Module 2 · a one-hour interlude on the conservation laws
Physics has a handful of quantities that the universe refuses to alter, no matter how violent the event. This hour is about why — not the formulas, which take five minutes, but the reasons, which are stranger and better than the formulas.
Walter Lewin's air track is a rail with a line of small holes that blow air, so the carts glide on a cushion and friction is very nearly zero — which is the whole point, since friction from an ordinary table would be a push from outside the pair. Two carts, a compressed spring between them held by a thread, and then a match to the thread. Nothing pushes on the pair from outside. The spring pushes on each cart from inside.
Cart A has mass 1 kg. Cart B has mass 3 kg. When the thread burns through:
Momentum has a direction as well as a size, so we keep track of it with a sign: motion to the right counts as plus, motion to the left as minus. Cart A ends up with momentum mAvA pointing one way, cart B with mBvB pointing the other. They are equal in size and opposite in sign, so their total mAvA + mBvB is still zero — exactly what it was before the thread burned. The speeds are unequal, the energies are unequal, but the momentum ledger balances. And the centre of mass — the black tick, the balance point of the two carts taken together, always closer to the heavier one — never moves. An internal force can rearrange a system as much as it likes; it cannot shift its balance point.
Write p for momentum, so p = mv. Newton's second law is really about momentum: a force is the rate at which momentum changes — physicists write that rate dp/dt, and F = ma is the same statement, since m times the rate of change of v is the rate of change of mv. Now: the spring pushes A with some force F, so pA changes at rate F. By Newton's third law it pushes B with −F, so pB changes at rate −F. Add up the second law for both carts:
That is correct, and it is also slightly unsatisfying, because it makes momentum conservation look like a consequence of a rule about forces — and rules about forces can be wrong. Feynman had a better argument, and it is the next bench.
Two identical carts. They approach each other at the same speed, collide, and stick. What happens?
Feynman's answer is not a calculation. It is: nothing can happen. There is no reason for the stuck pair to drift left rather than right — the setup is a perfect mirror image of itself, and any drift would break the mirror. So the pair must stop dead. That is not physics yet; it is just honesty about symmetry.
The principle that no one can tell the difference between moving steadily in a straight line and standing still is one of the most powerful ideas in physics: it converts one experiment into an infinite number of experiments, all done for free.the idea of Chapter 10, Volume I, in paraphrase
Now the trick. Watch the same collision from a train rolling past at speed u. From the train, cart A seems to move at v − u and cart B at −v − u. After the collision the stuck pair, which was at rest on the ground, seems to move at −u. Nobody on the train can tell they are moving, so what they see must also be a correct experiment. Add it up:
Feynman then extends it to unequal masses without any new physics. How?
Two things to take from the ledger. A frame is just a point of view you measure from — the ground is one, the moving train is another. Momentum is conserved in every frame, but its value is different in every frame; there is no such thing as "the" momentum of a system, only its momentum relative to someone. And kinetic energy is another matter entirely: with sticking carts it is lost in every frame (it became heat and sound), with bouncing carts it is kept in every frame, but its number, too, depends on who is watching. A collision that keeps kinetic energy is called elastic; one that loses some is inelastic, and the extreme case where the carts stick and travel on as one is perfectly inelastic. How elastic a real collision is — the fraction of kinetic energy it keeps — is called its elasticity, and it is the control the PhET card in the Library asks you to slide.
Newton's third law is false in electromagnetism: two moving charges push on each other with forces that are not equal and opposite, because the field takes time to carry the message across. Yet momentum is still conserved once you count the momentum carried by the field itself. The symmetry-and-relativity argument survives that; the "forces cancel in pairs" argument does not. When two arguments give the same answer, the one that still works after the other breaks is the deeper one.
"When no external force acts" sounds like a fact about the world. It is actually a fact about a decision you made: what to count as the system. Draw the box small and forces cross its wall; draw it big enough and nothing does.
Lewin fires a ball from a cannon mounted on a cart on the air track. The cart recoils. Then he points out that if the cart were bolted to the table, the building would recoil — and it does, by an amount too small to measure but not too small to be true.
With "ball only" ticked the momentum of the system jumps from zero to something. That is not a failure of the law: the cannon is outside the box and shoved the ball across its wall. Widen the box and the jump vanishes. There was never a moment at which momentum appeared from nowhere; there was only a moment at which your box was too small to see where it came from.
You jump straight up at 3 m/s. The Earth, by exactly the same argument as Cart B, recoils downward. Its mass is about 6 × 1024 kg — some 1023 times yours. Put your mass in:
The Earth moves. Not metaphorically — its centre of mass shifts by a distance smaller than the width of a proton, then comes back when you land. The law does not care that the number is absurd; it cares that the ledger balances. This is the mindset of every conservation law: the total is fixed, so if one thing changed, something else did too, and your job is to find it.
Feynman opens his chapter on energy with a story instead of a definition. A boy has 28 indestructible blocks. Every evening his mother counts them. Some evenings she finds 27, and after searching finds one under the rug. One day only 25 are visible, but the toy box is heavier than it should be. She weighs it, subtracts the empty box, and divides what is left by the weight of one block. The answer is exactly three. Another day the bathwater is higher than usual, and a little arithmetic on the water level accounts for the rest.
It is important to realise that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount.Feynman, Volume I, Chapter 4
What we have instead is a set of formulas — ½mv² for a mass m moving at speed v (note the square: double the speed and you get four times the energy, which is why the light cart in Bench 1 walked off with most of the energy while taking exactly half the momentum), mgh for a mass held at height h, ½kx² for a stretched spring, and separate rules for heat, chemical energy, and so on — each a "weigh the toy box" rule for a place where blocks can hide. Add them all up and the total never changes. That is the whole law. The blocks are a picture of the bookkeeping; energy itself is what those formulas compute, not a substance you can hold. Try it:
The window is the important one. When Dennis throws blocks out, the count in the room drops and no formula inside the room can find them. Either the law is broken, or the box must grow to include the garden. Physics has faced that choice several times — neutrinos were discovered because energy kept going missing in radioactive decays, and Pauli decided the law was more trustworthy than the assumption that he could see everything in the room. He was right.
A power bill is a Dennis ledger. Chemical energy in, spread out as light, heat, and the motion of a fan, then all of it eventually heat in the room — same total, less useful. "Energy consumption" is a strange phrase for something that cannot be consumed; what gets used up is the arrangement, not the amount. That distinction — the amount is kept, the arrangement is lost — is where thermodynamics begins, and Module 8 will come back for it.
Lewin sits on a stool that can rotate freely, holding a heavy bicycle wheel that is already spinning. He turns the wheel over. The stool — with him on it — starts to rotate the other way. Nobody pushed the stool. The rotation had to come from somewhere, and the somewhere was the wheel.
The quantity being kept is angular momentum, L = Iω. Here ω is the spin rate — radians per second — and I, the moment of inertia, is rotation's version of mass: for each small piece of the object, take its mass times the square of its distance from the axis, and add up every piece. The square is what matters. A piece of mass twice as far from the axis counts four times as much. With no external twist — no torque — L cannot change. Spinning also has a kinetic energy: it is the ordinary ½mv² with the rotational quantities put in their places, ½Iω². Since L = Iω, you can equally write it as L²/2I — the form to reach for when L is the thing being held fixed.
A skater spinning with arms out pulls her arms in. Her spin speeds up. Her kinetic energy:
Kepler found the same law in the sky two centuries before anyone had a name for it. A planet moving on an ellipse speeds up near the Sun and slows down far from it, in exactly the way that keeps the area its radius sweeps per second constant. Two things worth pausing on. A force twists about a point only if it acts off to one side of that point — push a door at its hinge and it will not turn — so gravity, pulling the planet dead through the Sun, exerts no torque about the Sun, and L cannot change. And in one second the Sun-to-planet line sweeps a thin triangle of area ½vr⊥, which is L/2m; if L is fixed, so is that area. Equal-areas-in-equal-times is L = constant, written in geometry.
Three conservation laws so far, each with its own argument. Feynman's Chapter 52 gives the one idea underneath all three, and it is the best-kept secret in school physics: every conservation law is a symmetry in disguise.
Suppose the laws of physics were slightly different in Chennai from what they are in Delhi — say gravity pulled a little harder in Chennai. Would momentum still be conserved?
Here is the argument at the level Feynman gives it. Take an isolated system and imagine shifting the whole thing one metre to the left. If the laws are the same everywhere, nothing about the system can tell it was moved: its potential energy — the energy it has stored because of where its parts are, the way a book on a high shelf has more stored than one on a low shelf — is unchanged, the forces inside it are unchanged, everything is identical. The step that does the work here is a rule worth stating on its own: if moving something a little way changes its stored energy, something is pulling on it, and the faster the energy changes with distance, the harder the pull. Steep energy hill, big force. Flat energy, no force. That is what "force is the slope of the energy" means. So: shift the isolated system, its stored energy stays the same, the slope is flat, no net force, no change in momentum. Momentum conservation is the statement that space has no special place.
Run the same argument with rotation instead of shifting and you get angular momentum: turn the whole system to face a new direction, nothing about it changes, so no twist can come from nowhere and L is fixed. Time is the third case, and worth being honest about: the argument there is not this one over again, and doing it properly takes more machinery than we have here — but the same kind of reasoning goes through, and the result is that if the laws tomorrow are the laws today, energy is conserved. This is Emmy Noether's theorem of 1918, one of the most beautiful results in all of physics, and at this depth it needs no equation at all.
| If you cannot detect… | …then this cannot change | The demo that shows it |
|---|---|---|
| where the whole system is (shift it) | momentum | the spring carts; the cannon on the cart |
| which way it faces (rotate it) | angular momentum | the stool and the wheel; the skater; Kepler's planets |
| when it happens (delay it) | energy | Dennis's blocks; every pendulum |
Read the table backwards, too. If some experiment ever found momentum not conserved in an isolated system, the conclusion would not be "the law is wrong"; it would be "space is not uniform — there is a special place in the universe." Nobody has ever found one. That is what the conservation of momentum is actually telling you about the world.
First: charge is conserved too. Which symmetry is that? (It is not a symmetry of space or time. The answer is in Volume II and is the doorway to all of modern particle physics.) Second: the universe is expanding, so the laws of physics as seen from inside it are not the same at all times — the background changes. Is energy conserved on the scale of the cosmos? The honest answer among physicists is "it depends how you define it, and possibly no." A law you were told was absolute has an edge. Finding edges is the job.
1. An astronaut floats motionless in space, out of reach of her ship, holding a spanner. Can she get back?
2. A ball dropped from 2 m bounces back to 1.5 m. Momentum of the ball was clearly not conserved during the bounce — it reversed. Energy was clearly not conserved either — it lost a quarter. Which statement is right?
3. A neutron star forms when a star's core, roughly the size of the Sun and rotating once a month, collapses to a ball 20 km across. What happens to its rotation?
Read Feynman for the arguments and watch Lewin for the demonstrations, in roughly this order. All free.