Angular Momentum Simulator

What Is the Angular Momentum Simulator?

How to use the angular momentum simulator

1. Load a case, or start from Reset. The five buttons under Load a case — Pull the masses right in, The masses never move, A light turntable, heavy masses, A heavy turntable, light masses and Pushed back out again — write all five sliders together and print the button's own words on the case line. Move any slider and that line reverts to “custom setting”. Reset returns to 2.00 kg m2, 3.00 kg, 0.80 m, 0.20 m and 2.50 rad/s, and restarts the spin.
2. Choose how much inertia the turntable has of its own. Turntable inertia runs from 0 to 6.00 in steps of 0.01 and reads back as “2.00 kg m2”. It is a number you pick rather than a measurement of anything, so if you want a defensible figure for a real disc or ring, work it out in the moment of inertia calculator from a mass, a radius and a shape factor first, and type the answer in here.
3. Set the mass and the starting radius. Mass at each radius covers 0.50 to 10.00 kg in 0.05 kg steps, and Starting radius covers 0.10 to 1.20 m in 0.01 m steps. Both move Angular momentum, because both change the moment of inertia the system begins with; the radius does it faster, since it enters as a square.
4. Drag the finishing radius and watch what stays still. Finishing radius has the same 0.10 to 1.20 m grid. Take it anywhere you like and Angular momentum does not stir, while Spin rate after, Spin-up factor, Rotational energy after and Work your arms do all follow it. Set it equal to the starting radius and the factor reads exactly “1.000”.
5. Choose how fast it is turning to begin with. Starting spin rate runs 0.20 to 20.00 rad/s in 0.01 steps. Radians per second is the only unit this lab will accept, which is worth remembering if your figure came off a nameplate in rpm; the angular velocity simulator prints its own disc's rate in rad/s with the rpm on the line underneath, and will do that conversion for you.
6. Read the four headline cards, the six stats and the three sentence lines. The cards give Spin rate after, Angular momentum, Spin-up factor and Rotational energy after; the stats add the moment of inertia in both states, the energy before, the work, the speed of each mass, the one-mass m v r and the pull each mass needs. Why restates the setting in words, The split reports the two masses' share, and What to notice changes as you cross the landmarks. Pause stops the discs without moving a number.

Worked example: change one thing at a time

Every row below starts from the state the lab boots in. Row 2 moves one slider and nothing else; rows 3 and 4 press a case button, which writes all five sliders at once. Each cell is a string the running lab printed at those control positions, and where a cell and the lab disagree, the lab is right. The setting column lists the five sliders in panel order.

Row 2 is the reason this lab exists. One slider moved, and the only column that did not budge is the one the page is named after: 14.60 kg m2/s in both rows. Everything else moved a long way — the spin rate back down to 2.500 rad/s, the factor to 1.000, the energy to 18.25 J, the work to 0.000 J. Do the same to any of the other four sliders and the angular momentum moves with them.

Row 3 flips the sign of the work. Starting at 0.20 m and letting the masses out to 0.80 m turns 71.68 J of rotational energy into 27.49 J, and Work your arms do reads “-44.19 J”: the energy is coming back out through your arms instead of going in. Spin-up factor reads “0.3836”, which is a slow-down, and the What to notice line switches to the branch that says so. The angular momentum is 17.92 kg m2/s either way.

Row 4 is where the lab stops describing a person. Eight-kilogram masses on a light turntable give a fifteen-fold spin-up, and Pull on each mass after reads “3581 N” — far more than a person could hold, which is what the note under the drawing says in those words. The arithmetic is exact; what it describes is a machine. If you would rather check one of these states on your own figures, the angular momentum calculator evaluates L = Iω at a single state; it is the same arithmetic on the same numbers, not a second opinion.

Formula and symbol reference

The lab builds the moment of inertia of the whole system as I = Ib + 2mr2 at each radius, multiplies the first of those by the starting spin rate to get the angular momentum, and divides that same angular momentum by the second to get the finishing spin rate. Everything else on the panel hangs off those four numbers. Ranges marked “in this lab” are the controls' own ends and the strings the lab prints there.

Two things in that table are worth a second look on screen. The lab writes its equations without Greek letters, so the cards read L = I omega and omega2 = L / I2 rather than using the symbol; and the unit is printed as “kg m2/s” throughout — a quantity you will also meet written as a joule second, or as a newton metre second. What the turntable-inertia slider stands for — why the same mass arranged as a ring and as a disc needs two different numbers — is the subject of the guide to moment of inertia, and why the spin-rate slider insists on radians per second is set out in the angular velocity formula.

The physics: why pulling in speeds you up

Each mass contributes mr2 to the moment of inertia, so where it sits matters far more than what it weighs. At the opening setting the Moment of inertia stat reads “5.840 to 2.240 kg m2” while the turntable's own contribution stays at the 2.00 kg m2 on its slider: the masses are the only thing that moved, and dropping their radius by a factor of four took most of the total with it. That is the square in mr2 doing the work.

Nothing outside the system is twisting it, so the product of the moment of inertia and the spin rate has to come out the same afterwards. The panel does that division for you: 14.60 divided by 2.240 kg m2 is the 6.518 rad/s on the first card. Spin-up factor reports the same thing as a ratio, and at this setting you can see where it comes from in the Moment of inertia stat — 5.840 divided by 2.240 is 2.607. The lab computes it from the unrounded values, so do not lean on that division at other settings.

The two bars are the bookkeeping. The upper one splits 14.60 kg m2/s into the turntable's 5.000 and the two masses' 9.600, which add to the printed total exactly; in the lower one the turntable's share has risen to 13.04 and the masses' has fallen to 1.564. Each bar is built from its own state's numbers rather than copied from the other, which is why the two ending level is a result and not a drawing convention. The lower pair is printed to four figures each, so adding those two will not give the printed total back — read the total off the card.

What does not survive the move is the energy. KE = ½Lω, so at a fixed angular momentum the energy rises in step with the spin rate: 18.25 J becomes 47.58 J, and 47.58 minus 18.25 is the 29.33 J in Work your arms do. That work is done pulling each mass inward against the force holding it on its circle, and Pull on each mass after is exactly that force at the finishing radius. The full guide to angular momentum takes the derivation further than a control panel usefully can.

One caution about the animation. Both discs turn while the lab is playing, and the caption under them says whether you are watching real time or a slowed version; the absolute rate is capped so that a 2880 rad/s setting is still watchable. What the drawing always keeps exact is the ratio of the two discs' rates, which is the spin-up factor on the third card. Treat the moving picture as a comparison between the two states, not as a clock.

If you want to see the other half of this story — the same mass rearranged into different shapes, and the same shape spun about a different axis — the moment of inertia simulator puts two bodies of equal mass side by side and lets you slide the axis away from the centre. That lab varies the I this one takes as a setting.

Where angular momentum conservation breaks down

The lab solves its own model exactly, so nothing on screen ever fails. Every limit below is a limit of that model, of the situation it stands for, or of the drawing, and each item says what the lab does about it.

There is no friction anywhere in it

The constants line under the canvas ends with the condition, in the lab's own words: “L is conserved only while the net external torque about the axis is zero.” Nothing here supplies one. Bearings rub, ice is not frictionless and air pushes back on a moving arm, so a stool, a chair or a skater all lose spin; how fast they lose it is not a question this lab can answer, because it keeps the total fixed by assuming the torque away rather than by showing there is none. Putting a number on what a real torque would take out needs a twist and a time, which is what the guide to torque covers.

The rotational energy is not conserved, and nothing here pretends it is

Two of the readings exist to make that impossible to miss. Energy before and Rotational energy after disagree at every setting where the radii differ, and Work your arms do is the difference between them with a sign on it. The commonest mistake on this topic is to read a rising spin rate as free energy; it is not, and the lab prices it for you in joules every time.

This is the scalar form, for a symmetry axis

Multiplying a moment of inertia by a spin rate gives the right answer here because the turntable is symmetric about the axis it turns on. Away from that case the quantity has a direction of its own, which need not line up with the axis at all, and one number will no longer stand in for the moment of inertia — a tensor is needed instead. Everything on this panel belongs to the scalar, fixed-axis case, and nothing here should be read as covering the general one.

Every value is about one axis, and only that axis

The axis here is the turntable's own, running through the hub. Change the point you measure about and every angular momentum on the panel changes with it: even a mass travelling dead straight carries some about any point its track misses, and none at all about a point the track runs through. None of the panel's readings is labelled with the axis it belongs to, so quote it yourself beside any figure you take away — the first row of the symbol table above names this one. Forgetting to do that is the commonest source of a wrong answer on this topic.

The two masses are treated as points

The mr2 each mass contributes is the formula for a point at distance r from the axis. A real hand weight is not a point: it turns about its own centre as well, and that extra share of the inertia is simply absent from this model. The omission hardly matters while the radius dwarfs the object, and it grows as the finishing radius comes down towards its 0.10 m minimum.

The sliders reach forces no person could exert

Take the mass to 10.00 kg, the radii from 1.20 m to 0.10 m and the spin to 20.00 rad/s and Pull on each mass after reads “7.840e+4 N”; with the turntable inertia at zero as well it reaches “8.294e+6 N”. Whenever the masses move inward and that pull reaches 500 N, the What to notice line names the figure and calls the setting a machine rather than a skater; let them out instead and a different branch of that line wins however big the pull is. Nothing about those numbers is wrong; they simply do not describe a body.

The turntable inertia is a setting, not a measurement

Nothing in this lab measures the moment of inertia of a person, a chair, a stool or a skater, and none of the five case buttons is named after a real object for exactly that reason. They are named after what they demonstrate. Whatever you put on the slider is what the arithmetic uses, so the answer is only ever as good as that number.

What the drawing leaves out when there is no room

Every string on the canvas is measured before it is drawn, then shortened or dropped rather than overprinted, and what survives depends on the width of its own band, not the width of your screen. In the bars' band the smallest segment's value goes first, then the caption shortens from “both bars end at the same place: that is conservation” to “both bars end together”, then the colour legend goes, and narrower still the short caption too. In the discs' band the caption shortens from “same angular momentum, different spin rate” to “same L, different spin”. The two bands need not be the same width, so neither order predicts the other; The split prints the same figures in words.

The lab's own arithmetic and display

Four significant figures on almost everything, two decimals on the slider labels. A number that looks exact on the panel has usually been rounded to get there, so a ratio or a difference worked out from two printed readings is not always the one the physics gives. Take the ratio from Spin-up factor, which is computed from the unrounded values, rather than dividing the two figures in the Moment of inertia stat.

Where angular momentum is actually used

Skaters, divers and gymnasts changing shape mid-rotation

Once a diver has left the board, nothing much can twist them, so the rotation rate is set entirely by how tightly they are tucked. A gentle pull-in is the setting that shows the effect with no violence in it: 0.60 m to 0.40 m at 1.50 rad/s takes the spin to 2.108 rad/s for a Pull on each mass after of 5.333 N. Nothing here measures a diver, and the inertia on the slider is a number you chose rather than anybody's body. Opening out again is the Pushed back out again case with the sign of the work reversed.

A helicopter's tail rotor

Spin a main rotor one way and the airframe underneath it wants to turn the other, because the two shares have to add up to what the machine started with. The tail rotor exists to supply the outside torque that stops it happening. The two bars on this canvas are the same bookkeeping drawn in a straight line: move some of the total from one segment to the other and the bar still reaches exactly as far.

Reaction wheels on a spacecraft

With nothing outside to push against, a satellite turns itself by spinning a wheel inside it in the opposite sense. The total stays where it was, exactly as the lower bar's end does here when you drag the finishing radius. It is the closest real system to this lab's assumption, because in orbit the external torques really are very small.

Flywheels, and why the spin rate is the design lever

A flywheel stores rotational energy, and the lab prices the trade-off directly: hold the angular momentum where it is and the energy rises in step with the spin rate. The light-turntable case shows it plainly — a 15.77-fold spin-up, and Rotational energy after at “1164 J”. Getting that energy back out means slowing the wheel down, which is the negative work the Pushed back out again case shows.

The inward pull a rotating machine has to survive

Every one of these systems needs something to hold the moving mass on its circle, and that is the reading this lab puts in newtons. A centrifuge rotor, a turbine blade root and a spinning tether are all designed around it. Setting a mass on a circle and varying the speed and radius on their own, without any conservation in the picture, is what the circular motion simulator is for.

The rotating-stool demonstration in a teaching laboratory

Somebody sits on a stool holding a weight in each hand, is set turning, and pulls the weights in. The measurement worth making is the spin rate before and after, and the check is that the product of the moment of inertia and the spin rate agrees. This lab is that experiment with the friction removed and the arithmetic done for you; a real stool will always give a slightly smaller answer afterwards.

Orbits, and why planets keep going round

There is no torque about the Sun on a planet's orbit, so its angular momentum about the Sun is fixed, which is why it moves fastest when it is closest. For the straight-line partner of this bookkeeping, and what survives a collision, read how momentum is conserved in a collision. No figure in this lab describes a planet.

Where to go next

The full account — what the quantity is, why the two formulas are one quantity, the axis it depends on and a set of worked problems — is in Angular Momentum Explained. To put your own figures through the same two formulas at a single state, the angular momentum calculator shows every step of the substitution and converts the units for you.

The two quantities this lab multiplies together each have a page of their own: moment of inertia, with the moment of inertia calculator and the moment of inertia simulator beside it, and the angular velocity formula, with the angular velocity calculator and the angular velocity simulator. For what happens when something does twist the system, read torque; the rest are in the library of physics simulations.