 − 3.1) ÷ 120 = 15.8 mA, a spread of about 10 percent instead of nearly four to one.

Series against parallel, for this job

For the reading light, parallel branches win: 5 V cannot drive two white LEDs in series, and a branch that fails leaves three LEDs working. The finished circuit has four branches, each a 120 Ω resistor and an LED. Typically each branch takes 16.7 mA, so the supply gives 4 × 16.7 = 66.7 mA, and the power drawn is 5.0 V × 0.0667 A = 0.33 W. Of that, the LEDs take 4 × 3.0 V × 0.0167 A = 0.20 W and the four resistors the other 0.13 W, as heat. Two-fifths of the energy never becomes light at all, a number lesson 17 will return to. The USB 2.0 standard defines a unit load as 100 mA, so the whole lamp draws less than one unit load.

Common misconceptions

• "A lower voltage makes an LED dimmer." Below its forward voltage an LED barely conducts at all. Two white LEDs in series on 5 V did not glow dimly; they stayed completely dark.
• "The resistor wastes power, so leave it out." Without it the current is limited by almost nothing; the LED drew 0.31 A, fifteen times its maximum, and died in seconds.
• "LEDs in parallel can share one resistor." Small differences in forward voltage make one LED take most of the current. Each branch needs its own resistor.
• "Calculate the resistor with the typical values and you are done." With 100 Ω, a high supply and a low-voltage LED give 22.5 mA, over the maximum. Design for the worst case: 120 Ω keeps every LED at or below 18.8 mA.

Where this leaves us

The two white LEDs in series stayed dark because together they need 6.0 V and USB supplies 5.0 V; the resistor formula gave an impossible −50 Ω. An LED is not ohmic: below its forward voltage it barely conducts, and just above it the current rises steeply, so a backwards LED, a voltmeter reading the full supply across a dark LED, and an LED without a resistor drawing 0.31 A are all the same curve seen from different sides. The series resistor sets the current: R = (V_supply − V_f) ÷ I = (5.0 − 3.0) ÷ 0.020 = 100 Ω typically, but USB's 4.75 to 5.25 V and an LED's 3.0 to 3.4 V make the worst case 22.5 mA, so the reading light uses 120 Ω, giving 11.3 to 18.8 mA and typically 16.7 mA. Four LEDs sharing one resistor split the current 34, 22, 15 and 9 mA; a resistor per branch keeps them within about 10 percent. The finished lamp has four parallel branches drawing about 67 mA, 0.33 W, of which 0.13 W heats the resistors.

In short: check polarity, check that the supply exceeds the forward voltages, give every LED its own resistor, and size that resistor for the worst case.

Sources

1. SparkFun Electronics. (n.d.). Light-emitting diodes (LEDs) [tutorial], and LED: super bright white (COM-00531) [product specifications]. SparkFun. SparkFun tutorial; SparkFun product
2. Adafruit Industries. (n.d.). Super bright white 5mm LED (25 pack), product 754, technical details. Adafruit
3. Urone, P. P., & Hinrichs, R. (2022). 21.1 Resistors in series and parallel. In College physics 2e. OpenStax. OpenStax
4. Ling, S. J., Sanny, J., & Moebs, W. (2016). 9.7 Semiconductor devices. In University physics volume 3. OpenStax. OpenStax
5. Wikipedia contributors. (2026). LED circuit. In Wikipedia, The Free Encyclopedia. Wikipedia

224 kWh, $28, and a small printed assumption

Taped to the lid of an Avanti CF70B0W, a compact chest freezer with 7.0 cubic feet of space, is a yellow tag. In the middle, in large type: Estimated Yearly Energy Cost, $28. Under it: 224 kWh, Estimated Yearly Electricity Use. And at the bottom, in small type: estimated energy cost based on a national average electricity cost of 12.4 cents per kWh. The EnergyGuide label is required by federal law in the United States, and the Federal Trade Commission lists the appliances that carry it: refrigerators and freezers, dishwashers, clothes washers, water heaters, televisions, room and central air conditioners, heat pumps, furnaces, boilers, pool heaters and ceiling fans. The FTC is candid about the dollar figure: it is only an estimate, based on typical use and a national average price for energy, and your cost will depend on how you use the appliance and your local energy price.

So the $28 is a calculation, and the small print gives you its inputs. This lesson redoes it, step by step, for your own electricity price; then changes one input, the price, and watches the better of two freezers change places; then runs the same procedure on a device with no label at all, the reading light, to find its yearly energy and to settle the brief's requirement R3, at least 20 hours per charge.

Step 1: separate power from energy

Power is a rate, how fast energy is used, in watts: one watt is one joule per second. Energy is an amount, power multiplied by time. The bill counts energy in kilowatt-hours, and OpenStax gives the conversion: 1 kWh = 3.6 × 10⁶ J, because 1,000 W running for 3,600 s is 3,600,000 J. Its own example: a 60 W bulb used for 1,000 hours uses 60 W × 1,000 h = 60,000 Wh = 60.0 kWh, which at 12 cents per kWh costs 60.0 × $0.12 = $7.20. Watts tell you how hard something works; kilowatt-hours tell you how much work it did.

Step 2: check the label's arithmetic

Yearly cost = yearly energy × price. On the label: 224 kWh × $0.124 per kWh = $27.78, which rounds to the $28 printed. The label is consistent with its own assumption. The question is whether the assumption is yours.

Step 3: use your own price

The U.S. Energy Information Administration publishes average electricity prices every month. In its table released on 24 September 2026, the average U.S. residential price for July 2026 was 18.31 cents per kWh, a preliminary figure for a single summer month; a year earlier, July 2025, it was 17.45. The same 224 kWh therefore costs:

The freezer is the same in every row. Its real running cost spans a factor of almost three depending on where it is plugged in, and in every row it is more than the label's $28. For your own figure, find the price per kWh on a household electricity bill, which includes the delivery charges, and multiply.

Step 4: turn the yearly energy into an average power

A year has 365 × 24 = 8,760 hours, so the freezer's average power is 224 kWh ÷ 8,760 h = 0.0256 kW, or 25.6 W. Its compressor actually switches on and off, drawing much more while running and nothing while off, but averaged over the year it uses the energy of a 26 W load running all the time, 224 ÷ 365 = 0.61 kWh a day. Average power is the single number that lets you compare anything with anything: the freezer, a phone charger, the reading light.

A second run: a changed price flips the better buy

Now compare two freezers of the same size. Freezer A is the CF70B0W, at 224 kWh a year. Freezer B, an illustrative alternative, uses 190 kWh a year, 15 percent less, but costs $50 more to buy. Assume both last 10 years; that assumption is part of the answer, so write it down. Freezer B saves 224 − 190 = 34 kWh a year, and the payback time is the extra cost divided by the yearly saving.

Same two freezers, same use, a different price, and the better buy changes. At the label's 12.4 cents the efficient freezer never pays back its extra $50 within ten years; at California's price it pays back in under five. That is why the label's small print matters, and why a decision about energy needs the local price as an input, like a weight in lesson 3's matrix. ENERGY STAR adds two facts that change the inputs again: certified freezers must be at least 10 percent more efficient than the federal minimum standard, and manual-defrost freezers, like the CF70B0W, use about half the energy of automatic-defrost models.

The upshot: a label's yearly cost is energy times an assumed price. Redo it with your own price, and when you compare appliances, compare first cost plus running cost over the life you expect.

The reading light: no label, the same procedure

The reading light has no yellow tag, so build the numbers from its parts. Lesson 16 gave its power: four LED branches drawing 66.7 mA in total from 5.0 V, so 5.0 V × 0.0667 A = 0.333 W.

Energy per year. The brief's use is 8:30 to 10:30 p.m., 2 hours a day: 0.333 W × 2 h × 365 = 243 Wh, or 0.243 kWh a year at the lamp.

Running time, R3. The power bank's label says 5,000 mAh. The plausible calculation divides by the lamp's current: 5,000 mAh ÷ 66.7 mA = 75 hours. That is wrong twice. Wikipedia's article on power banks notes that the advertised capacity is usually that of the internal cells, which run at about 3.7 V, not at the 5 V the bank delivers; and that converting up to 5 V loses energy, with the average power bank in 2023 delivering around two-thirds of its battery's energy to the device. So work in energy, not milliamp-hours:

• Energy stored: 5.000 Ah × 3.7 V = 18.5 Wh.
• Energy delivered at 5 V, at about two-thirds: 18.5 Wh × 0.67 = 12.4 Wh.
• Running time: 12.4 Wh ÷ 0.333 W = 37 hours.

Thirty-seven hours passes R3's 20 hours with a margin of 37 ÷ 20 = 1.9, while the naive 75 hours was twice too optimistic. The article adds a warning worth testing: some power banks switch off by themselves when the current drawn stays below a threshold for a while. The calculation says the bank has enough energy; only a timed test can say whether this bank keeps supplying a 67 mA load, which is why the brief listed R3 as analysis, then test.

Energy from the wall, and cost. If the bank delivers two-thirds of what it stores, the cells must take in about 0.243 ÷ 0.67 = 0.36 kWh a year, plus whatever the charger loses; call it 0.4 kWh. At 18.31 cents per kWh, that is 0.4 × $0.1831 = $0.07: about seven cents a year. Recharging takes about 243 Wh ÷ 12.4 Wh = 20 charges a year, roughly one every two and a half weeks. For comparison, the freezer's 224 kWh is about 560 times as much energy. The reading light's electricity is almost free; what it costs to own lies in the power bank itself, which wears out and must be disposed of properly, a point lesson 22 takes up.

Efficiency along the chain

OpenStax defines efficiency as useful energy out divided by energy in, and a device's overall efficiency is the product of the efficiencies of each stage it passes through. Follow the reading light's energy. Of the energy in the bank's cells, about two-thirds reaches the 5 V output. Of that, lesson 16 found 0.20 W of 0.333 W, 60 percent, goes into the LEDs and 40 percent heats the resistors. So only about 0.67 × 0.60 = 0.40, 40 percent, of the stored energy reaches the LEDs at all, before the LEDs' own conversion of electricity to light and before the hood's losses. Each stage looked reasonable; multiplied together, they lose more than half. That is where an engineer would look first to make the lamp run longer on the same bank: the resistors and the converter, not the LEDs.

Common misconceptions

• "The label's dollar figure is what I will pay." It is the label's energy times a national average price of 12.4 cents per kWh. At July 2026's U.S. average of 18.31 cents, the same freezer costs $41 a year, and $75 in California.
• "Watts and watt-hours are the same thing." Watts are a rate; watt-hours are an amount. The freezer averages 25.6 W and uses 224 kWh a year; the reading light draws 0.333 W and uses 0.243 kWh a year.
• "A 5,000 mAh power bank runs a 67 mA lamp for 75 hours." The milliamp-hours are at the cells' 3.7 V, and about a third of the energy is lost converting to 5 V. Working in watt-hours gives about 37 hours.
• "The more efficient appliance is always the better buy." It depends on the price difference, the energy saved, the electricity price and the life. At 12.4 cents per kWh, the efficient freezer in the example never paid back its extra $50 in ten years.

What you now know

The CF70B0W chest freezer's EnergyGuide label shows 224 kWh a year and $28, the product of the energy and a national average price of 12.4 cents per kWh; the FTC itself says the real cost depends on use and the local price. Energy is power times time, and 1 kWh is 3.6 million joules. At EIA's U.S. average residential price for July 2026, 18.31 cents per kWh, the freezer costs $41.01 a year, and anywhere from $35.57 in Texas to $75.29 in California. Its average power is 224 kWh ÷ 8,760 h = 25.6 W. A freezer using 34 kWh a year less but costing $50 more pays back in 11.9 years at 12.4 cents, 8.0 at 18.31 and 4.4 at 33.61, so the better buy flips with the price. The reading light uses 0.333 W, 0.243 kWh a year at the lamp; its 5,000 mAh bank stores 18.5 Wh at 3.7 V, delivers about 12.4 Wh, and runs it for about 37 hours, passing R3 with a margin of 1.9 where the naive 75 hours was twice too high; its electricity costs about seven cents a year, and only about 40 percent of the stored energy reaches the LEDs.

The upshot: energy is power times time, cost is energy times your price, and every conversion along the way takes its share, so calculate in watt-hours, use local prices, and test what the label or the arithmetic cannot promise.

Sources

1. Avanti Products. (n.d.). EnergyGuide label: Chest freezer, model CF70B0W. Retrieved from AJ Madison. EnergyGuide label (PDF)
2. Federal Trade Commission. (2022, August). How to use the EnergyGuide label to shop for home appliances. FTC Consumer Advice. FTC
3. U.S. Energy Information Administration. (2026, September 24). Table 5.6.A: Average price of electricity to ultimate customers by end-use sector, by state. In Electric power monthly. EIA
4. Urone, P. P., & Hinrichs, R. (2022). 20.4 Electric power and energy. In College physics 2e. OpenStax. OpenStax
5. Wikipedia contributors. (2026). Power bank. In Wikipedia, The Free Encyclopedia. Wikipedia

1788: two iron balls on a spinning shaft

In 1788 James Watt adapted a device to keep his steam engines from running away: a centrifugal governor. Two heavy balls hang from a vertical spindle driven by the engine. As the engine speeds up, the spindle turns faster, the balls swing outward and upward, and a linkage they pull partly closes the throttle valve that lets steam into the cylinders, so the engine slows. If it slows too much, the balls drop, the valve opens and it speeds up again. Nobody stands at the valve. The machine measures what it is doing and corrects itself.

The governor was not perfect. It held a speed range rather than one exact speed, and at higher engine speeds the simple Watt-type governor tended to over-compensate, oscillating between opening and closing the valve. In 1868 the physicist James Clerk Maxwell analysed why in a paper titled "On Governors", now widely considered a classic of feedback control theory. A loop that checks itself can still make mistakes, and the mistakes have a pattern.

The reading light needs a small loop of its own. Lesson 1 promised a lamp that switches on at dusk, and the brief needs one that does not drain its battery if the reader falls asleep. This lesson builds the ideas, comparing three ways of controlling the lamp side by side; lesson 19 writes the program.

Sensor, controller, actuator

Every control system has three jobs. A sensor measures something and turns it into a signal. A controller compares that signal with what is wanted, the setpoint, and decides what to do. An actuator does it, turning a signal back into action in the world. In Watt's governor the spinning balls are the sensor, the linkage is the controller and the valve is the actuator. In the reading light:

• Sensor: a photoresistor, also called a light-dependent resistor or LDR, whose resistance falls as light on it increases: as high as several megaohms in the dark, as low as a few hundred ohms in bright light. And a push button, a sensor for the reader's wishes.
• Controller: a small Arduino microcontroller board, powered from the same 5 V USB supply, running a program.
• Actuator: a transistor that switches the four LED branches from lesson 16 on and off.

If the controller acts without measuring the result, the system is open loop: a toaster with a timer does not look at the bread. If it measures and corrects, it is closed loop, or a feedback system: a thermostat looks at the temperature it is trying to control.

From a resistance to a number the controller can use

A microcontroller reads voltages, not resistances, so the photoresistor goes in a voltage divider: the LDR from the 5 V supply to an input pin, and a fixed 10 kΩ resistor from that pin to ground. The same current flows through both, so, as with resistors in series in lesson 16, the supply's voltage divides between them in proportion to their resistances. The voltage at the pin is 5 V × 10 kΩ ÷ (10 kΩ + R_LDR). More light means a smaller R_LDR and a higher voltage.

Arduino's reference describes what the input pin does with it. An Arduino Uno contains a 10-bit analog-to-digital converter that maps input voltages from 0 to the operating voltage, 5 V, onto whole numbers from 0 to 1023, a step of about 4.9 mV. So the reading is 1023 × 10 ÷ (10 + R_LDR), with R_LDR in kilohms, and the supply voltage cancels out: if the battery sags, divider and converter sag together. With illustrative resistances:

Keep the last row in mind. It is the most important number in the table.

Three ways to switch the lamp on at dusk

Reading the table: why a timer is not enough

The open-loop timer switches on at 8:30 p.m. whether or not anyone has turned the ceiling light off. It cannot tell, because it never looks. It is not useless: a toaster works because bread varies little from slice to slice. Open loop is fine when nothing important changes between runs. A bedroom's evening changes every day, which is the case for closing the loop.

Reading the table: why on-off control needs two thresholds

On-off control, also called bang-bang control, is a feedback controller that switches abruptly between two states; most household thermostats work this way, running the heater or not according to the measured temperature. The naive version for the lamp uses one threshold: on below 300, off above 300. At dusk the reading drifts slowly past 300 and wobbles as clouds pass or a car's headlights sweep the ceiling, so it crosses 300 again and again and the lamp flickers on and off. The article on bang-bang control notes that a narrow hysteresis gap leads to frequent on-off switching, which is often undesirable. The cure is a gap, a deadband: switch on below 300 but off only above 400. Once on, the lamp ignores any wobble smaller than 100 counts. A thermostat does the same with temperature, turning the heat on a little below the setting and off a little above it.

Check the gap against the table. The ceiling light gives 706, well above 400, so the lamp stays off while the room is lit. Dusk gives 292, below 300, so the lamp comes on. The dark room's 54 is far below 300. And a reading that wanders between 290 and 390 changes nothing once the lamp has decided.

Reading the table: proportional control and its two flaws

A proportional controller does more than switch: it sets its output in proportion to the error, the difference between the setpoint and the measurement. For the lamp, the sensor would watch the page and the controller would raise the LEDs' brightness, by pulse-width modulation, in proportion to how far the page falls short of 300 lux. Cruise control in a car works on the same principle; when the car meets a hill and slows, the controller adds engine power. The article on PID controllers states the two flaws. Because a non-zero error is needed to drive it, a proportional controller generally settles with a steady-state error: the page ends up a little below 300 lux, not at it. And if the proportional gain is too high, the system can become unstable, which is what Watt's governor did at high speed. Engineers add integral and derivative terms, the I and D of a PID controller, to deal with both. For a reading light, on-off control with a deadband is enough.

So what?: the right controller is the simplest one that copes with the disturbances you actually have. A timer copes with none, a deadband copes with wobble, and proportional control copes with steady changes at the price of tuning.

The trap: a sensor that can see its own actuator

Now the last row of the reading table. Suppose the photoresistor sits on top of the hood, where the lamp's own light spills onto it. At dusk the reading falls below 300 and the lamp comes on. Its light hits the sensor, the reading jumps to 947, above 400, and the controller switches the lamp off. The room is dark again, the reading falls below 300, and the lamp comes back on. The lamp blinks for as long as it is plugged in. The deadband cannot help, because the lamp moves the reading far more than 100 counts.

The controller is working perfectly. The loop is closed around the wrong thing: it is controlling the light the sensor sees, which includes the lamp itself, instead of the light in the room. The fix is physical. Put the sensor on the back of the base, facing the ceiling and away from the hood, where the hood's visor keeps the lamp's light off it, and test it in the dark with the lamp on: the reading must stay below 300. The photoresistor also has a delay worth knowing: its resistance falls within about 10 ms when light arrives, but can take as long as a second to rise again in the dark.

Common misconceptions

• "Feedback always makes a system steadier." Too much gain made Watt's governor over-compensate, and a lamp that can see its own light switches itself off and on forever. Feedback steadies a system only when the loop is designed for it.
• "Open-loop control is always worse." When conditions hardly change, as with a toaster and bread, an open-loop timer is simpler, cheaper and good enough.
• "A sensor measures the thing you care about." It measures whatever reaches it. The photoresistor on the hood measured the lamp, not the room.
• "One threshold is enough for on-off control." A single threshold makes the lamp flicker at dusk as the reading wobbles across it. Two thresholds, with a deadband between them, stop the chatter.

Putting it together

Watt's governor of 1788 measured engine speed with spinning balls and closed the throttle when the engine ran fast, a closed loop of sensor, controller and actuator; it held a speed range and could over-compensate, which Maxwell analysed in "On Governors" in 1868. The reading light's sensor is a photoresistor in a voltage divider with a 10 kΩ resistor, read on the Arduino's 0 to 1023 scale as 1023 × 10 ÷ (10 + R_LDR), so the ceiling light reads about 706, dusk 292 and darkness 54. An open-loop timer cannot see whether the room is lit. On-off control with one threshold flickers at dusk, so the lamp switches on below 300 and off only above 400. Proportional control holds a setpoint more closely but leaves a steady error and can oscillate if its gain is too high. And a sensor that sees its own lamp, reading 947, turns the loop into an oscillator, so the sensor goes on the back of the base, facing away.

So what?: a feedback loop is only as good as what its sensor actually sees and the margin built into its decisions, so place the sensor first, give on-off control a deadband, and test the loop in the conditions it will meet.

Sources

1. Arduino. (n.d.). analogRead(). In Arduino language reference. Arduino Docs
2. Wikipedia contributors. (2026). Centrifugal governor. In Wikipedia, The Free Encyclopedia. Wikipedia
3. Wikipedia contributors. (2026). Bang-bang control. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). PID controller. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Maxwell, J. C. (1868). On governors. Proceedings of the Royal Society of London, 16, 270-283.

Ivrea, 2005: a board for people who are not engineers

In 2005, at the Interaction Design Institute Ivrea in northern Italy, a small team of teachers and engineers, Massimo Banzi, David Cuartielles, Tom Igoe, Gianluca Martino and David Mellis, began a project to give design students simple, low-cost tools for creating digital projects. They named it after the bar where some of them used to meet, the Bar di Re Arduino. Arduino became open-source hardware: a small board with a microcontroller on it, pins to connect sensors and actuators, and a free program on a computer for writing code and sending it to the board. A program written for it is called a sketch.

This lesson follows one sketch, the reading light's, from a six-line first attempt to the version that ships, through the three faults that testing found on the way. The code is shown here to be read, line by line; it is not run on this page. To run it you need an Arduino board, the free Arduino software and the circuit described near the end of the lesson. Every function it uses is checked against Arduino's own language reference. (The test results in this lesson are illustrative.)

Version 1: six lines that almost work

void setup() {
  pinMode(9, OUTPUT);
}

void loop() {
  if (analogRead(A0) < 300) {
    digitalWrite(9, HIGH);
  } else {
    digitalWrite(9, LOW);
  }
}

Every Arduino sketch has these two parts. Arduino's reference describes setup() as the function that runs once, when the board powers up or is reset, used to set pin modes and starting values; loop() then runs over and over, as long as the board has power. Here setup() makes pin 9 an output. Then, thousands of times a second, loop() reads the light sensor on analog pin A0, a number from 0 to 1023 as lesson 18 described, and sets pin 9 HIGH, 5 V, switching the LEDs on, if the reading is below 300, or LOW, 0 V, if not.

On the bench, with a hand cupped over the sensor, it worked perfectly. At real dusk it flickered for ten minutes as the reading drifted across 300, exactly the chattering lesson 18 predicted for a single threshold. And when the sensor was taped beside the hood, the lamp blinked on and off without stopping, because the lamp's own light reached the sensor. Two fixes went in: the sensor moved to the back of the base, facing away from the hood, and the program gained a memory and a deadband.

Version 2: a memory, a deadband and a timer that froze everything

To use two thresholds, the program must remember what it last decided, because a reading of 350 means "stay dark" if the room was dark and "stay light" if it was light. That memory is a variable, roomDark, which changes only when the reading falls below 300 or rises above 400.

The brief also needed a limit, so that the lamp would not drain the battery if the reader fell asleep. The first attempt was one line after switching on: delay(7200000), two hours in milliseconds. Testing found the flaw within a minute. Arduino's reference for delay() warns that no other reading of sensors, calculation or pin manipulation can occur during the delay, so in effect it brings most other activity to a halt; its Blink Without Delay example adds that if a button is pressed while the program is paused, the program will miss the press. For two hours the lamp ignored the ceiling light and everything else.

The fix is to note the time when the lamp comes on and check, on every pass through loop(), how long it has been. millis(), the reference says, returns the number of milliseconds since the board began running the current program, as an unsigned long, a type that holds whole numbers from 0 to 4,294,967,295, and the count goes back to zero after about 50 days. The program keeps the switch-on time in a variable and compares millis() minus that time with two hours. Because both are unsigned long, the subtraction still gives the right answer when the count passes its limit and starts again from zero.

Writing two hours as a number held one more trap. The obvious expression, 2 * 60 * 60 * 1000, equals 7,200,000. But Arduino's reference notes that whole numbers typed into a sketch are treated as int by default, and on an Arduino Uno an int holds only −32,768 to 32,767. The multiplication overflows partway through and produces a wrong number. Adding UL to each number, 2UL * 60UL * 60UL * 1000UL, makes every one an unsigned long, the reference's own remedy, and the product fits.

Version 3: a button that counted every press three times

The reader must be able to switch the lamp on and off by hand. The button connects pin 2 to ground, and the pin is set to INPUT_PULLUP. Arduino's page on digital pins explains that this connects a built-in resistor of 20 kΩ or more from the pin to 5 V, so the pin reads HIGH when the switch is open and LOW when it is pressed: the logic is inverted, and no extra resistor is needed.

The first version toggled the lamp whenever the pin read LOW. Pressing once often switched the lamp on, off and on again. Arduino's Debounce example explains why: push buttons often make spurious open and close transitions when pressed, because the metal contacts bounce, and these may be read as several presses in a very short time. Its remedy is to check twice, a short time apart, to make sure the button really is pressed. The shipped version waits 50 ms after seeing LOW, checks again, toggles only if the button is still down, then waits for the release and lets that settle too.

The program that ships

// Reading light controller. Shown for reading; it is not run on this page.
const int SENSOR_PIN = A0;      // light sensor divider
const int BUTTON_PIN = 2;       // push button to ground
const int LAMP_PIN = 9;         // transistor that switches the LEDs

const int DARK_BELOW = 300;     // the room counts as dark below this
const int LIGHT_ABOVE = 400;    // and as light again above this
const unsigned long ON_TIME = 2UL * 60UL * 60UL * 1000UL;  // two hours in ms

bool roomDark = false;          // the sensor's last decision
bool lampOn = false;            // what the lamp should be doing
unsigned long litSince = 0;     // when the lamp last came on

void setup() {
  pinMode(BUTTON_PIN, INPUT_PULLUP);
  pinMode(LAMP_PIN, OUTPUT);
}

void loop() {
  int light = analogRead(SENSOR_PIN);

  if (!roomDark && light < DARK_BELOW) {         // the room has just gone dark
    roomDark = true;
    lampOn = true;
    litSince = millis();
  } else if (roomDark && light > LIGHT_ABOVE) {  // the room is light again
    roomDark = false;
    lampOn = false;
  }

  if (digitalRead(BUTTON_PIN) == LOW) {          // the button may be pressed
    delay(50);                                   // wait out the contact bounce
    if (digitalRead(BUTTON_PIN) == LOW) {        // still pressed: a real press
      lampOn = !lampOn;
      litSince = millis();
      while (digitalRead(BUTTON_PIN) == LOW) {   // wait for the release
      }
      delay(50);                                 // let the release settle
    }
  }

  if (lampOn && millis() - litSince >= ON_TIME) {  // two hours are up
    lampOn = false;
  }

  if (lampOn) {
    digitalWrite(LAMP_PIN, HIGH);
  } else {
    digitalWrite(LAMP_PIN, LOW);
  }
}

Reading it line by line

Worth holding on to: a good program separates deciding from doing. Every rule in the brief, dark means on, light means off, a press toggles, two hours is the limit, lives in its own few lines, and only the last lines touch the hardware.

The circuit it runs

The Arduino board is powered from the lamp's USB power bank. The photoresistor runs from 5 V to pin A0, with a 10 kΩ resistor from A0 to ground. The push button connects pin 2 to ground. Pin 9 does not drive the LEDs directly: Arduino's page on digital pins says an ATmega pin can supply up to 40 mA, enough for an LED but not more, and the four LED branches of lesson 16 together draw about 67 mA. So pin 9 feeds the base of a small NPN transistor through a 1 kΩ resistor, which limits the pin's current to about (5.0 − 0.7) ÷ 1,000 = 4.3 mA, and the transistor switches the LED branches' return to ground. The transistor takes a few tenths of a volt itself, so each LED's current comes out a little below lesson 16's 16.7 mA; the light test in lesson 26 checks that the page still gets enough.

What testing found that the code could not show

Each behaviour the program promises became a test, with ON_TIME set to ten seconds so that the timer could be checked in minutes, not hours, and then set back. Ceiling light on: lamp off. Ceiling light off: lamp on within a second, the photoresistor's slow recovery in the dark. A hand waved over the sensor at dusk: no flicker. One press: one toggle. The timer: off after ten seconds. A press in a lit room: on, then off after ten seconds.

The last test was the one nobody had planned. The power bank went flat after three days, although the lamp had been lit for only six hours. The board itself draws current whenever it is powered: measured with the meter in series, 45 mA, lamp off. At 5 V that is 0.225 W, and 12.4 Wh ÷ (0.225 W × 24 h) = 2.3 days to empty the bank doing nothing. Two changes followed. A slide switch in the base now cuts power to the whole lamp, and the reader turns it to on when they sit down to work, after which the program does the rest. And R3 was recalculated with the board included: 67 mA plus 45 mA is 112 mA, or 5.0 V × 0.112 A = 0.56 W, so the running time is 12.4 Wh ÷ 0.56 W = 22 hours. That still passes R3's 20 hours, but with a margin of only 1.1, not lesson 17's 1.9. Software runs on hardware, and the hardware has an energy bill too.

Common misconceptions

• "delay() is a harmless way to wait." During delay() the program does nothing else. A two-hour delay made the lamp deaf to its button and blind to the room; millis() lets the program keep watching while it waits.
• "A number in the code is just a number." On an Arduino Uno, whole numbers default to int, which stops at 32,767. 2 * 60 * 60 * 1000 overflows; 2UL * 60UL * 60UL * 1000UL does not.
• "A button press is one clean signal." Contacts bounce, and an unprotected program can read one press as several. Check twice, a short time apart.
• "If the program is right, the device is right." The code passed every logical test, and the lamp still emptied its battery in three days because the board draws 45 mA whenever it has power.

Summing up

Arduino began at Ivrea in 2005 as a low-cost tool for people who are not engineers, and a sketch has two parts: setup(), run once, and loop(), run over and over. The reading light's first six-line sketch compared one reading with 300 and flickered at dusk; moving the sensor away from the hood and adding a remembered state with two thresholds, 300 and 400, fixed it. A two-hour delay() froze the program, so the shipped version records litSince with millis(), an unsigned long that wraps after about 50 days, and compares millis() minus litSince with ON_TIME, written 2UL * 60UL * 60UL * 1000UL because an Uno's int stops at 32,767. INPUT_PULLUP makes the button read LOW when pressed, and a 50 ms double check stops contact bounce counting one press as several. Deciding and doing are separated, so only the last lines touch pin 9, which drives the 67 mA of LEDs through a transistor because a pin can supply only up to 40 mA. Testing with a ten-second timer checked every behaviour, and a three-day battery found the board's own 45 mA: a master switch went in, and R3 became 22 hours, a margin of 1.1.

Worth holding on to: read code as a list of promises, test each promise on the real hardware, and remember that the controller's own power is part of the design.

Sources

1. Arduino. (n.d.). millis(). In Arduino language reference. Arduino Docs
2. Arduino. (n.d.). delay(), and Blink without delay. In Arduino language reference and Built-in examples. delay(); Blink Without Delay
3. Arduino. (n.d.). Digital pins. In Arduino documentation. Arduino Docs
4. Arduino. (n.d.). int, and Integer constants. In Arduino language reference. int; Integer constants
5. Arduino. (n.d.). Debounce on a pushbutton. In Built-in examples. Arduino Docs

1863: $10,000 for something that was not ivory

In 1863 Michael Phelan offered a prize of $10,000 for a substitute for the ivory in billiard balls, because ivory was expensive and growing scarce. John Wesley Hyatt, who had begun working as a printer at sixteen, took up the challenge. His eventual result was a commercially workable way to make a solid, stable nitrocellulose plastic, patented in the United States in 1869 as celluloid. A new material needed a new way of shaping it, and in 1872 Hyatt and his brother Isaiah patented one of the first injection moulding machines. It worked like a large hypodermic needle, a plunger pushing plastic through a heated cylinder into a mould.

The way a part is made is not an afterthought to its design. It decides what shapes are possible, how accurate the part can be, how much material is wasted, how long each part takes and, above all, what it costs to make one, a hundred or a million. This lesson puts four processes side by side on one part, the reading light's hood, then reads the table row by row.

The part: one hood, four ways

From lesson 5's drawing, the hood is an open cone 80 mm tall, 40 mm across at the top, where the LED board closes it, and 140 mm across the rim, with a visor 40 mm deep around the half of the rim that faces the bed. In plastic with walls 1.2 mm thick, its surface of about 35,500 mm² (26,700 mm² of cone and 8,800 of visor) gives a volume of 35,500 × 1.2 = 42,600 mm³, or 42.6 cm³. In ABS, which the Cambridge data book gives a density of 1.01 to 1.21 g/cm³, call it 1.1, the hood weighs 42.6 × 1.1 = 47 g; with its LED board and wires it makes the 60 g hood of lesson 10. Here are four ways to make it. The costs and times are illustrative, of the kind a quote might give for this part; real quotes vary widely with supplier, material and quantity.

Reading the first row: add, subtract or form

Every process in the table does one of three things. Additive processes build the part up: in fused filament fabrication, the commonest kind of desktop 3D printing, thermoplastic filament is fed from a spool through a moving, heated extruder head and laid down one horizontal layer at a time; the technique was developed by S. Scott Crump, a co-founder of Stratasys, in 1988. Subtractive processes start with more material than the part and cut away the rest: in computer numerical control (CNC) machining, a computer drives a cutting tool along a programmed path, a technology that grew from the numerically controlled machines of the 1940s and 1950s. Formative processes force material into shape: casting pours a liquid into a mould with a cavity of the desired shape and lets it solidify, and injection moulding forces melted plastic into a cold mould where it solidifies almost as soon as the mould is filled.

The waste row follows directly. A hood is a thin shell, so machining one from a solid block measuring 140 by 140 by 120 mm, 2,352 cm³, keeps 42.6 cm³ and turns the other 98 percent into chips. Additive and formative processes put material only where the part is. For a thin, hollow plastic part, subtractive is the wrong family to start from; for a small, solid, precise metal part, such as the pin of lesson 6, it is often the right one.

Reading the cost rows: tooling against cost per part

The two cost rows pull in opposite directions. The printer needs no tooling at all, so the first hood costs the same $2.50 as the hundredth, but each takes four hours. The injection mould costs about $12,000 before a single hood exists, and then each one takes 40 seconds and costs about 45 cents. Wikipedia's article on the process says it plainly: because moulds have been expensive to manufacture, they were usually used only in mass production, and the process is ideal for producing high volumes of the same object. Which process is cheapest therefore depends on how many you make, and lesson 21 finds exactly where the lines cross.

The core of it: every process trades a one-off cost against a cost per part. Processes with no tooling win for a few parts; processes with expensive tooling win only when the tooling is shared across thousands.

Reading the design rules: each process has its own

A part drawn without a process in mind can be easy to make one way and nearly impossible another.

• 3D printing: overhangs and layers. Each layer must sit on the one below. Printed rim-down, the hood's cone walls lean only 32 degrees from vertical, since their radius grows 50 mm over 80 mm of height, so each layer is mostly supported; a feature that juts out sideways needs support material, removed afterwards. And the part's strength across its layers depends on how well each layer bonds to the one below, which depends on printing temperature and other settings, so a printed part should be oriented so that its main load runs along the layers, not across them, much like wood's grain in lesson 8.
• CNC machining: the tool must reach. Every surface must be reachable by a spinning cutter, and an inside corner can be no sharper than the cutter's own radius.
• Casting: the part must come out. A flexible silicone mould can be peeled off shapes a rigid mould would trap, which is why it suits small batches of complicated parts.
• Injection moulding: draft and undercuts. The two halves of a steel mould pull straight apart. Sides parallel to that direction are angled slightly, called draft, so the part releases, and mould features must not overhang one another in the direction the mould opens unless moving parts such as lifters are added to the mould. The hood's cone already has plenty of draft. But a hole through its side wall, for a clip to the arm, is an undercut: it points sideways, so a sliding core would have to be built into the mould, adding cost. Moving the attachment to the top of the hood, where it can be moulded straight along the opening direction, removes the problem before the mould is made. That is design for manufacture.

Shrinkage matters too. Plastic shrinks as it cools, and the amount depends on the resin but can be fairly predictable, so the mould cavity is made slightly larger than the part. Lesson 6 met the same effect from the other side, when holes printed 0.30 mm undersize.

Safety, process by process

• 3D printers have a nozzle and often a bed hot enough to burn: do not touch them while printing or just after. Printing also releases emissions. A NIOSH study of 3D printing in eleven university and school settings, published in 2025, detected volatile organic compounds and ultrafine particles during printing, found that printers are often placed in spaces not designed for them, and recommended engineering controls, administrative controls and personal protective equipment. In practice: print in a well-ventilated room, ideally with the printer in an enclosure, not on a bedroom desk.
• CNC machines and power tools throw chips and can catch hair, sleeves and fingers. They need guards, eye protection and training, and are used in a school workshop under supervision, never reached into while running.
• Resin casting uses chemicals that can irritate skin and lungs: wear gloves, work with good ventilation, and follow the maker's instructions. Molten-metal casting needs a foundry and training; it is not a home process.
• Injection moulding happens in factories, behind guards, at high temperatures and pressures.

Common misconceptions

• "3D printing will replace injection moulding." For one hood the printer wins: no tooling, $2.50. For ten thousand, four hours each is more than four years of continuous printing on one machine, against under five days at 40 seconds each in a mould.
• "The most accurate process is the best process." CNC machining gives the best accuracy and finish, and wastes about 98 percent of a block to make a thin hood. Best depends on the part, the quantity and the requirement.
• "A part that prints can be moulded as it is." A side hole that prints easily is an undercut in a mould. Parts are redesigned for their process, with draft, without undercuts where possible, and with an eye on shrinkage.
• "A desktop 3D printer is just an appliance." It has burn hazards and releases volatile organic compounds and ultrafine particles while printing; NIOSH recommends controlling the exposure.

The takeaway

John Wesley Hyatt's celluloid of 1869, made in answer to a $10,000 prize for an ivory substitute, led in 1872 to one of the first injection moulding machines, a plunger forcing hot plastic into a mould. Processes add material layer by layer, as fused filament 3D printing does; subtract it, as CNC machining does; or form it in a mould, as casting and injection moulding do. For the reading light's 47 g ABS hood, a printer needs no tooling but about four hours and $2.50 a part; machining wastes 98 percent of a 2,352 cm³ block; a $80 silicone mould gives about 25 resin casts; and a $12,000 steel mould makes a hood every 40 seconds for about 45 cents (illustrative figures). Each process imposes design rules: printed overhangs and layer bonding, cutter reach and corner radii, mould release, and for injection moulding draft, shrinkage and no undercuts without moving cores. Each has hazards, including the volatile compounds and ultrafine particles a NIOSH study found around school 3D printers.

The core of it: choose a process by the quantity, the material, the accuracy and the shape, and design the part for that process from the start, because the cheapest part is the one that was drawn to be made.

Sources

1. Glassford, E. K., & Li, J. F. (2025). Three-dimensional (3D) printing in non-industrial spaces: A summary of emissions evaluations in 11 school settings. Journal of School Health, 95(4-5), 307-316. NIOSH, via CDC Stacks. CDC Stacks
2. Wikipedia contributors. (2026). Injection moulding. In Wikipedia, The Free Encyclopedia. Wikipedia
3. Wikipedia contributors. (2026). Fused filament fabrication. In Wikipedia, The Free Encyclopedia. Wikipedia
4. Wikipedia contributors. (2026). Casting. In Wikipedia, The Free Encyclopedia. Wikipedia
5. Cambridge University Engineering Department. (2003). Materials data book (2003 ed.). University of Cambridge. University of Cambridge

$13.20 on the receipts, and a question about ten thousand

The receipts for the reading light on the shelf add up to $13.20: $2.50 for the printed hood of lesson 20, $4.50 for the microcontroller board of lesson 19, $1.00 for four white LEDs, and the rest in card, dowel, plywood, resistors, a switch and wire. (Every price, quote and time in this lesson is illustrative, of the kind a small maker might be quoted; real prices vary widely with supplier and quantity.) The power bank is not on the list, because the brief's C1 lets the lamp run from a bank the household already owns, and the prototype used one. So the prototype meets C2, materials for one lamp no more than $15, with $1.80 to spare.

Now suppose a small company likes the lamp and wants to sell 10,000 of them in its first year. Its first question sounds simple: what will each one cost to make? Is it $13.20, like the prototype? More, because a company pays wages, rent and testing that a bedroom workshop does not? Or less? A second question follows straight from lesson 20: should the company pay $12,000 for a steel mould for the hood, when a printer makes a hood for $2.50? This lesson works both questions to a number. The answers rest on three ideas, fixed and variable cost, the break-even point and the learning curve, and together they explain why something made by the hundred thousand can cost less than the parts of one made by hand.

What you pay once, and what you pay every time

Every cost of making something is one of two kinds. OpenStax's Principles of Economics calls them fixed costs, the costs of inputs that do not change with the quantity made, and variable costs, which do. For a product, fixed costs are paid once, or once a year, whatever the quantity: the hood mould, the design of a circuit board, the jigs that hold parts during assembly, the safety testing a product must pass before a shop will stock it. Variable costs are paid again for every unit: the plastic in each hood, the LEDs on each board, the minutes of work to put each lamp together. Total cost is the sum of the two, and dividing it by the quantity gives the average cost, the number the company really wants:

average cost per unit = F / N + v

Here F is the fixed cost in dollars, N the number of units made and v the variable cost of each unit. The first term is the one that moves. A fixed cost shared by more units is a smaller share for each, which OpenStax puts in one line: average total cost declines as the fixed costs are spread over an increasing quantity of output. Here is the moulded hood, with F = $12,000 for the mould and v = $0.45 for the plastic and machine time in each hood:

Read down the last column. Each tenfold step in quantity divides the mould's share by ten, and the average falls toward $0.45 without ever reaching it. At 100,000 hoods the plastic and machine time are 0.45 / 0.57 = 79 percent of the cost; at 100 hoods they are less than half of one percent. That one table holds most of the economics of mass production.

Where the lines cross: printing against moulding the hood

Now the second question. The printed hood needs no tooling, so its total cost for N hoods is 2.50 N dollars. The moulded hood's total is 12,000 + 0.45 N. Printing is cheaper for a few hoods and moulding for very many, so somewhere the totals are equal. Set them equal and solve:

2.50 N = 12,000 + 0.45 N
2.05 N = 12,000
N = 12,000 / 2.05 = 5,853.7, so 5,854 hoods

Check it by putting 5,854 back in. Printing costs 5,854 × $2.50 = $14,635.00, and moulding costs $12,000 + 5,854 × $0.45 = $14,634.30, within a dollar of each other, as they should be. This quantity is the break-even point between the two processes, and each part of the formula has a meaning. The top, $12,000, is the extra fixed cost the mould adds. The bottom, $2.05, is what each moulded hood saves against a printed one. The mould pays for itself one $2.05 saving at a time, and it takes 5,854 of them.

The picture shows why the break-even is the right question to ask. Below 5,854 hoods the printed line lies under the moulded one. Above it, the moulded line, climbing only 45 cents a hood, stays underneath for good. At the company's 10,000 hoods, printing would cost $25,000 and moulding $12,000 + $4,500 = $16,500, a saving of $8,500. A company that expected to sell 3,000 would be wrong to buy the mould: 3,000 printed hoods cost $7,500, and 3,000 moulded ones $13,350.

Bottom line: the break-even quantity is the extra fixed cost divided by the saving on each unit, and a process with expensive tooling is the cheaper choice only if you are confident of making more than that many.

No tooling is not the same as no fixed cost

Lesson 20's $2.50 a hood quietly assumed a printer that was already there with nothing else to do. For one hood that is fair. For 10,000 hoods at about 4 hours each, it is 40,000 hours of printing. A year has 365 × 24 = 8,760 hours, so a single printer running day and night without one failed print would need 40,000 / 8,760 = 4.6 years. To deliver in one year the company needs at least five printers, someone to clear and restart them, and room to put them: fixed costs, all of them, which the "no tooling" row in lesson 20's table did not show. The mould makes the same 10,000 hoods at 40 seconds each in 400,000 seconds, which is 111 hours, under five days.

The general lesson is that a cost which looks variable at one scale arrives in steps at another. One printer is enough for a prototype; a second, third, fourth and fifth appear only as demand grows past what the others can make. When engineers cost a product, they write down which assumption they used, so that the next person can see where it stops being true.

Why the thousandth lamp takes less time than the first

Materials are only part of a lamp's cost. Someone has to put it together, and the person who assembles the first lamp in a factory, working from the drawings, will be slow: finding parts, checking the drawing, fitting the arm, soldering, testing. Call it 60 minutes. Nobody expects the ten-thousandth to take 60 minutes too. The question is how fast the time comes down.

The engineer who first put a number on it was the American aeronautical engineer Theodore Paul Wright, in a paper called "Factors Affecting the Cost of Airplanes", published in the Journal of the Aeronautical Sciences in February 1936. An operations management textbook by F. Robert Jacobs and colleagues sums up what the aircraft industry found: as output doubled, the direct labour hours per airplane fell by about 20 percent between doubled units. If the first airplane took 100,000 hours, the second took 80,000, the fourth 64,000 and the eighth 51,200. Because each doubling leaves 80 percent of the time, this is called an 80 percent learning curve. The pattern has turned up far beyond airplanes. In 2013 Béla Nagy, J. Doyne Farmer, Quan Bui and Jessika Trancik compared rules for forecasting how the costs of many technologies fall over time, and found that Wright's rule, cost falling as a power of the total number made so far, gave the best forecasts, with Moore's law, cost falling steadily with the calendar, not far behind.

A time per unit that falls to 80 percent at each doubling can be written as a formula:

T(n) = T₁ × n^b, where b = log₂(0.80) = −0.322

T₁ is the time for the first unit and n is the unit's number. Check that the formula does what the rule says: for n = 2, 2^−0.322 = 0.80, so T(2) = 60 × 0.80 = 48 minutes; for n = 4, 4^−0.322 = 0.64, so T(4) = 38.4 minutes. Which percentage should you use before anything has been built? The textbook calls estimating it at that stage "enlightened guesswork" and offers rules of thumb: work that is about three-quarters hand assembly and one-quarter machining tends to follow about an 80 percent curve, while work that is three-quarters machining is nearer 90 percent, because a machine does not get faster with practice. A lamp put together by hand fits the first description, so use 80 percent.

The 1,024th lamp, ten doublings on, takes 60 × 0.80¹⁰ = 6.4 minutes, and the 10,000th about 3.1. The average over all 10,000 is higher, about 4.56 minutes, because the slow early lamps pull it up; adding the times lamp by lamp, the whole batch takes about 10,000 × 4.56 / 60 = 760 hours of assembly. At an illustrative $24 an hour for wages and overheads together, that is 4.56 / 60 × $24 = $1.82 of labour per lamp, against $24.00 for the first one.

Two warnings come with the curve. It is an observed pattern, not a law of nature, and its percentage has to be measured from production records or estimated from similar work. And the improvement is not only the workers getting quicker. The textbook separates individual learning, people gaining skill by repeating a task, from organisational learning, which also comes from changes in management, equipment and the design of the product: a better jig, a part redesigned to snap into place, a test that runs itself. A fully automated line, it adds, may barely improve at all once it is installed.

Highland Park, 1913 to 1914: the same arithmetic on a car

The most famous demonstration of falling labour per unit happened at Ford's Highland Park plant in Michigan. The Henry Ford museum's research guide records that Ford began building magnetos on a crude moving assembly line in April 1913, and then moved the idea on to complete chassis. Instead of a gang of workers building a car where it stood, the car moved past workers who each did a few operations, with parts delivered to them. The method combined interchangeable parts, divided labour and a steady flow of materials, and the first of those depends on lesson 6: a part that must fit any car on the line without filing needs a tolerance that is written down and checked. The guide compares the result by 1914, a Model T every 93 minutes, with 12½ hours per car under the old stationary methods.

Read those 93 minutes carefully, because they are easy to misread in exactly the way lesson 7 warned about, as a number whose unit has gone missing. The same guide says that by April 1914 three parallel chassis lines were running and workers could put together more than 1,200 chassis in eight hours. That is 480 minutes / 1,200 = one chassis every 24 seconds, or every 72 seconds on each line. One car every 93 minutes cannot be squared with that. What can be squared with it is 93 minutes of work in each car, summed over everyone who touched it, with many cars on each line at once, each a little further along: 1,200 chassis in 8 hours is 150 an hour, and 150 × 1.55 hours of work each keeps about 233 people busy at any moment.

The work per car fell by a factor of 12.5 × 60 / 93 = 8.1. The price followed, more slowly. Ford's own history gives the Model T's price as $825 in 1908 and $260 by 1925, a factor of 825 / 260 = 3.2, and the museum counts more than 15 million Model Ts built from 1908 to 1927. Labour was only part of a car's cost. The steel, glass, rubber and the factory itself did not become eight times cheaper, just as the reading light's plastic does not become cheaper because someone assembles it faster.

The whole lamp at ten thousand

Back to the company. Making the lamp as a product changes three things from the prototype. The hood and the base become injection moulded. The microcontroller development board, made to be plugged into breadboards and reprogrammed over USB, is replaced by a bare microcontroller chip soldered onto the lamp's own circuit board along with the LEDs, resistors, transistor, sensor and button. And a lamp sold in a shop has to come with its own power, so a power bank module, a lithium-ion cell with its charging and 5 V circuits, goes into the base. Here is the bill of materials, the list of every part and its cost, at both scales:

The materials per lamp fall from $13.20 to $7.70, although the product carries a power bank and a box that the prototype did not. Each part got cheaper for one of the reasons already met: it is bought by the thousand, made by a process whose tooling is shared over many parts, or redesigned so that one part does the work of several. The single biggest item is now the power bank, $3.50 of $7.70, or 45 percent of the materials. Lesson 22 asks whether it belongs in the box at all. The product still meets C2, comfortably.

Then the fixed costs, each one illustrative:

• Hood mould: $12,000.
• Base mould, for two halves: $18,000.
• Circuit board design, production setup and a test fixture that checks every board: $4,000.
• Assembly jigs and setting up the line: $2,000.
• Safety testing of the finished lamp and its battery: $6,000.

That is $42,000 before the first lamp exists. Put the three kinds of cost together, using the learning curve for the labour, and the company's first question gets its answer:

At 1,000 lamps the moulds and the testing swamp everything, and each lamp costs four times the prototype's receipts. At 10,000 it costs $13.72, so the honest answer to the company is "about $13.70 each, on these assumptions". At 100,000 the fixed share has almost vanished and labour is under a dollar, and what is left is mostly materials. That is the floor. No quantity makes the plastic, the LEDs or the lithium cell free, which is why, at high volume, engineers spend their effort on the bill of materials. OpenStax calls the falling part of this pattern economies of scale, the situation where the cost per unit goes down as the quantity of output goes up. Its alarm-clock example has a small factory making 1,000 clocks at $12 each, a medium one 2,000 at $8 and a large one 5,000 at $4. It also warns of the opposite, diseconomies of scale, when a factory or firm grows so large that it becomes hard to manage and its cost per unit rises again.

Bottom line: at low volume a product's cost is mostly its fixed costs shared out; at high volume it is mostly its materials, so the question an engineer should ask changes with the quantity.

A changed forecast flips the plan

The $42,000 assumed moulds for the hood and base. Now cost a plan without them: print the hoods on five printers bought for $800 each, $4,000 in all, have the bases cut from plywood by a supplier at $1.50 each, and keep everything else. The plan's fixed cost is 42,000 − 12,000 − 18,000 + 4,000 = $16,000, and its materials per lamp are 7.70 − 0.45 − 0.70 + 2.50 + 1.50 = $10.55. The labour is the same in both plans, so it can be left out of the comparison. The break-even between the plans uses the same formula as the hood:

N = (42,000 − 16,000) / (10.55 − 7.70) = 26,000 / 2.85 = 9,123 lamps

At the forecast of 10,000 the moulds win, by 25 cents a lamp. That is about 2 percent, smaller than the uncertainty in any of these quotes, and lesson 3 had a rule for a margin like that: treat it as a tie. Change the input and the answer moves decisively. If sales come in at 3,000, the moulds would have cost the company $5.82 × 3,000 = $17,460 more than printing; if they reach 30,000, the moulds save $1.98 × 30,000 = $59,400. So the right reply to "should we buy the moulds?" is a question back: how sure are you of the forecast? A cautious company might start with printers and plywood, watch how the lamp sells, and order the moulds once sales are clearly heading past 9,123, paying a little more on the first few thousand lamps rather than betting $30,000 of moulds on a guess.

Common misconceptions

• "A part costs what its material costs." The first moulded hood costs $12,000.45, nearly all of it mould. Material sets the cost only after the fixed costs are shared across a large quantity.
• "Making more always makes each one cheaper." Only while fixed costs are still being spread and learning continues; below the break-even, expensive tooling was a bad bet, and OpenStax notes that a firm can grow large enough for its costs per unit to rise again.
• "No tooling means no fixed costs." Printing 10,000 hoods takes 40,000 printer-hours, five printers for a year. Costs that look variable at one scale arrive in steps at another.
• "The learning curve is just workers getting faster." Practice is part of it. Better jigs, redesigned parts and changed methods are the rest, and work paced by machines learns more slowly than hand assembly.
• "Ford's line built a car every 93 minutes." Ninety-three minutes was the work in each car. More than 1,200 chassis came off three lines in eight hours, about one every 24 seconds.
• "What it costs to make is what it sells for." A shop's price must also pay for design, marketing, shipping, the shop's own costs and some profit, so a lamp that costs $13.72 to make cannot sell for $13.72.

What to remember

The reading light's receipts came to $13.20, inside C2's $15 with the household's own power bank. A cost is fixed, paid once, or variable, paid for each unit, and the average cost is F / N + v, so a $12,000 mould adds $12,000 to a single hood and $1.20 to each of 10,000. Printing at $2.50 and moulding at $12,000 + $0.45 a hood break even at 12,000 / 2.05 = 5,854 hoods, and printing 10,000 would need 40,000 printer-hours, a fixed cost the "no tooling" label hides. T. P. Wright's 1936 paper on airplane costs gave the learning curve, about 20 percent less labour per unit at each doubling of the number made; at 80 percent, the lamp's assembly falls from 60 minutes to about 3 by the 10,000th, an average of 4.56 minutes and $1.82. Ford's Highland Park lines cut the work in a car from 12½ hours to 93 minutes by 1914, and the Model T's price fell from $825 in 1908 to $260 by 1925. With $42,000 of fixed costs and $7.70 of materials, the lamp costs $53.51 each at 1,000, $13.72 at 10,000 and $8.99 at 100,000, and the moulds beat printing and plywood only above 9,123 lamps.

Bottom line: cost per unit is the fixed cost shared out plus what each unit takes, so decide on tooling by the break-even quantity, check that your variable costs stay variable at the scale you plan, and treat a thin margin on an uncertain forecast as a tie.

Sources

1. Greenlaw, S. A., Shapiro, D., & MacDonald, D. (2022). Principles of economics (3rd ed.), sections 7.3, Costs in the short run, and 7.5, Costs in the long run. OpenStax. OpenStax
2. Jacobs, F. R., Chase, R. B., Grando, A., & Sianesi, A. (2020). Learning curves (chapter 4A, pp. 90-105). In Operations management (4th ed.). McGraw-Hill Education. Cites Wright, T. P. (1936). Factors affecting the cost of airplanes. Journal of the Aeronautical Sciences, 3(4), 122-128. McGraw-Hill
3. Nagy, B., Farmer, J. D., Bui, Q. M., & Trancik, J. E. (2013). Statistical basis for predicting technological progress. PLOS ONE, 8(2), e52669. PLOS ONE
4. The Henry Ford. (2026). K-12 research guide: Assembly line. AskUs, The Henry Ford. The Henry Ford
5. Ford Motor Company. (n.d.). The moving assembly line and the five-dollar workday. Ford Motor Company

One cotton bag, three answers: 131, 52 and 7,100

In February 2011 England's Environment Agency published a report by Chris Edwards and Jonna Meyhoff Fry with an unexciting title, Life cycle assessment of supermarket carrier bags, and one number from it travelled round the world. A cotton shopping bag, it found, would have to be used 131 times before its contribution to global warming fell below that of a conventional thin plastic carrier bag used once. A paper bag needed 3 uses, a thick plastic bag for life 4, and a heavier bag of non-woven polypropylene 11.

In February 2018 the Danish Environmental Protection Agency published its own study, carried out at the Technical University of Denmark. For climate change alone, it found that a conventional cotton bag must be used 52 times to match the plastic bag Danish shops always stock. Counting every one of the fifteen kinds of environmental impact it measured, the figure became 7,100 uses: more than a shopping trip a day for nineteen years. An organic cotton bag needed 20,000.

So is it 131, 52 or 7,100? None of the three numbers is wrong. They are careful answers to three slightly different questions, and seeing exactly how they differ is the quickest way into the subject of this lesson: life-cycle thinking, the habit of judging a product by everything it does from the moment its raw materials are dug up to the moment it is thrown away. The lesson then turns that habit on the reading light, where lesson 21 left a question hanging: should the lamp a company sells come with its own power bank built in?

What a life cycle assessment counts

A life cycle assessment (LCA) follows a product through every stage of its life: extracting the raw materials, processing them, manufacturing the product, transporting it, using it, and dealing with it at the end, whether by reuse, recycling, burning or landfill. Following all of it is called cradle to grave. Both bag studies were built to the international standard for the method, ISO 14040, and reviewed against it, and both are built from the same parts:

• Goal and scope. What question is being asked, for whom, and what is inside the system boundary, the line around what gets counted.
• A functional unit. The amount of service being compared. You cannot fairly compare one cotton bag with one thin plastic bag, because they do not do the same amount of carrying. The English study compared the bags needed for carrying one month's shopping, 483 items, from supermarket to home in 2006 and 2007. The Danish study compared bags for carrying one trip's shopping, 22 litres and 12 kilograms, from a Danish supermarket to a home in 2017.
• An inventory. Every material and every unit of energy that goes in at every stage, and every emission and waste that comes out.
• An impact assessment. The inventory translated into impact categories: climate change, ozone depletion, acidification, the over-fertilising of water called eutrophication, toxicity, the using up of resources, and others.
• Interpretation. What the numbers mean, how sensitive they are to the assumptions, and what they do not show.

The functional unit is the idea most people skip and the one that matters most. It is lesson 3's matrix rule in another form: compare designs on the job they do, not on the object. Lighting studies use the same trick. A three-part U.S. Department of Energy study of household lamps compared incandescent, compact fluorescent and LED lamps per 20 million lumen-hours of light delivered, because an LED lamp lasts far longer than an incandescent bulb and one of each would not be a fair comparison.

Why careful studies of the same bags disagree

Now the three numbers can be read. The English 131 compared the cotton bag with a thin bag made of high-density polyethylene that was not reused at all. When the comparison allowed for 40.3 percent of the thin bags being reused as bin liners, the cotton bag needed 173 uses. The Danish study compared it with a different reference, the average low-density polyethylene bag sold in Danish shops, reused as a waste bin bag before being burned. Change the reference and the number of uses needed to beat it changes too, and the Danish figure for climate change alone, 52, is the closer cousin of the English 131.

The 7,100 is a different kind of number. The Danish team calculated the uses needed for each of its fifteen impact categories and reported, as the figure for all indicators, the highest of them. For cotton that highest value came, the report's footnote says, from ozone depletion alone, for which the dataset on cotton production gives large impacts. Leave ozone depletion out and the conventional cotton bag needs between 50 and 1,400 uses, depending on the category. So 7,100 is honest, and it is also the result of one impact category and one dataset. A newspaper headline that says "cotton bags are 7,100 times worse" has thrown away everything that makes the number mean something.

Underneath the disagreement, the studies agree on more than they differ. The English report found that the impact of every type of bag was dominated by the resource use and production stages, while transport, secondary packaging and end-of-life management generally had a minimal influence. Both reports said the key is to reuse a bag as many times as possible. And the English report found that reusing the thin bags as bin liners did more good than recycling them, and that recycling or composting generally gave only a small reduction in global warming potential.

Why this matters: an LCA's answer depends on its functional unit, its reference, its impact categories and its data, so before you quote one, find out what was compared with what, per what, and which category produced the headline.

The reading light, from cradle to grave

Here is a first inventory of the lamp the company plans to sell, by mass. The masses of the hood and of the post and arm come from lessons 20 and 10; the others are illustrative estimates for parts that have not been weighed.

The lamp itself, without its box and cable, comes to 316 g, of which the power bank is 110 g, 35 percent. Lesson 21 found the bank was also 45 percent of the materials cost. Now look at the use stage. Lesson 17 worked out that the lamp's LEDs use about 0.24 kWh a year, about 0.4 kWh from the wall once the bank's and charger's losses are counted, roughly 560 times less than the chest freezer's 224 kWh. The Department of Energy's lamp study found the opposite pattern for ordinary household lamps: across the ten earlier assessments it reviewed, the use stage accounted for about 90 percent of the lamps' total life-cycle energy, with manufacturing and transport next. A household lamp burns for thousands of hours at several watts. The reading light draws a third of a watt for two hours a night.

That contrast tells you where to look. For a freezer or a ceiling lamp, the best environmental decision is usually the most efficient one, because using it is most of its life. For the reading light, using it is almost nothing, so the making and the throwing away are almost everything, as they were for the carrier bags. And among the things made, one part is the heaviest, the most expensive, the most processed and the one that, as lesson 15 said, must never go into a household bin or a recycling box because a damaged lithium-ion cell can start a fire: the power bank.

The dispute: should the power bank be in the box?

There are two credible positions, and each rests on real evidence.

Build it in. A lamp sold in a shop must work when it comes out of the box, and the person who buys it may not own a power bank, or may own one that is not suitable. Lesson 17 found that some power banks switch themselves off when the current drawn stays low for a while, and the lamp's steady 67 mA is a small load of exactly the kind that lesson 17 said only a timed test could check; a module built into the base and chosen for the lamp removes that risk. A built-in module can also be matched to the job instead of being a general-purpose bank with a case, a display and a spare port. And the bag studies' central finding cuts in this position's favour: a made thing is worth its making if it is used for a long time, and a battery used every night for the lamp's whole life is used a great deal.

Leave it out, or make it replaceable. The power bank is the part with the most at stake, and sealing it in ties the lamp's life to the battery's: when a sealed cell stops holding a charge, the hood, base, arm and circuit board, all still working, go out with it. The end of life is where products are most often lost. The Global E-waste Monitor 2024, from the International Telecommunication Union and the United Nations Institute for Training and Research, found that a record 62 million tonnes of electronic waste was produced in 2022, and that less than a quarter of it, 22.3 percent, was documented as properly collected and recycled. Regulators have taken a side. Article 11 of the European Union's Batteries Regulation of 12 July 2023 requires anyone who sells products containing portable batteries in the EU to make those batteries readily removable and replaceable by the end user at any time during the product's lifetime, from 18 February 2027, with narrow exceptions such as appliances designed to be washed. A battery counts as readily removable if commercially available tools can remove it, without specialised or proprietary tools, heat or solvents. And if the household already owns a power bank, as the prototype's did, a lamp without one avoids making a second.

What would settle it

Neither position wins by argument, because both rest on a number nobody in the dispute has measured: how long the battery lasts in this lamp, compared with everything else in it. A proper LCA would settle it, and you now know how it would have to be set up. Its functional unit would be the job, not the object: 300 lux on an A4 page for 2 hours a night, every night, for, say, 12 years. It would compare three designs: a sealed-in bank, a cell the user can replace with a screwdriver, and no battery at all. And it would need the life of the cell at about 20 charges a year, the life of the rest of the lamp, the fraction of buyers who already own a suitable bank, and how many lamps are actually collected at the end.

Even without the impact data, counting shows how the answer turns on those inputs. Suppose, as an assumption to test rather than a measured fact, that the cell lasts 4 years and everything else lasts 12. Over 12 years of reading:

Change the input and the table changes. If the cell lasts 12 years, as long as the lamp, the sealed design makes one lamp and one cell, and the case for building it in is strong. If the cell lasts 4 years, the sealed design makes three of everything, and only the replaceable or battery-free designs make sense. So the measurement that would settle the dispute is not an opinion about plastic or about batteries. It is a test: run cells through this lamp's duty, 20 charges a year in a room at bedroom temperature, and measure how their capacity falls with time, which is the same move lesson 4 made when it stopped arguing and built a wind tunnel's worth of evidence.

Why this matters: a design dispute that seems to be about values is often a dispute about one unmeasured number, and finding that number tells you which test to run.

Design moves that hold up either way

Some decisions do not depend on how the dispute ends, and an engineer can make them now:

• Make the battery replaceable even if it is supplied. A standard cell in a holder behind a screwed cover satisfies both positions: the lamp works out of the box, and a worn cell does not end the lamp. It costs a door, two contacts and a screw, and the instructions must say how to replace the cell safely, which Article 11 also requires.
• Join parts so that they come apart. Screws instead of glue let the base be opened, the circuit board replaced and the plastic, wood, steel and electronics separated at the end. A part that can be taken apart can be repaired.
• Make it from less. Every gram not moulded is a gram not made from oil, transported or thrown away. Thinner walls where the stress is low, as lessons 8 and 9 taught you to check, and no part that does no work.
• Use one material where you can. The box is card and paper only: no plastic window, no foam, sized to the lamp. Card is light, and a single material is easier to recover than a laminate.
• Say what it is made of. Moulding the plastic type into the hood and base tells a recycler what they are holding.

Common misconceptions

• "Natural or reusable materials are always greener." In the English study a cotton bag had to be used 131 times to beat a thin plastic bag on global warming. What counts is the impact per use, which depends on how much material the thing takes and how many times it is used.
• "For anything electrical, the electricity is the big impact." For household lamps the use stage was about 90 percent of life-cycle energy; for a lamp drawing a third of a watt for two hours a night, the use stage is tiny and the making and disposal dominate.
• "Recyclable means it will be recycled." Only 22.3 percent of the world's electronic waste in 2022 was documented as properly collected and recycled, and the English bag study found reuse did more good than recycling.
• "A life cycle assessment gives one objective number." The Danish 7,100 was the highest of fifteen category results, driven by ozone depletion alone; the English 131 used a different reference bag. The numbers are only as meaningful as the functional unit, reference and categories behind them.
• "A product whose battery has died is worn out." Often everything else still works. Whether it is waste depends on whether the battery can be replaced.

Looking back

England's Environment Agency found in 2011 that a cotton carrier bag had to be used 131 times, or 173 if thin bags were reused as bin liners, to beat a thin plastic bag on global warming; Denmark's 2018 study found 52 uses for climate change and 7,100 across fifteen impact categories, the highest of which came from ozone depletion alone. Both followed ISO 14040: a goal and scope with a system boundary, a functional unit that compares the job rather than the object, an inventory, an impact assessment and an interpretation. The English study found that making the bags dominated their impact, and both urged reuse as many times as possible and ranked reuse as a bin bag above recycling for the light bags. The Department of Energy found household lamps the other way round, with the use stage about 90 percent of their life-cycle energy, but the reading light uses only about 0.4 kWh a year, so its making and disposal matter most, and its power bank is 35 percent of its mass and 45 percent of its materials cost. Whether to seal the bank in turns on one unmeasured number, the cell's life in this duty against the lamp's: with a 4-year cell and a 12-year lamp, a sealed design makes three lamps where a replaceable one makes one. EU law from 18 February 2027 requires portable batteries in products to be removable and replaceable by the user, and only 22.3 percent of 2022's 62 million tonnes of e-waste was documented as properly recycled.

Why this matters: judge a design by the whole of its life and by the job it does, find the stage and the part where its impact really lies, and when two good arguments collide, find the number that decides between them and measure it.

Sources

1. Edwards, C., & Fry, J. M. (2011). Life cycle assessment of supermarket carrier bags: A review of the bags available in 2006 (Report SC030148). Environment Agency. GOV.UK
2. Bisinella, V., Albizzati, P. F., Astrup, T. F., & Damgaard, A. (Eds.). (2018). Life cycle assessment of grocery carrier bags (Environmental Project No. 1985). Danish Environmental Protection Agency. Danish EPA
3. U.S. Department of Energy, Building Technologies Office. (2013). Life-cycle assessment of energy and environmental impacts of LED lighting products [Fact sheet]. U.S. Department of Energy
4. European Parliament & Council of the European Union. (2023). Regulation (EU) 2023/1542 of 12 July 2023 concerning batteries and waste batteries, Article 11 and Article 96. Official Journal of the European Union, L 191. EUR-Lex
5. International Telecommunication Union & United Nations Institute for Training and Research. (2024). The global e-waste monitor 2024. Global E-waste Monitor

29 August 1907: eighty-six men on the bridge

On 29 August 1907 eighty-six men were at work on the half-built Quebec Bridge, high over the St. Lawrence River near Quebec City. Before the day was out, the steelwork they stood on had fallen into the river. "Out of eighty-six men on the work," the Royal Commission that investigated wrote the following year, "only eleven escaped with their lives." Seventy-five men died. Many of them, The Canadian Encyclopedia records, were Mohawk ironworkers from Kahnawake, famous for their work on high steel.

Thirty-three years later and on the other side of the continent, a very different bridge failed in front of cameras. Nobody died when the Tacoma Narrows Bridge fell on 7 November 1940, but the film of its deck twisting in the wind became the most famous footage in engineering. The board of engineers who investigated it opened their report by calling it the most notable failure in the history of bridge building since the Quebec Bridge. This lesson follows both bridges through their official investigations, because between them they hold two of the hardest lessons in this course: a structure is only as safe as the numbers it was checked against, and a structure strong enough for every load you calculated can still be destroyed by one you did not think to calculate.

Quebec, 1900: a span stretched by 200 feet

The Quebec Bridge was to be a cantilever bridge. Picture each half as a giant see-saw on its main pier: an anchor arm reaching back to the shore, tied down at its end, balancing a cantilever arm that reaches out over the river. The two cantilever arms reach toward each other, and a suspended span is lifted in to join their tips. Over the main pier the see-saw is bent like a beam over a support, so its top chords are pulled in tension and its bottom chords pushed in compression, and the bottom chords of the anchor arm nearest the main pier carry some of the largest compressive forces in the whole bridge.

The design was made by the Phoenix Bridge Company of Phoenixville, Pennsylvania, whose designing engineer was P. L. Szlapka, and was examined and approved by Theodore Cooper, a leading bridge engineer with an office in New York, engaged as consulting engineer by the Quebec Bridge and Railway Company. On 5 May 1900, the Commission's report records, the company's directors resolved to change the main span "from 1,600 to 1,800 feet", from 488 m to 549 m, a lengthening of 12.5 percent.

The weights nobody recalculated

Every structural calculation starts with the load, and for a long-span bridge the largest load is its own weight, the dead load. Weight depends on the size of the members, and the size of the members depends on the forces, which depend on the weight. So a designer has to assume the dead load, size the members, then work out what those members really weigh and go round again. The Royal Commission found that at Quebec the second time round never properly happened. Here are the dead loads the 1904 calculations assumed, against the figures in a revision made in June 1907, while the bridge was being erected:

Work the anchor arm's row yourself: 17,318,000 / 13,317,600 = 1.300, so the real dead load was 30 percent more than the one the members had been sized for. The report records that Szlapka had advised in April 1904 that the bridge's weight "would not be more than five per cent above the estimate", and that Cooper, when the error came to light, estimated that it raised the stresses in the members by seven to ten percent, and the work went on. The Commission's finding was blunt: "A grave error was made in assuming the dead load for the calculations at too low a value and not afterwards revising this assumption." The error, it said, was large enough by itself to have required the condemnation of the bridge even if the chords had been strong enough, because the finished bridge would have been stressed considerably beyond its specifications.

Key idea: an assumption made to start a calculation has to be checked once the design it produced exists; a design that is never checked against its own weight is checked against nothing.

27 to 29 August 1907: a bend that grew

By August 1907 the anchor arm and cantilever arm stood complete and the fourth panel of the suspended span was being built out from the cantilever's tip. On 27 August a bend was found in a lower chord of the anchor arm, the one labelled 9-L. McLure, the inspector for erection on the Quebec Bridge, wrote to Cooper in New York that day: "Only a little over a week ago, I measured one rib of the 9-L chord of anchor arm here shown, and it was only 1-inch out of line." Now it was out by more. His letter said that "the erection will not proceed until we hear from you and from Phoenixville."

It did proceed. The Phoenix company's erection superintendent, Yenser, had announced he would stop work until the matter was referred to Phoenixville, but, the report says, he "had changed his mind during the night and in the morning continued erection." The Quebec company's chief engineer, Hoare, was on site and had the power to stop it. The Commission found that he "did not order Mr. Yenser to discontinue erection which he had power to do", and put it down to "indecision and to a habit of relying upon Mr. Cooper for instructions." Hoare sent McLure to New York to see Cooper in person.

McLure was waiting in Cooper's office shortly after 11 a.m. on 29 August. Cooper sent a telegram to the Phoenix Bridge Company: "Add no more load to bridge till after due consideration of facts. McLure will be over at five o'clock." It arrived in Phoenixville at 1:15 p.m. The company's chief engineer, John Sterling Deans, arranged for Szlapka and another engineer to meet McLure but "otherwise took no action", and McLure, who had promised to telegraph Cooper's decision to the site, did not. At Phoenixville, the engineers looking at the reports decided to postpone action until the morning. "This decision", the Commission wrote, "was made almost at the minute that the bridge fell."

What the Royal Commission found

The Royal Commission, Henry Holgate, J. G. G. Kerry and John Galbraith, reported on 20 February 1908. Its findings still read as a list of the ways an engineering project fails:

• "The collapse of the Quebec bridge resulted from the failure of the lower chords in the anchor arm near the main pier", and their failure "was due to their defective design." The stresses that broke them were not caused by abnormal weather or accident but were those expected in the ordinary course of erection.
• The chords were designed by Szlapka and approved by Cooper, and the failure could not be attributed to any cause other than errors in judgment on the part of these two engineers. Those errors, the Commission stressed, came neither from a lack of ordinary professional knowledge, nor from neglect of duty, nor from a desire to economise: their ability "was tried in one of the most difficult professional problems of the day and proved to be insufficient for the task."
• The loss of life "might have been prevented by the exercise of better judgment on the part of those in responsible charge of the work" for both companies, though nothing done after 27 August could have saved the bridge itself.
• The Quebec company's failure to appoint an experienced bridge engineer as chief engineer was a mistake, which resulted in "loose and inefficient supervision".
• The fabrication and erection were good and "the steel used was of good quality. The serious defects were fundamental errors in design."
• Professional knowledge "concerning the action of steel columns under load" was not sufficient to design such a structure economically, and the experimental studies needed to confirm the designers' judgment "were not made."

What was wrong with the chords? Each was a built-up member, four tall steel plates, called ribs, held together by a lattice of diagonal bars riveted across them, so that the four ribs would act as one wide column. Lesson 11 showed why a column in compression is dangerous: it fails by buckling, at a load that falls with the square of its length and rises with its stiffness. Four ribs acting together form a wide, stiff column. Four ribs acting separately are four slender plates, each of which buckles at a small fraction of the load. Everything depends on the lattice. The Commission's appendices call chord A 9 L "an example of an insufficient lattice system": its "webs bent and the lattices failed under a load only three-fourths of the specified maximum working load." Model chords tested for the inquiry made the point with numbers. One failed "with explosive violence, by the shearing of the outer rivets of the latticing" at about 22,110 lb per square inch, 152 MPa; a second, with latticing strong enough to hold its ribs together, carried 37,000 lb per square inch, 255 MPa, before its webs buckled. Same steel, same ribs, and 37,000 / 22,110 = 1.67 times the strength, from the bars that joined them. No column of that form had ever been tested before the bridge was built.

The bridge was redesigned with far heavier members and built again. In September 1916 the new centre span fell into the river while it was being hoisted into place, killing 13 men. The bridge was finished in 1917 and opened in 1919. It still stands.

Tacoma, 1940: the narrowest bridge of its size

The Tacoma Narrows Bridge in Washington State was a suspension bridge, the third longest suspension span in the world, with a centre span of 2,800 feet, 853 m. The Washington State Department of Transportation's history of the bridge records that the state's own preliminary design, by Clark Eldridge, stiffened the deck with trusses 25 feet deep. The New York engineer Leon Moisseiff substituted a solid plate girder 8 feet deep, and told the federal Public Works Administration that his changes would cut the estimated cost from $11 million to $7 million. With only two traffic lanes and 39 feet between its cables, the bridge had a ratio of centre span to width of 1 to 72, against the Golden Gate Bridge's 1 to 47, and a ratio of girder depth to span of 1 to 350.

Lesson 8 taught you that strong is not stiff, and this is the place to use it. The stiffness of a deck in bending grows roughly with the square of its depth, for the same amount of steel in its top and bottom chords. On that rough rule, trusses 25 feet deep would have been (25 / 8)² = 9.8, about ten times, as stiff in bending as an 8-foot girder. A narrow deck is also easy to twist. The bridge that opened on 1 July 1940 was strong enough to carry its loads and very much less stiff than any bridge of its size.

It moved from the start. Vertical waves ran along the deck during construction and continued after opening, enough to earn the bridge the nickname Galloping Gertie. Remedial ties and guys were added. In September 1940 the state toll bridge authority hired Professor F. B. Farquharson of the University of Washington to study a scale model of the bridge in a wind tunnel, and in November, the WSDOT timeline records, he found a twisting motion that could destroy the span.

7 November 1940: from waves to twisting

The board's report sets out the morning. The wind was clocked at 42 miles per hour, 18.8 m/s, by an anemometer near the bridge at about 9 a.m., and Farquharson, who was there, believed it was somewhat stronger when the bridge went down. Until 10 a.m. the deck moved as it had moved for months: vertical waves with eight or nine nodes, at 36 to 38 cycles a minute, the two cables moving together, no twisting. Then, at about 10 a.m., the centre span began to twist, with a node at mid-span, so that one half rolled one way while the other rolled the other. The frequency dropped suddenly from 37 cycles a minute to 14, one twist about every 4.3 seconds. At the quarter points the deck rotated nearly 45 degrees each way, and the cables moved through about 28 feet.

The board found it "reasonably certain that the first failure was the slipping of the cable band on the north side of the bridge" at mid-span, where diagonal ties had been fixed to restrain the deck, and that "this slipping probably initiated the torsional oscillations." At about 10:30 a section of the concrete roadway fell from the centre. At about 11:00 some 600 feet of the suspended structure dropped into the water; at 11:10 most of the rest of the centre span followed, and as its weight was released the towers were pulled about 25 feet toward the shores. WSDOT's history records that the only life lost was a dog, Tubby, left in a car on the bridge.

Not resonance: what the board and later analyses found

The board, Othmar Ammann, Theodore von Kármán and Glenn Woodruff, reported to John Carmody of the Federal Works Agency on 28 March 1941. Its first conclusion was that the bridge "was well designed and built to resist safely all static forces, including wind, usually considered in the design of similar structures. Its failure resulted from excessive oscillations caused by wind action." Its second blamed "the extraordinary degree of flexibility of the structure and of its relatively small capacity to absorb dynamic forces." And its tenth is the sentence engineers remember: "The criteria usually considered for rigidity against static forces do not necessarily apply to dynamic forces." The materials and workmanship, it found, were "of a high order."

For decades afterwards, physics textbooks taught Tacoma Narrows as the classic example of resonance. A 2015 article in Physics Today by Donald Olson, Steven Wolf and Joseph Hook notes that influential textbooks of the early 1960s explained that the wind produced a fluctuating force in resonance with a natural frequency of the structure. The idea is that air flowing past a bluff deck sheds swirling vortices from alternate edges, a Kármán vortex street, and that these pushed the deck up and down in step with its natural frequency, the way well-timed pushes build up a swing.

The board itself, with von Kármán, the scientist those vortices are named after, as a member, rejected that story in 1941. "It is very improbable", it wrote, "that resonance with alternating vortices plays an important role in the oscillations of suspension bridges." Its reasons are worth following. Vortices are shed at a frequency that rises in proportion to the wind speed, so resonance with them would happen only near one particular wind speed for each mode of the bridge, and the board found "no sharp correlation between wind velocity and oscillation frequency such as is required." It also found no evidence that a deck of Tacoma's shape shed alternating vortices at all while it was not moving, and concluded that "the vortex formation and frequency is determined by the oscillation of the structure", not the other way round. The twisting, it said, came from aerodynamic instability: with the torsional motion once started, "the suspended structure being aerodynamically unstable with reference to torsional motions, the oscillations would build up very rapidly."

Later analyses gave the mechanism its name, flutter. The engineers K. Yusuf Billah and Robert Scanlan argued in a 1991 paper written against the textbook story that the twisting was self-excited, not forced, and, as Olson and his co-authors summarise, the wind-driven amplification they described, "unlike a resonance, increases monotonically with increasing wind speed." WSDOT's account of the failure names it torsional flutter: once the twisting became self-excited, the deck's own movement generated the forces. Here is the difference, step by step.

• Forced resonance needs a push that arrives with its own rhythm, independent of the structure, at the structure's natural frequency. Its amplitude is limited by damping, and it peaks at one forcing frequency and falls away on either side.
• Self-excited flutter needs only a steady wind. When the deck twists, it changes the angle at which the wind meets it, and above a critical wind speed the twisting moment the wind then produces pushes the deck the way it is already turning. The motion makes the force that feeds the motion. If the wind adds more energy in each cycle than damping removes, the amplitude grows every cycle, and it keeps growing until something breaks.

Growth from cycle to cycle is what makes flutter so fast. As an illustration, not a measurement: if each twist were 5 percent larger than the one before, a twist of 1 degree would become 45 degrees after n cycles where 1.05ⁿ = 45, so n = log 45 / log 1.05 = 78 cycles, under 6 minutes at 14 cycles a minute. The swing analogy works for resonance and fails for flutter, because nobody is pushing the bridge in time. The nearer picture is a flag flapping in a steady breeze: the breeze has no rhythm, and the flag makes its own.

What changed

The replacement bridge, which opened on 14 October 1950 on the old piers, had four lanes, open steel grating between the lanes to let air through, and a stiffening system of Warren trusses 33 feet deep; WSDOT describes it as 58 times more rigid than its predecessor. Its design was tested in a wind tunnel at the University of Washington by the same Professor Farquharson, and wind tunnel testing of models for aerodynamic effects became commonplace. The Quebec Bridge left its lesson in the Commission's finding that the experiments needed to confirm the designers' judgment were never made, and in a profession reminded that a famous name approving a drawing is not the same as a check.

Common misconceptions

• "Tacoma Narrows was resonance, like a singer shattering a glass." The 1941 board found it "very improbable" that resonance with vortices played an important role, and later analyses describe self-excited flutter, which grows with wind speed instead of peaking at one rhythm. A steady 42 mph wind has no rhythm of its own to match.
• "Both bridges fell because the steel was bad." Both investigations found the materials and workmanship good. Quebec failed in design and judgment; Tacoma failed in a behaviour its design never considered.
• "Quebec was one arithmetic slip." The dead-load error was serious, but the chords failed through an untested lattice design, and the deaths came from a chain of decisions: a dead load never recalculated, no experienced bridge engineer as chief engineer, erection resumed after the inspector said it would stop, and a stop order that reached Phoenixville and not the bridge.
• "A structure that passes its strength calculations is safe." Tacoma Narrows passed every static check. Its problem was stiffness and damping against a dynamic load, which is exactly the board's tenth conclusion.

Where this leaves us

The Quebec Bridge's main span was lengthened from 1,600 to 1,800 feet in 1900, and its dead load was never properly recalculated: the anchor arm turned out 30 percent heavier than assumed. On 27 August 1907 inspector McLure reported chord 9-L bending and wrote that erection would not proceed; it proceeded. Cooper's telegram, "Add no more load to bridge", reached Phoenixville at 1:15 p.m. on 29 August and never reached the site, and 75 of the 86 men on the bridge died. The Royal Commission found that the lower chords failed by defective design, their latticing too weak to hold four ribs together, and model chords proved it: 22,110 lb per square inch with weak latticing, 37,000 with strong. The Tacoma Narrows Bridge, 2,800 feet long, 39 feet wide and stiffened by girders 8 feet deep, was strong enough for every static load; on 7 November 1940, in a wind of about 42 mph, its deck switched from vertical waves to twisting at 14 cycles a minute after a cable band slipped, and it fell within about an hour and ten minutes. The board rejected resonance with vortices, and later analyses describe self-excited torsional flutter.

Key idea: check every assumption once the design it produced exists, test what nobody has tested before you rely on it, design for the loads that move as well as the loads that sit still, and make sure a warning has a path to the person who can stop the work.

Sources

1. Royal Commission Quebec Bridge Inquiry. (1908). Report of the Royal Commission on the Quebec Bridge inquiry (Sessional Paper No. 154). King's Printer. Internet Archive
2. Ammann, O. H., von Kármán, T., & Woodruff, G. B. (1941). The failure of the Tacoma Narrows Bridge: A report to the Honorable John M. Carmody, Administrator, Federal Works Agency. Federal Works Agency. CaltechAUTHORS
3. Washington State Department of Transportation. (n.d.). Tacoma Narrows Bridge history: The 1940 bridge, Lessons from failure, Aftermath, and Timeline. WSDOT
4. Olson, D. W., Wolf, S. F., & Hook, J. M. (2015). The Tacoma Narrows Bridge collapse. Physics Today, 68(11), 64. Physics Today
5. Marsh, J. H. (2013). Quebec Bridge disaster. In The Canadian Encyclopedia. Historica Canada. The Canadian Encyclopedia

17 July 1981: two walkways, 142,000 pounds

On 17 July 1981, between 1,500 and 2,000 people were in the atrium lobby of the Hyatt Regency Hotel in Kansas City, Missouri, when two of the three walkways hanging across it fell. The fourth-floor walkway came down onto the second-floor walkway directly beneath it, and both, weighing 142,000 pounds together, landed on the lobby floor. One hundred and fourteen people died and at least 186 were injured. The National Bureau of Standards, now NIST, which investigated, concluded that in loss of life and injuries it was the most devastating structural collapse ever to take place in the United States.

This lesson works backwards from wrong answers, as lesson 7 did with units. Each case has a plausible explanation that is wrong in an instructive way, and tracing exactly where it goes wrong is how you learn what the investigators learned. The second case is the space shuttle Challenger, which broke apart 73 seconds after launch on 28 January 1986. The two failures look nothing alike, a hotel walkway and a rocket booster, and the investigations found the same thing underneath both.

Wrong answer 1: "Two rods are basically the same as one"

The walkways hung from the atrium roof on steel rods 1¼ inches thick. Each walkway's floor rested on cross beams called box beams, and at each end of each box beam a threaded rod passed through it, held by a nut underneath. As designed, the Missouri Court of Appeals later summarised, six rods, three on each side, ran from the roof straight down through the fourth-floor walkway to the second-floor walkway below, so that "each walkway would receive its support from the steel rods and the second floor walkway would not be supported by the fourth floor walkway."

Because of what the court called "certain fabricating problems", the steel fabricator, Havens Steel Company, proposed a "double rod" system: the original rods would stop at the fourth-floor walkway, and a second set of rods would hang the second-floor walkway from the fourth-floor box beams. The proposal went to the structural engineers. According to the findings of the Missouri commission that later heard the case, a technician at the engineering firm, G.C.E. International, raised questions about the strength of the rods and the change from one rod to two, and the project engineer, Daniel Duncan, told him the change was "basically the same as the one rod concept."

It looks the same: the same rods, the same beams, the same nuts. Draw the free-body diagrams, as lesson 10 taught, and it is not. Let P stand for the load one walkway puts on one hanger connection: its own weight's share plus the people standing on it.

As designed, cut the fourth-floor box beam free with its nut. Down on it acts its own walkway's load, P. Up on it acts the nut, P. The rod passes through the beam and carries the second-floor walkway's P straight past it to the roof, so the rod above the fourth floor carries 2P, but the connection between the nut and the beam carries only P.

As built, cut the same box beam free. Down on it act its own walkway's P and the pull of the lower rod, whose nut now sits on top of the beam, carrying the whole second-floor walkway's P. Up on it acts only the upper rod's nut, which must therefore push up with 2P. The rods still carry the same forces they did before, P below and 2P above. What changed is the one place that was weakest: the nut bearing on the underside of the fourth-floor box beam now carried twice the load. The court put it in one sentence: "The effect of this change was to double the load on the fourth floor walkway and the box beam-hanger rod connections on that walkway."

Remember: a change that keeps every part the same can still change every force. Draw the free-body diagram of the part where the load goes in before you call two designs equivalent.

Wrong answer 2: "The rods were too thin"

It is the obvious suspect, and the investigation cleared it. "The rods themselves did not fail," the appeals court noted. NBS concluded that the most probable cause was insufficient load capacity of the box beam-hanger rod connections, and that the failure began at the connection on the east end of the fourth-floor walkway's middle box beam, where the beam gave way around the rod and the nut beneath it pulled through. The commission found that the engineer's drawings did not show the need for a special weld, for stiffeners or for bearing plates at these connections, and specified rods of regular-strength steel rather than high-strength. The connections were also what engineers call non-redundant: if one failed, no other load path could take over, and the whole system came down.

Wrong answer 3: "The original design was safe; the change killed them"

This is the comfortable version, because it puts the whole fault on one change. It is wrong too. The commission found, and the engineers did not contest, that the hanger rods and box beam-hanger rod connections shown on the original structural drawings did not meet the design specifications of the Kansas City Building Code. As the appeals court reported, NBS found that, as originally designed, the connection capacity was 60 percent of that required by the Building Code, and as constructed, 31 percent.

Check those two numbers against the free-body diagrams. Doubling the load on the same connection halves its capacity relative to what is required: 60 percent / 2 = 30 percent, and NBS's figure was 31. The arithmetic agrees, which is a useful sign you have understood the mechanism. Then remember lesson 9's factor of safety. A building code requires a connection to be able to carry well more than the load expected on it, to cover everything the calculation does not know. The original connection had little more than half of that required capacity; the one actually built had less than a third. The change roughly halved a margin that was already too small.

Wrong answer 4: "Nobody could have caught it"

There were at least three chances, and the commission's findings describe each.

• The shop drawings. A steel fabricator prepares detailed shop drawings of every part it will make, and the structural engineer reviews them. Havens's drawings, showing the double-rod connection, went back to the engineers. Their firm's own procedures called for a detailed check of all special connections, but Duncan did not review the fourth-floor box beam connection shown on them, and the drawings were approved.
• The technician's question. It was raised and answered with "basically the same", without the calculations that, as the court put it, it was a reasonable inference were never made.
• The roof collapse. While the hotel was being built, part of the atrium roof collapsed, from poor construction workmanship. The owner and architect asked the engineers to check the design of the entire atrium. Their report said, "we then checked the suspended bridges and found them to be satisfactory." The commission found they had checked only the atrium roof steel.

Who was responsible for a connection the fabricator had detailed? The court was clear: an engineer may, by custom, let the fabricator design connections, but "the adequacy of the connection design remains the responsibility of the structural engineer." In February 1984 the Missouri Board for Architects, Professional Engineers and Land Surveyors brought a complaint. After twenty-seven days of hearings, the state's Administrative Hearing Commission found Duncan grossly negligent and guilty of misconduct for assuring the architects that the connection was safe without the calculations to know it; it found Jack Gillum, the engineer whose professional seal was on the drawings, responsible for Duncan's work and himself grossly negligent; and it held their firm responsible for both. The Board revoked all three licences. The Missouri Court of Appeals affirmed on 26 January 1988, reversing only one finding of misconduct against Gillum concerning the atrium review.

28 January 1986: 36 degrees on the launch pad

Challenger's crew were Francis R. (Dick) Scobee, Michael J. Smith, Ellison S. Onizuka, Judith A. Resnik, Ronald E. McNair, S. Christa McAuliffe and Gregory B. Jarvis. The Presidential Commission on the Space Shuttle Challenger Accident, chaired by William P. Rogers, reported on 6 June 1986. It concluded that "the cause of the Challenger accident was the failure of the pressure seal in the aft field joint of the right Solid Rocket Motor. The failure was due to a faulty design unacceptably sensitive to a number of factors": the effects of temperature, physical dimensions, the character of materials, the effects of reusability, processing, and the reaction of the joint to dynamic loading.

Each solid rocket booster was built in segments, and the joints between them were sealed by rubber O-rings. When the motor ignites, the joint flexes and its gap opens slightly, and the O-ring has to spring back fast enough to keep sealing it. Cold makes rubber slow. The Commission found that a compressed O-ring at 75 degrees Fahrenheit "is five times more responsive in returning to its uncompressed shape than a cold O-ring at 30 degrees Fahrenheit." The air temperature at launch was 36 degrees Fahrenheit. A leak of hot combustion gas through the right booster's aft field joint began at or shortly after ignition, and it weakened or penetrated the external fuel tank, starting the breakup of the vehicle.

Wrong answer 5: "The cold was a surprise nobody could have foreseen"

The Commission's chapter on the history of the joint is titled "An Accident Rooted in History." The first in-flight erosion of a primary O-ring was found after the second shuttle flight, in November 1981. As damage turned up on later flights, the report says, NASA and Thiokol, the booster's maker, responded by working to "increase the amount of damage considered 'acceptable,'" and Thiokol's stated position became that "the condition is not desirable but is acceptable." On 31 July 1985 a Thiokol engineer, Roger Boisjoly, wrote to his vice president of engineering, Robert Lund, that if a field joint failed the result could be "a catastrophe of the highest order - loss of human life." The physicist Richard Feynman, a member of the Commission, put the logic of acceptance into one sentence in his appendix to the report: "The fact that this danger did not lead to a catastrophe before is no guarantee that it will not the next time."

Wrong answer 6: "The data showed no link with temperature"

On the night of 27 January the engineers and managers did look at O-ring damage against temperature, and the comparison they used seemed to show nothing. The Commission explains why. "The managers compared as a function of temperature the flights for which thermal distress of O-rings had been observed, not the frequency of occurrence based on all flights." Plot only the flights with damage, and they are spread across joint temperatures from 53 to 75 degrees Fahrenheit, damage at the warm end as well as the cool end, with "nothing irregular" in the pattern.

The flights with no damage are data too. Put them back and the picture changes completely. Here are the Commission's own counts:

Every cold flight had been damaged; most warm flights had not. "Consideration of the entire launch temperature history", the Commission wrote, "indicates that the probability of O-ring distress is increased to almost a certainty if the temperature of the joint is less than 65." Challenger launched with the air at 36 degrees, colder than any flight in the table. The mistake is the one lesson 7 warned about in another form: a number without its context. Seven flights with damage means nothing until you know how many flights there were at each temperature. The denominator, and the zeros in it, carried the whole answer.

Remember: when you look for a cause, count the cases where it did not happen as carefully as the cases where it did.

Wrong answer 7: "The engineers never warned anyone"

They did, the night before. In a teleconference between Thiokol and NASA's Marshall Space Flight Center on 27 January, Thiokol's engineers presented charts, with Lund presenting the conclusion, and recommended against launching below 53 degrees, the coldest launch in their experience. As Boisjoly testified, "The conclusion was we should not fly outside of our data base, which was 53 degrees." Lund testified that "Mr. Mulloy said he did not accept that, and Mr. Hardy said he was appalled that we would make such a recommendation", referring to Lawrence Mulloy and George Hardy of Marshall. Mulloy later explained his concern that a new 53-degree limit meant "we may not be able to launch until next April."

Thiokol's managers asked for time to talk among themselves. Boisjoly testified that during that caucus Jerald Mason, a senior vice president, "turned to Bob Lund and asked him to take off his engineering hat and put on his management hat." Lund described what had changed: "We had to prove to them that we weren't ready." Thiokol's managers reversed the recommendation, and Joe Kilminster signed a written recommendation to launch, faxed late that night. Allan McDonald, Thiokol's representative at the launch site, argued for delay and was told, the report records, that it was not his concern. The people who made the final decision to launch were never told. The Commission's chapter opens: "The decision to launch the Challenger was flawed. Those who made that decision were unaware of the recent history of problems concerning the O-rings and the joint and were unaware of the initial written recommendation of the contractor advising against the launch at temperatures below 53 degrees Fahrenheit and the continuing opposition of the engineers at Thiokol after the management reversed its position." Its finding was that Thiokol management "reversed its position and recommended the launch of 51-L, at the urging of Marshall and contrary to the views of its engineers in order to accommodate a major customer."

The same failure under both

Read across the rows. In both cases the weakness was in the design before anything changed. In both, someone raised a concern in the right place. And in both, the concern was answered by a judgment that no calculation or complete data set supported, and the burden of proof had quietly reversed: instead of the design having to be shown safe, the person worried had to show it unsafe.

Common misconceptions

• "An engineer's stamp means somebody checked every detail." Gillum's seal was on drawings he had not reviewed, and the commission found him grossly negligent for exactly that. A seal is a promise of responsibility, and it means nothing unless the work behind it was done.
• "If the fabricator designs a connection, the fabricator is responsible for it." The court held that the adequacy of the connection design remains the responsibility of the structural engineer.
• "It worked last time, so it is safe." Earlier flights survived O-ring erosion, and that history was used as evidence of safety. Feynman's point was that it was evidence of luck.
• "Disasters like these come from one person's mistake." Both investigations found chains: a weak original design, a change or a condition that made it worse, skipped checks, and a warning that went unheeded or never travelled upward.

The takeaway

The Hyatt Regency walkways fell on 17 July 1981, killing 114 people. As designed, one rod carried both walkways and the nut under the fourth-floor box beam carried only that walkway's load P; as built, with two rods, the same nut carried 2P. The rods did not fail; the connections did, and NBS found them at 60 percent of the Kansas City Building Code's required capacity as designed and 31 percent as built, close to the half that doubling predicts. A technician's question was answered with "basically the same", the shop drawings went unreviewed, and a promised check of the bridges covered only the roof. Missouri revoked the licences of the engineers and their firm, and the court held that the engineer stays responsible for a connection the fabricator details. Challenger broke apart 73 seconds after launch on 28 January 1986 at 36 °F, when a booster field joint's O-rings, five times less responsive at 30 °F than at 75 °F, failed to seal. O-ring damage had been accepted for years; the night before, the comparison that showed no temperature effect left out the undamaged flights, which put back gave 3 of 20 warm flights and 4 of 4 cold flights damaged; and Thiokol's engineers' recommendation against launch below 53 °F was reversed by their managers and never reached the people who decided.

Remember: a safety question is answered by a calculation or a complete set of data, never by "basically the same" or "it worked last time", and the person who raises it should not have to prove the danger; the design has to prove it is safe.

Sources

1. Marshall, R. D., Pfrang, E. O., Leyendecker, E. V., Woodward, K. A., Reed, R. P., Kasen, M. B., & Shives, T. R. (1982). Investigation of the Kansas City Hyatt Regency walkways collapse (NBS Building Science Series 143). National Bureau of Standards. NIST
2. National Institute of Standards and Technology. (n.d.). Walkway collapse, Kansas City, Missouri, 1981. NIST
3. Duncan v. Missouri Board for Architects, Professional Engineers and Land Surveyors, 744 S.W.2d 524 (Mo. Ct. App. 1988). CourtListener
4. Presidential Commission on the Space Shuttle Challenger Accident. (1986). Report to the President by the Presidential Commission on the Space Shuttle Challenger Accident (Vol. 1, chapters IV to VI). NASA
5. Feynman, R. P. (1986). Personal observations on the reliability of the Shuttle. In Report to the President by the Presidential Commission on the Space Shuttle Challenger Accident (Vol. 2, Appendix F). NASA

46 grams a day: where one capstone started

For seven mornings, one student weighed the family's pot of basil on the kitchen scale at 7 a.m., before anyone watered it. The pot lost 41, 46, 52, 38, 55, 44 and 46 grams in the seven days, an average of 322 / 7 = 46 grams of water a day. (This lesson follows that student's capstone as a worked example; its numbers are illustrative, of the kind your own measurements will give.) The family was going away for ten days in the summer, the neighbour who usually watered the plant would be away too, and the basil had died the previous summer. That is the start of a capstone: a real problem, a real user, and a number you measured yourself before designing anything.

The capstone is your own project, run through the whole design process this course has taught, and it ends in a design report like the one lesson 26 shows for the reading light. The Next Generation Science Standards, used by many US states, describe the skill it practises: to design a solution to a complex real-world problem "by breaking it down into smaller, more manageable problems that can be solved through engineering," and to evaluate it "based on prioritized criteria and trade-offs" that account for cost, safety, reliability and aesthetics. This lesson is the procedure for the first half: choosing, defining and planning. It walks the basil student's project through eight steps, then changes one fact and shows the choice flip.

Step 1: list problems you can measure

Start with complaints, not inventions, the way lesson 2 started with "Your light keeps me awake". Spend a week noticing small, real problems at home or school that annoy someone. Then keep only those where something can be measured with what you have: a ruler, a kitchen scale, a timer, a thermometer, a phone light meter. The basil student's list had four:

• A. A planter that keeps the basil watered for ten days while the family is away.
• B. A phone stand that holds a phone at eye level for a grandparent's video calls, who now props the phone against a mug and shows the ceiling.
• C. A draught stopper for the gap under a bedroom door that lets cold air in.
• D. A pad under the grandparent's pillow that vibrates when the smoke alarm sounds, because the grandparent is hard of hearing.

Step 2: screen every candidate against the capstone's constraints

As in lesson 3, constraints come first and are pass or fail. Every capstone in this course has the same five:

• K1 Safety. It can be built and tested safely at home or in a school lab with ordinary supervision: no mains electricity, no hazardous chemicals, power tools only under supervision in a workshop, and nothing whose failure could hurt someone.
• K2 Cost. Materials cost no more than $20.
• K3 Time. The whole project, including at least two rounds of testing, fits in the time available: here six weeks, 42 days, at about four hours a week, 24 hours of work.
• K4 A user. Someone other than you will use it and can judge it.
• K5 A measurement. At least one requirement can be measured with tools you have.

A, B and C pass all five. D fails K1, and not on a technicality. A device that someone's life might depend on, in a fire, at night, must be a tested and certified product, because if a homemade one fails, the person it was meant to protect sleeps through the alarm. A problem like D is real, and the right engineering answer is to buy the right product, not to make a capstone of it.

Step 3: score the survivors

Set the weights before scoring anything, as lesson 3 insisted, with a reason for each. The basil student chose: measurable results, 30, because a capstone is judged on its data; a real user who will test it, 25; uses what the course taught, 20; chance of finishing well, 15; and the student's own interest, 10. Scores run from 1 to 5.

A leads B by 25 points of 500, 5 percent: a real lead, though not a large one. C's low score on measurability is the lesson of lesson 6 applied to choosing a project: if your instrument cannot resolve the effect, your project cannot show that it worked. The student chose A.

A second run: three weeks instead of six

Now change one fact. Suppose exams move and the capstone must be finished in three weeks, 21 days. Go back to step 2, because time is a constraint. The planter's key test cannot be hurried: a plant uses its water at its own rate, and checking that a reservoir lasts ten days takes ten days, however hard you work. Count its calendar: 7 days measuring water use, 2 building, 10 for the first test, 2 rebuilding, 10 for the second test, 3 writing up, 34 days in all. Thirty-four is more than 21, so A now fails K3 and is out before any scoring. The phone stand's tests take minutes: 3 days researching, 3 building, 1 testing, 2 rebuilding, 1 retesting, 3 writing, 13 days. B passes, beats C by 405 to 310, and becomes the project.

The point: count a project's calendar time as well as its hours, because some tests take as long as they take, and a constraint that a project fails removes it, whatever its score.

Step 4: write the brief

Back to six weeks and the planter. Lesson 2's five parts, with numbers:

Two of those numbers came from measurements and arithmetic. For R1, the student weighed the pot freshly watered, 720 g, and again when the top 2 cm of soil felt dry to a finger, 610 g. The plant is in trouble somewhere below 610 g, so 640 g leaves a margin before trouble starts. For R2: 46 g a day for 10 days is 460 g, and since 1 g of water is 1 mL, that is 460 mL. The wettest day used 55 g, and 55 × 10 = 550 g, so the average alone has no margin for a hot spell. Multiplying by 1.5, a working margin like the factors of safety in lesson 9, gives 460 × 1.5 = 690 mL, which also covers ten days at the highest rate measured. Notice too what the brief does not say: no wick, no bottle, no pump. Those are answers, and the brief is the question.

Step 5: break the work into tasks and give them hours

A large job becomes manageable when it is broken into tasks small enough to estimate, a list engineers call a work breakdown:

About 21 hours of the 24 available leaves 3 hours spare, about one hour in eight. Keep it. In every project something takes longer than planned, and spare time written into the plan is the cheapest way to absorb it.

Step 6: put the tasks on a calendar with milestones

A schedule laid out as bars against weeks is called a Gantt chart. A table does the same job:

Read the schedule for its longest chain. The two ten-day tests and the week of measuring fill most of the six weeks, so they decide when the project can finish, and anything that delays them delays everything. That is why the wick test, a cheap experiment with three wicks in three jars, comes early in week 2: if no wick can lift 46 g a day, the student needs to know before prototype 1, not after test 1.

Step 7: list the risks before they happen

NASA's Systems Engineering Handbook describes a risk as a set of three things: a scenario that could go wrong, its likelihood, and its consequence, and it tells the technical team to keep identifying risks on a continuing basis, writing each as a clear risk statement. For a capstone, score likelihood and consequence from 1 (low) to 3 (high) and multiply, so the scores run from 1 to 9:

Deal with the highest scores first. The 6 goes first because it would end the project late, after a ten-day test. Notice that two of the five risks are not about the design at all: a capstone can fail on its plant, its calendar or its cut finger as easily as on its engineering.

Step 8: write the test plan before building anything

Every requirement gets a test, a pass mark and a record, written now, while you cannot yet know the results. NASA's handbook names four ways to verify a requirement: analysis, by calculation; demonstration, by showing that it works; inspection, by looking at the product; and test, using the product to obtain detailed data.

• R1, pot mass. Test: weigh the pot alone at 7 a.m. on each of 10 days, with no watering; pass if every reading is at least 640 g. Run the control plant beside it.
• R2, capacity. Analysis from the container's size, then test: pour in measured water to the fill line; pass at 690 mL or more.
• R3, refill. Demonstration: the parent refills it, timed, without moving the plant; pass under 2 minutes.
• R4, look. The parent's rating out of 5 at the end of test 2; pass at 4 or more.
• C3 and C4. Inspection with a ruler; the paper towel under the planter, checked daily.

Writing the pass marks first protects you from yourself. A student who sees a pot fall to 625 g on day 9 is tempted to decide that 625 g was fine after all. The plan, written in week 1, says 640.

What the reading light's plan looked like

The reading light in this course went through the same steps, spread across many lessons, and lesson 26 shows the report that resulted. Use it as your map:

Common misconceptions

• "The best capstone is the most impressive one." The best is the one whose results you can measure and whose user can judge it. A modest planter with ten days of weighings beats an ambitious robot that is never tested.
• "Planning is time taken from building." The basil student's schedule showed that the tests, not the building, set the finishing date. Without the plan, the first ten-day test would have started in week 4 and there would have been no time to improve anything.
• "Risks are for big projects." The planter's top risk, a wick too weak to lift 46 g a day, costs ten days if discovered in test 1 and one afternoon if discovered in a jar.
• "You can set the pass marks once you see the results." Then every result passes. Pass marks written in advance are what make a test a test.

What to carry forward

One student's basil lost an average of 46 g of water a day over 7 days, a measured number that started a capstone. The procedure runs in eight steps: list problems you can measure; screen them against fixed constraints, safety, $20, the time including two test rounds, a user and a measurement; score the survivors with weights set beforehand; write the brief with numbers and verification methods; break the work into tasks with hours; schedule them with milestones and spare time; score the risks by likelihood and consequence and act on the highest first; and write the test plan, with pass marks, before building. A smoke-alarm pad failed the safety constraint outright. The planter won the matrix 430 to 405, needed a 690 mL reservoir (46 × 10 × 1.5), about 21 of 24 hours and 34 of 42 calendar days, and lost to the phone stand when three weeks left only 21 days for its 34.

The point: choose a problem you can measure for a user who can judge it, write its numbers and its tests before you build, and plan by the calendar the tests need as well as by the hours you have.

Sources

1. NGSS Lead States. (2013). Next Generation Science Standards: HS-ETS1 Engineering Design. Achieve. NGSS
2. National Aeronautics and Space Administration. (2016). NASA systems engineering handbook (NASA/SP-2016-6105 Rev2), section 6.4, Technical risk management. NASA
3. National Aeronautics and Space Administration. (2016). NASA systems engineering handbook (NASA/SP-2016-6105 Rev2), section 5.3, Product verification. NASA
4. National Aeronautics and Space Administration. (2016). NASA systems engineering handbook (NASA/SP-2016-6105 Rev2), Appendix C, How to write a good requirement. NASA
5. Wikipedia contributors. (2026). Gantt chart. In Wikipedia, The Free Encyclopedia. Wikipedia

288 lux in the bottom right corner

On the first full test of the finished reading light, the phone light meter read 288 lux at the bottom right corner of an A4 page: 12 lux short of the 300 lux that requirement R1 demands at every one of five points. (The test results in this lesson are illustrative, of the kind the finished lamp would give; the calculations behind them are worked from the earlier lessons.) Lesson 19 had warned that this might happen. The transistor that switches the LEDs takes a few tenths of a volt for itself, so each LED's current comes out below lesson 16's 16.7 mA, and "the light test in lesson 26 checks that the page still gets enough." It did not.

That reading is the most useful number in this lesson, for the reason lesson 4 gave: a test that disagrees with your prediction is the only kind that can tell you something you did not already believe. This lesson does three things with the finished lamp. It compares the first full test with the final one, requirement by requirement, and reads what changed in between. It compares the two kinds of checking every design needs, verification and validation. And it lays out the reading light's complete design report, section by section, as the model for yours.

Two tests of one lamp, requirement by requirement

Here is the whole brief from lesson 2, tested twice: once on the first complete lamp, and once on the final lamp after the changes this lesson describes.

Reading the R1 row: a transistor's 0.2 volts

Start with the failure, and with arithmetic, not guesses. Measured with the meter across it, the transistor took 0.2 V. Each LED branch now shares the 5.0 V supply among the LED's 3.0 V, the transistor's 0.2 V and the resistor, so with lesson 16's 120 Ω the current is (5.0 − 3.0 − 0.2) / 120 = 0.0150 A, 15.0 mA instead of 16.7 mA, 90 percent as much. SparkFun's guide to LEDs puts the dependence plainly: the brightness of an LED is directly dependent on how much current it draws. Taking the light as roughly proportional to the current, a first approximation that the retest will check, the page loses about 10 percent of its light, and a corner that would have been about 320 lux becomes 288.

The fix is lesson 16's calculation done again with the new voltage in it. The worst case is still a high supply and a low forward voltage, so the smallest safe resistor is (5.25 − 3.0 − 0.2) / 0.020 = 102.5 Ω. The standard value 100 Ω is too small, because at 5.25 V it would let (5.25 − 3.0 − 0.2) / 100 = 20.5 mA through, over the LED's 20 mA maximum. The next standard value up is 110 Ω, which gives a typical (5.0 − 3.0 − 0.2) / 110 = 16.4 mA and a worst case of 18.6 mA. Prediction: every reading rises by 16.4 / 15.0 = 1.091, so the corners become 323, 328, 314 and 333 lux. Measured: 321, 327, 312 and 330. The prediction was within 1 percent, which also confirms the proportional approximation over this small range.

Notice that exactly one thing changed: four resistors, from 120 Ω to 110 Ω, costing $0.40, which is where C2's $13.60 comes from. If the hood lining had been changed at the same time, the 24 lux gained at the worst corner could not have been shared out between the two changes, the trap lesson 4 described.

Reading the other rows

R3 shows why a requirement verified by analysis still needs a test. With 110 Ω the LEDs draw 4 × 16.4 = 65.5 mA and the board 45 mA, 110.5 mA in all, 5.0 × 0.1105 = 0.552 W, so lesson 17's 12.4 Wh predicts 12.4 / 0.552 = 22.5 hours. The timed test gave 21.5 hours, 4 percent less. The likeliest reason is that the bank delivered about 64 percent of its stored energy rather than the two-thirds lesson 17 assumed, and the test also settled lesson 17's other worry: this bank kept supplying a 110 mA load without switching itself off. A pass by 8 percent is a pass, and the report should say how slim it is.

R4 holds a trap of its own. The five first-time users' times average (1.9 + 2.4 + 4.6 + 2.1 + 2.8) / 5 = 2.76 seconds, under 3. But the requirement says a first-time user shall find and operate the switch within 3 seconds, and one user took 4.6, groping along the back of the base where the push button sat. That is a fail, and averaging it away would have hidden the one result that mattered. The change: the button moved to the front edge of the base, where a hand reaching in the dark lands first, with a raised ring of card around it to be found by touch. Five new first-time users all succeeded, the slowest in 2.2 seconds.

R5 and the constraints passed both times, and the report says so in one line each. A passed row still gets its number: 309 g, not "light enough."

The core of it: test every requirement against the pass mark written before the test, read each failure with the calculation that predicts it, change one thing, and show the effect with numbers on both sides.

Verification is not validation

Every row in that table passed in the end, and the lamp could still have failed. NASA's Systems Engineering Handbook separates the two kinds of checking, verification and validation, with two questions. Verification asks, "Was the end product realized right?": does it meet its requirements? Validation asks, "Was the right end product realized?": does it do what the users need in the place where they will use it, a check the handbook says should be done with the anticipated operators or users whenever possible.

The validation week found two things no requirement had covered. On the first night the sleeper reported "a tiny red dot" visible from the bed: the development board's power indicator LED, shining through a gap at the back of the base. It was far too faint for the phone meter, which cannot resolve less than 1 lux, and perfectly visible to a dark-adapted eye in a dark room. A square of black tape over it fixed it, and the sleeper slept through on the other six nights. And once, the two-hour timer of lesson 19 switched the lamp off in the middle of homework. The reader pressed the button twice to get it back, which works, but the report recommends a single blink a minute before the timer ends as a warning. Both findings came from users doing the real job, which is why validation is done with them and not on the bench.

The reading light: the complete design report

Here is the report the whole course has been building, section by section. Yours follows the same plan.

Title. A reading light for a shared bedroom: design report. Author, date, version.

Summary. Two people share a bedroom; one reads until 10:30 p.m. while the other sleeps about 2 m away. The brief required at least 300 lux at five points of an A4 page, a rise of no more than 1 lux at the pillow, at least 20 hours per charge, a switch a first-time user finds in 3 seconds in the dark, a mass under 400 g, battery or USB power only, materials under $15, a 30 cm by 20 cm footprint and no surface above 40 °C. Of four concepts, a lamp with a deep hood on a hinged arm scored 440 of 500 in a weighted matrix. The final lamp uses four white LEDs, each with a 110 Ω resistor, switched by a microcontroller that turns the lamp on at dusk and off after two hours. It meets every requirement: 312 lux at the dimmest point, 0 lux measurable at the pillow, 21.5 hours per charge, 2.2 seconds for the slowest first-time user, 309 g and $13.60. A week's use by both people confirmed it after one fix. The main limitation is that page brightness depends on the supply voltage.

1. The problem and the brief. The complaint, the measured room, the need statement, the stakeholders, and the brief's table of constraints, criteria and verification methods from lesson 2, including the estimate that a bare lamp giving 300 lux on the page would put 12 lux on the pillow, which made shielding the central problem.

2. Concepts and choice. Four concepts; the constraint screen that removed the floor screen; the weighted matrix, 440 against 375 and 290; the sensitivity check; and the reason the hooded lamp went forward, from lesson 3.

3. The design. The three-view drawing of the hood at 1:2, 80 mm tall, 40 mm across the top and 140 mm across the rim, with its 40 mm visor on the bed side (lesson 5). The arm joint: a 5.10 ± 0.10 mm hole on a pin measured at 4.83 ± 0.06 mm, for clearance between 0.11 and 0.43 mm (lesson 6). The arm: an 8 mm pine dowel, chosen from five materials for its 3 mm droop and 11 g (lesson 8). The base: widened to 200 mm so that the stability ratio with the arm swung sideways is 1.8 (lesson 10). The friction joint: a wing nut clamping with 26.8 N, twice the force needed to hold 0.193 N·m (lesson 13). The circuit: four parallel branches with a resistor each (lessons 16 and 26). The energy budget (lesson 17), the light sensor's thresholds of 300 and 400 (lesson 18), and the program with its two-hour timer (lesson 19). Each calculation appears once, with its units, and each assumption is stated where it is used.

4. Verification. The table above: every requirement, its method, its pass mark and its result, including the first results that failed.

5. Iterations. One entry per change, each with the reason, the single thing changed and the before-and-after numbers: the card hood's four versions (lesson 4), the wider base (lesson 10), the tightened joint (lesson 13), the sensor moved to the back of the base (lesson 18), the master switch after the three-day flat battery (lesson 19), the 110 Ω resistors and the moved button (this lesson).

6. Validation. The week of use: the sleeper's nights, the reader's rating of 4 out of 5, the red indicator light and its fix, and the timer that ended mid-homework.

7. Cost and life cycle. $13.60 for the prototype's materials; for a product, about $13.70 a lamp at 10,000 with moulds that pay only above about 9,000 lamps (lesson 21); and the open question of whether the power bank should be built in, sealed, replaceable or left out, with the measurement that would settle it (lesson 22).

8. Limitations and recommendations. R1 passes by 4 percent with this bank and these LEDs. With a supply at the bottom of the USB range, 4.75 V, and LEDs at the top of their forward-voltage range, 3.4 V, the current would fall to (4.75 − 3.4 − 0.2) / 110 = 10.5 mA and the dimmest corner to roughly 200 lux; a constant-current LED driver would remove that sensitivity. R3 passes by only 8 percent, most of it lost to the development board's 45 mA, so a product should use a bare microcontroller chip. A warning blink before the timer ends. A replaceable battery, as lesson 22 argued.

9. References. Every source cited in the report, in one style.

Appendices. The test logs with dates, the program listing, the full drawings and the receipts.

How to write a result

The same measurement can be reported in ways that range from useless to complete:

The complete version names the requirement, the pass mark, the method, the number and the verdict. The middle column is the commonest mistake in student reports, and the R4 row shows why it is dangerous: the average was true and the conclusion it suggests was false.

Common misconceptions

• "If every requirement passes, the design is finished." The reading light passed every requirement and still had a red light visible from the bed. Validation with the real users in the real place is a separate check.
• "A failed test means a failed project." The 288 lux reading led directly to the 110 Ω resistors and a measured 312. Reports with no failures in them usually describe projects that were not tested hard.
• "Change several things at once to save a test." Then the data cannot say which change did what. One change per version is slower on paper and faster in practice.
• "A report should show the design at its best." A report is a record someone else will rely on. Its limitations section, the 4 percent margin and the supply voltage it depends on, is the part a reader needs most.
• "An average shows whether a requirement passed." Only if the requirement is about an average. R4 was about every first-time user, and the average hid the one who failed.

Looking back

The finished reading light's first full test read 288 lux at one corner, because the switching transistor took 0.2 V and cut each LED's current from 16.7 to 15.0 mA. Resizing for the worst case, (5.25 − 3.0 − 0.2) / 0.020 = 102.5 Ω, gave 110 Ω, a predicted rise of 9.1 percent and a measured dimmest corner of 312 lux. R3 predicted 22.5 hours and measured 21.5; R4's average of 2.76 seconds hid a 4.6-second failure, fixed by moving the button; R5 was 309 g, C2 $13.60 and C4 29 °C. Verification asked whether the lamp was built right; validation, a week with the reader and the sleeper, asked whether it was the right lamp, and found a red indicator light and a timer that cut out mid-homework. The design report sets it all out in order: a summary, the brief, the concepts and choice, the design with its calculations, verification, iterations, validation, cost and life cycle, limitations and recommendations, references and appendices. Your capstone report follows the same plan.

The core of it: a design is finished when it has been tested against every requirement, checked with its real users, changed one thing at a time where it failed, and written up so that someone else could trust every number in it, including the ones that show how close it came to failing.

Sources

1. National Aeronautics and Space Administration. (2016). NASA systems engineering handbook (NASA/SP-2016-6105 Rev2), section 5.3, Product verification. NASA
2. National Aeronautics and Space Administration. (2016). NASA systems engineering handbook (NASA/SP-2016-6105 Rev2), section 5.4, Product validation. NASA
3. SparkFun Electronics. (n.d.). Light-emitting diodes (LEDs). SparkFun Learn. SparkFun
4. NGSS Lead States. (2013). Next Generation Science Standards: HS-ETS1 Engineering Design. Achieve. NGSS