What a scale reads while a chain falls onto it
At its defaults it draws what a scale reads while a chain falls onto it. The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.
chain-fall is one function in lib/figures/mechanics.js —
motion, force, energy and rotation. Everything below came out
of it during this build, at parameters taken from the essays rather than invented for this
page. A figure here is the figure a reader meets in an essay, and if the generator changes,
this page changes with it.
At its defaults
Drawn even though every essay passes options, because a default nothing exercises is a trap for the next essay to call this with none — which has happened here twice.
The reading of a scale, in units of the whole chain's weight, against the length of chain that has already landed, for two ways of putting the same chain down. Lowered gently, the scale reads the weight of what is resting on it and nothing else, so the reading climbs along the diagonal to one and stops. Dropped from rest with its lower end just touching, the scale reads three times that at every instant of the fall: one part is the pile's weight and two parts is the force needed to stop the links that are arriving, which is λv² with v² = 2gx and is therefore exactly twice λgx however far the fall has got. The peak, read off the drawn curve, is 3.00 chain weights. It is reached at the instant the last link lands, and the reading then falls discontinuously to one, because the momentum flux stops all at once. The discontinuity is the part a real experiment does not show — a real chain has links of a finite size and a scale has a response time — and it is the reason a chain dropped into a bucket on a kitchen scale reads high and then settles.
A collision with restitution 0.6
The options are the ones Collisions are easier than forces, and momentum is the reason passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.
An elastic collision
The options are the ones Collisions are easier than forces, and momentum is the reason passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.
An elastic collision
The options are the ones Collisions are easier than forces, and momentum is the reason passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.
A perfectly inelastic collision
The options are the ones Collisions are easier than forces, and momentum is the reason passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.
A collision with restitution 0.6
The options are the ones Collisions are easier than forces, and momentum is the reason passes. A branch drawn at its own defaults instead would be a picture no essay asked for and no assertion has been run against.
Two bodies before and after a head-on collision. Momentum is the same on both rows by construction; kinetic energy is only preserved when the collision is elastic.
What checks it
physicscheck asserts something about chain-fall that
could fail — it draws it and measures the result against a value reached some other
way.
Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
Collisions are easier than forces, and momentum is the reason
Nobody knows what happens inside a collision. Momentum conservation makes that ignorance irrelevant, which is the whole trick — and energy, deliberately, is not conserved.
MechanicsFive balls, and the law that does not choose
The usual account of a Newton's cradle says that momentum and energy conservation force one ball out at the striking speed. For three balls or more they do no such thing: the two laws leave a whole curve of possible outcomes, and what picks one is the shape of the force between two touching spheres.
MechanicsThe bounce an elastic plate cannot give back
Drop a steel ball on a thick steel block and it bounces almost to where it started. Drop it on a steel plate five millimetres thick and it comes back at barely half its speed, although steel on steel is about as elastic as materials get and nothing in the impact is plastic, sticky or damped. The energy has not been turned into heat. It has been turned into waves, running out through the plate faster than the ball can take them back — and a ball striking a longer rod loses energy the same way, by leaving it ringing.
MechanicsThe bounces that add up to a stop
A ball dropped on a hard floor bounces, and bounces again, each time a fixed fraction lower. The bounces never run out — there is always another, smaller one — and yet the ball is lying still four seconds later. The flight times form a geometric series with a finite sum, the floor's infinitely many kicks add up to exactly the ball's weight times that time, and the series is only broken where each flight becomes shorter than the impact that launched it.
MechanicsThe chain that climbs out of the jar
Pull the end of a long bead chain over the rim of a jar and let it fall to the floor, and the rest follows, as a siphon does. That much is unsurprising. What a video showed in 2013 was stranger: the chain does not run over the rim. It leaps up from the jar in a standing arch, tens of centimetres above it, with nothing supporting it. The explanation took a year and turned on something a string does not have: a link is stiff, so the pile has to push back.
MechanicsThe collisions that count the digits of π
Slide a heavy block towards a light one resting near a wall. The light block is batted between the heavy one and the wall until the heavy block turns round and leaves. Count the collisions. With equal masses there are 3; with one block a hundred times heavier, 31; ten thousand times, 314; a million, 3141. Nothing in two blocks and a wall mentions a circle, but conservation of energy puts their velocities on one, and each collision steps round it by the same small angle. The count is how many such steps fit in half a turn.
MechanicsThe crack at the end of a whip
An arm moves a whip's handle at ten metres a second, and a fifth of a second later its tip passes the speed of sound. Nothing in between is a lever or a spring. The arm throws a loop into the whip, the part behind the loop comes to rest, and the energy stays with the part still moving, which grows lighter as the loop runs out towards the tip. Energy shared by less and less mass means speed, and what caps it is the tip's own mass, the tuft of string at the end. The crack is a sonic boom, made by something the size of a shoelace's end.
MechanicsThe energy that depends on the observer
A moving train has kinetic energy. Watched from a second train alongside it, it has none. Both statements are correct, neither can be corrected, and the whole of mechanics still works — because what conservation laws constrain is not how much energy there is but how much of it changes.
MechanicsThe floor that does no work
A jumper leaves the ground with three hundred joules of kinetic energy, supplied by a floor that does exactly zero work — because work is a force times the displacement of its own point of application, and the patch of floor under the foot never moves. Newton's second law integrated over the centre of mass gives the right kinetic energy and is not the work-energy theorem, and telling the two apart is what the first law of thermodynamics is for.
MechanicsThe pile that lands heavier than it weighs
Drop a chain onto a scale and the reading is three times the weight of the part that has landed — not approximately, exactly, all the way through the fall. The extra two parts are the force needed to stop links that are still arriving, and the same arithmetic run backwards says that picking a chain up wastes exactly half the energy it takes to get it moving.
MechanicsThe point that keeps moving as if nothing had happened
Newton's third law makes every internal force cancel against its own partner, which leaves the external sum governing a single mass-weighted average of positions. In the collision below the total momentum stays at 4.00 kg·m/s while 63 per cent of the kinetic energy leaves, and the average travels at 1.00 m/s throughout, before and after.
MechanicsThe push that needs nothing to push against
A rocket engine produces more thrust in vacuum than at sea level — fifteen per cent more, for the engine drawn here, and the extra is exactly the ambient pressure times the nozzle's exit area. The account in which the exhaust shoves against the atmosphere does not merely overstate a coefficient. It has the sign wrong, and every engine ever fired has said so.
MechanicsThe quantity that survives a change of shape
A skater pulls her arms in and spins four times faster. Angular momentum is conserved, which is the usual explanation, and it accounts for only half of what happened — because the kinetic energy has gone up by the same factor, and something had to pay for it.
MechanicsThe raindrop that falls at a seventh of g
A drop of water falling through a still mist, gathering every droplet in its path and with no air to slow it, does not fall at g. It settles to an acceleration of exactly a seventh of g, whatever the mist, whatever the drop, and its radius grows by the same amount for every metre it falls. Of the work gravity does on it, three-sevenths goes into the gathering and never comes back. The same reasoning gives a third of g for a chain paid out from a heap, and a single rule — g/(2n + 1) — for anything that sweeps up mass as the n-th power of the distance it falls.
MechanicsThe spring whose bottom does not know it has been dropped
Hang a soft spring from its top, let go, and its bottom end stays exactly where it was — not falling slowly, not falling at all — while the top races down to meet it. A spring that hangs a metre takes about a quarter of a second to collapse, and only then does the bottom start to fall. Nothing holds it up. The centre of mass falls freely from the first instant, as it must, and the bottom simply has not been told: the news that the top has been released travels down the spring as a front of collapsing coils, and nothing below the front can feel anything until the front arrives.