The Bookkeeping That Always Balances
1. Work is a narrower word than you think
Hold a heavy barbell perfectly still above your head. You are shaking, your arms are burning, and you are doing no work on the barbell at all. In physics you do work on something only when your force actually moves it, and only the part of the motion that goes along your push counts.
That will annoy people, and it should be allowed to for a minute before it is resolved. The weightlifter is certainly using energy: her muscles are twitching and rebuilding tension thousands of times a second and turning most of it into heat. She is doing a great deal of work inside her own body. She is doing none on the bar, because the bar is not going anywhere.
Carrying a suitcase along a level corridor is the same trick in a different shape. You hold it up, which is a vertical force, and you walk, which is horizontal motion. The two have nothing to do with each other, so no work is done on the suitcase however tired your arm gets.
2. Lift something and you have banked the work
Do work against gravity and it is not gone. It is stored, ready to be handed back, and the store is called potential energy. How much you have depends on how heavy the thing is and how far you raised it, and on nothing else at all.
That last clause is the useful one. Take a long gentle ramp or take a ladder, and the store at the top is identical. What the ramp saved you was force, not energy: you pushed less hard, over a longer distance, and did the same total work. Anyone who says the ramp saved energy has found the misconception this chapter exists to break.
A crate going up to the same loading bay two ways. Change how long the ramp is and watch the push you need fall away. The two bars underneath are the work done and the store at the top, and neither one moves however gentle you make the slope.
Gravity is not the only place to keep a store. A drawn bow, a wound spring, a compressed gas, a battery and the food you had at lunch are all stores of energy of position, in the sense that something is being held somewhere it would rather not be. Water behind a dam is the one people picture, and it is exactly right.
3. Energy of motion, and why speed counts twice
Something already moving carries energy too. Get in its way and it will do work on you, which is a useful way to think about why being hit hurts.
Here mass and speed do not count equally, and this is the one place in the chapter where that matters more than anything else. Double the mass and you double the energy. Double the speed and you get four times as much, because the speed enters the account twice over.
The everyday version is braking distance. A car at 60 does not need twice the road that a car at 30 needs. It needs about four times as much. From 90 it needs about nine times. Every increase in speed costs more than it looks like it should, and that single sentence is worth more to a seventeen-year-old than anything else on this page.
4. Following it through
Energy is never made and never destroyed. It only moves around and changes form. That sounds like a slogan until you notice what it lets you do, which is follow it: out of a store, into motion, into warmth and sound, and the books still balance at the end.
So a ball at the top of a ramp has a full store and no motion. Partway down it has some of each. At the bottom it is nearly all motion, minus whatever friction has already turned into heat. And a ball that stops has not lost its energy; it has warmed the floor and made a noise, and if you could measure carefully enough you would find every bit of it.
A ball let go at the top of a ramp, round the loop and along to a stop. Drag the position slider to walk it through, and watch the three bars trade with each other. The total is the whole height of the stack, and it never changes.
Two things are worth finding on purpose. Set the roughness to zero and walk the ball round: the grey bar never appears, and the ball arrives at the far end still moving. Then set the start height low and try to get it round the loop. It will not go, and the bars say why before the picture does.
The demonstration to remember is the pendulum. A heavy bob is released from the tip of your nose and swings away and back, and it returns to just short of your nose every time, because it cannot arrive with more energy than it left with. Do not push it. Pushing it is the one way to make the demonstration a lie, and it is also how the injury happens.
5. Machines trade force for distance
A lever, a ramp, a pulley and a gearbox all do the same thing: they let you apply a small force where a large one was needed. What none of them do is make energy, and the trade is always honest. Quarter the force and you move about four times the distance. Push a piano up a ramp with a third of the force and you push it three times as far.
Then friction takes its cut, so what comes out is always a little less than what went in. Efficiency is the name for how much survived the trip, and it is why perpetual motion is not a clever engineering problem waiting for a clever engineer. A machine that ran itself forever would have to give out at least as much as it took in, and no machine does.
Check yourself
1. For each of these, say whether work is done on the object and why: pushing a wall that does not move; lifting a box onto a shelf; carrying that box across the room at a steady walk; lowering it back down.
The wall: none, because nothing moved. The lift: yes, force up and motion up. The carry: none on the box, because the motion is sideways and the force holding it is vertical. The lowering: yes, and this time the force and the motion point opposite ways, so the box gives energy back rather than taking it. That last case is worth saying out loud even though nothing has to be calculated.
2. Two identical crates reach the same loading bay, one hauled up a long gentle ramp and one lifted straight up. Compare the store at the top, and say what the ramp actually saved.
The stores are identical, because both crates ended at the same height and the store depends on nothing else. The ramp saved force, not energy: a smaller push over a longer distance is the same total work. Anyone who says the ramp saved energy should be asked what happened to the extra distance.
3. Rank these by kinetic energy, most first, and justify the order without arithmetic beyond doubling: a 2 kg ball at 1 m/s; a 1 kg ball at 2 m/s; a 1 kg ball at 1 m/s; a 2 kg ball at 2 m/s.
The 2 kg ball at 2 m/s first, on its own. Then the 1 kg ball at 2 m/s, then the 2 kg ball at 1 m/s, then the 1 kg ball at 1 m/s last. The middle two are the whole question, and they are not a tie: doubling the speed beats doubling the mass, because speed counts twice and mass counts once.
4. A dropped ball never bounces back to the height you dropped it from. Say where the missing energy went, and say why that does not count as destroying it.
Into heat, in the ball and in the floor, and into the sound of the bounce. The ball really is very slightly warmer afterwards. Nothing was destroyed, because you could in principle account for every bit of it as warmth and noise. Missing and destroyed are different claims, and only one of them is ever true.
5. A student says a pulley system gives you extra energy, because it lets you lift something you could never lift on your own. Say what they have confused with what.
They have confused force with energy. The pulley gives you more force and takes back more rope pulled through your hands, and the two cancel out to the same work, minus whatever friction took. Nothing was added; it was rearranged. This is the same answer as the ramp in question two, arriving from a different direction.
Chapter 8, Rotational Motion. Momentum and energy both came out of asking what happens when things move in straight lines. The next chapter asks the same questions about things that spin, and gets answers that are recognizably the same shape.