Energy transfers · Model
Conservation of energy
A pendulum comes back almost as high, but only almost, and after a few hundred swings it hangs straight down. Energy is supposed to be conserved. What happened?
Start here
The swing that gives up.
Pull a pendulum to one side and let go. It comes back almost as high — but only almost. Every swing is a little lower than the last, and after a few hundred swings it is hanging straight down, motionless.
Commit. Energy is supposed to be conserved. What happened?
Nothing happened to the total. Every joule that left the swinging is now in the air of the room and the pivot of the pendulum, as a rise in temperature far too small to feel. The pendulum stopped; the energy did not. This lesson gives you an instrument that keeps score, so you can watch that being true instead of taking it on trust.
Conservation of energy is the most tested claim in physics and it has never once failed. It is not a rule about how machines behave, or an ideal that real systems fall short of — it is exact, every time, with no exceptions found in two hundred years of looking. What makes it feel false is that the stores energy ends up in are usually invisible. So the instrument below shows them all.
The running total · keep score through the whole swing
Three stores. One total that never moves.
Release the pendulum and watch the top of the bar rather than the bob. Then try hiding the thermal store, and see what the law looks like without it.
Commit first. At the very bottom of the swing, which store holds the most?
Gravitational
Kinetic
Thermal, surroundings
Total
Held at 120 J, all of it gravitational. Press start and keep your eye on the top of the bar rather than the pendulum.
Watch the two lower blocks trade places twice a swing while the orange one creeps up and never falls. Nothing takes the total past the line, and nothing lets it drop below.
Stopped — and the bar still reaches the top line. Every joule that was in the swinging is now in the thermal store of the air and the pivot. The pendulum ran down; the total did not. Nothing here can set it swinging again on its own — to go again, pull it back and release.
Friction off. The swing never dies down and nothing enters the thermal store, so gravitational and kinetic simply trade back and forth forever. No real pendulum does this — but it makes the trade visible without the complication.
With the thermal store hidden, the bar no longer reaches the total line and the law looks false. This is exactly the mistake behind “the energy was lost” — the store is real, it is just not one you would have thought to look at.
Fresh run — pulled back to the top, and the count starts at 120 J again. The room keeps the energy from the last run; none of it has come back. You put in a new 120 J by lifting the bob.
Key fact
Energy is never created and never destroyed. In a closed system the total before a change equals the total after; all that alters is which stores are holding it.
Writing it down · the shape of the relationship matters
Conservation is a balance, not a triangle
You will meet formula triangles in this course and they are genuinely useful — but only for a relationship built from multiplying and dividing. Conservation of energy is not one of those. It is a sum on each side of an equals sign, and a triangle drawn over it would teach you a relationship that does not exist.
120 J, before
the same 120 J, shared out
always level
Move energy between the stores on the right — the beam stays level
All of it in the gravitational store, at the top of the swing. The beam is level because 120 = 120 + 0 + 0.
Split between two stores, with a few joules already in the air. Still level: 62 + 55 + 3 is still 120. Notice the beam does not care how many stores you use.
Gravitational empty, kinetic nearly full. This is the moment most people get wrong — the pendulum is lowest and fastest at the same instant.
Everything in the thermal store of the room. The pendulum is motionless and the sum is still exactly 120. Nothing about the beam has changed — which is the whole point of drawing it as a balance rather than a triangle.
Think again
“The car has stopped, so it has run out of energy.”
First: a stopped car has not run out of anything. Its kinetic store is empty, which is a different statement — its fuel tank may be full, and the brake discs are now hot enough to boil water. Second, and this is the deeper one: “ran out” treats energy as a supply that gets consumed. It is not. It is a number that moved.
You have met this exact belief before wearing different clothes. When a sealed bag of ice melted, the mass did not change — and the temptation was to say some of it had gone. Same instinct, same error: a quantity stops being visible and we conclude it has stopped existing. A balance answered it for mass. A thermometer on the brake discs answers it for energy.
“Efficient machines conserve energy and wasteful ones do not.”
Both conserve it exactly. A 20% efficient petrol engine and a 90% efficient electric motor obey the same law to the same precision — the difference is where the energy ends up, not how much of it survives. Efficiency is the fraction that arrives in the store you wanted; the rest still exists, warming the engine, the road and the air. Conservation is not something a machine can be good or bad at.
Key fact
A machine that seems to lose energy has dissipated it into thermal stores. “Lost” names your attention, not the energy.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Recall
Complete the law: energy cannot be created or destroyed, only…
Rung 2 · The one that catches people
A pendulum swings until it hangs still. What is true of the total energy at the end compared with the start?
Rung 3 · Explain
A skateboarder drops into a half-pipe from the top of one side and rises up the other. They never quite reach the height they started from. Explain what has happened to the energy, and why the law of conservation is not broken.
Rung 4 · Take it somewhere new
Someone advertises a machine with magnets that, once started, turns forever and powers a light with no fuel. Using conservation of energy, explain why it cannot work — and say what you would measure to test it.
Key note
The total is fixed. A store emptying is never a store disappearing — find where it went. “Stopped” and “out of energy” are different statements, and only the first one is ever true.
Going further
Twice in history the sum has looked broken, and both times the gap turned out to be a discovery rather than a mistake. In the 1920s energy appeared to go missing during a particular kind of radioactive decay — always a bit short, always by a varying amount. Rather than abandon the sum, Wolfgang Pauli proposed in 1930 that an undetected particle was carrying the difference away, an idea so hard to test he apologised for it. The neutrino was finally detected in 1956, twenty-six years later, and the sum balanced exactly. Conservation of energy has been trusted enough to predict a new particle from nothing but a shortfall in the arithmetic — that is how strong the evidence for it is.
Before this lesson
At GCSE this becomes
- Closed systems and dissipation quantified — efficiency as a calculated fraction, and the idea that dissipated energy is no longer useful even though it is still there.
Where to next
Ask Mr Badmus AI
Still not sure why a stopped pendulum has not run out of energy?
Lesson content © MrBadmusAI.