Angular Momentum Calculator

What Is the Angular Momentum Calculator?

How to use the angular momentum calculator

1. Pick an arrangement. The Calculate buttons open on L from inertia and spin. The other four are L from mass, speed, radius; Spin rate; Moment of inertia; and Speed. Only the boxes that arrangement needs stay on screen, so nothing hidden can steer the answer.
2. Enter the quantities you have. Each box carries its own unit menu. A moment of inertia can go in as kg·m² or g·cm², a spin rate as rad/s, rev/s, rpm or deg/s, and a radius as m, cm, mm or km.
3. Measure every radius from the axis you are quoting. Angular momentum has no meaning until an axis is named, and the same motion gives a different answer about a different point. Keep the axis fixed while you change anything else.
4. Read the headline and the four extras. The headline is always in SI. Moment of inertia behind it is L divided by the spin rate, Spin rate behind it gives that rate in rad/s and rpm, Kinetic energy of the rotation is half of L times the spin rate, and Rotation period is the time for one turn.
5. Open Show working. The steps restate the arrangement and substitute your figures in SI, whatever the menus are set to, so the arithmetic can be checked by hand.

The moment of inertia is the input most people have to find first, because it depends on shape as well as mass. A ring, a disc and a sphere of the same mass and radius all have different values, which is what the moment of inertia calculator is for; the guide to moment of inertia sets out where each shape factor comes from. Feed its answer straight into the first box here.

The spin rate has to be in radians per second before it multiplies anything, which is the second common stumble. A wheel quoted at 500 rpm is not turning at 500 rad/s, so either pick rpm from the menu and let the tool convert, or convert first with the angular velocity calculator; the angular velocity formula explained covers the radian-per-second convention in full. The extras always report the rate both ways so the conversion is visible.

The third slip is mixing the two arrangements in one sum. L from inertia and spin describes a whole body about its axis; L from mass, speed, radius describes one point mass on one circle. Adding a whole system's L to one of its own parts double-counts that part, and the worked table below shows the two figures side by side so the difference is obvious.

Worked example: change one thing at a time

The table below starts at the defaults and moves one thing at a time: the arrangement, then the units, then which quantity is the unknown. Each Result and Rotation energy cell was copied out of the running widget rather than worked out by hand, so where a cell and the tool ever part company, believe the tool. A dash marks a box the chosen arrangement does not show.

Rows 1 and 2 are the two ways of writing the same idea, applied to two different objects. The whole system carries 14.6 kg·m²/s; one of the two 3 kg masses on its rim carries 4.8 kg·m²/s of that total. Two such masses account for 9.600 kg·m²/s of it, and the turntable underneath them carries the remaining 5.000.

Rows 3 and 4 change the question and then the units. Asking for the spin rate that 14.6 kg·m²/s implies on 2.24 kg·m² returns 6.518 rad/s, the figure the simulator reaches when the masses come in to 0.2 m. Retyping the original 5.84 kg·m² as 5.84e7 g·cm² changes nothing at all, because the menu converts and the physics does not care which unit you thought in.

Rows 5 and 6 solve backwards and land exactly where rows 1 and 2 started: 2 m/s for the mass, and 2.24 kg·m² for the inertia. That round trip is the check worth doing whenever an answer looks surprising. Rows 7 and 8 close the last two loops — the arms-in pair multiplied out gives 14.6 kg·m²/s again, and the 1.920 kg·m² the extras report as the inertia behind one mass, typed back in beside that mass's own 4.8 kg·m²/s, returns it to 2.5 rad/s.

Rotation energy is the column that does not stay put. The whole system reads 18.25 J at 2.5 rad/s and 47.58 J at 6.518 rad/s, for the same 14.6 kg·m²/s in both rows, and the difference of 29.33 J is work that has to be done by something. Watching where that work comes from is the job of the Angular Momentum Simulator, which moves the masses for you; this page only ever evaluates one state at a time, and those two rows were typed separately.

Formula and symbol reference

The calculator uses one relation in five arrangements: L = Iω, L = mvr, ω = L/I, I = L/ω and v = L/(mr). The extras come from L/ω, from ω itself, from ½Lω and from 2π/ω. No measured constant of nature enters any of them: apart from π in the period and in the rpm conversion, every figure here is arithmetic on the numbers you type.

The physics: how L = Iω and L = mvr say the same thing

Take one mass m going round a circle of radius r at speed v. Its angular momentum about the centre is L = mvr, and its speed is tied to the spin rate by v = ωr. Substitute the second into the first and you get L = m(ωr)r = (mr2)ω, where mr2 is the moment of inertia of a point mass at radius r — so the two formulae are one formula written twice.

A real body is a great many such masses at a great many radii, so its moment of inertia is the sum of all their mr2 terms. That single number is what I stands for, and it is why the shape matters as much as the mass: move material outward and I grows as the square of the distance. The turntable in the opening case reads 5.84 kg·m² with its two masses at 0.8 m and 2.24 kg·m² with them at 0.2 m: only the masses move, so their 2mr2 share falls 16-fold from 3.84 to 0.24 kg·m² while the 2.00 kg·m² of the turntable itself stays put, which is why the total drops by 2.607 and not by 16.

What makes the quantity worth defining is that it only changes when something twists the object. With no net external torque about the axis, L stays put whatever the object does to itself — which is the argument behind every spinning-up skater, and the whole point of the full guide to angular momentum. Because I and ω must then trade off exactly, halving the moment of inertia doubles the spin rate.

The energy does not follow suit. KE = ½Iω2 = ½Lω, so at fixed L the energy rises in step with ω: the 2.607-fold spin-up between the two states above costs a 2.607-fold rise in rotational energy, from 18.25 to 47.58 J.

That extra 29.33 J is work done pulling the masses inward against the force that was holding them on their circles, and letting them back out returns it. Nothing is created, and the Angular Momentum Simulator is where you can watch the two bars move while the total stays where it was.

One more thing is worth saying plainly. Conservation is a statement about the net external torque being zero, not a promise about real objects: a real chair has bearing friction and a real skater has friction and air drag, and both slow down. This page never claims otherwise, because it never models a change at all.

Where the angular momentum calculator breaks down

Multiplying two numbers is hard to get wrong. What fails is the axis the numbers were measured about, the model the object is being squeezed into, or an expectation the arithmetic was never making.

An axis that moved between the two numbers

A moment of inertia is about one axis and a radius is from one point. Take I about the centre of mass and r from the pivot and the answer is a mixture of two different problems. Fix the axis first, then take every quantity about it.

Symptom: an answer that changes when you re-derive it a second way.

Rotation that is not about a symmetry axis

L = Iω is the scalar form for spinning about a symmetry axis. In general angular momentum is a vector, it need not point along the rotation axis, and the moment of inertia is a tensor rather than a single number. This tool handles the scalar case only, and says so rather than pretending to the general one.

Treating a real object as a point mass

The mass, speed and radius arrangement uses mr2, which treats whatever you are holding as a point at distance r. A real dumbbell also has a moment of inertia of its own about its own centre, which this model ignores. The error is small while r is much larger than the object, and not otherwise.

Expecting it to conserve anything

The calculator evaluates one state. It never asks for a second radius and it never compares two states, so it cannot show angular momentum being conserved and does not claim to. Changing a radius and watching what happens is what the simulator does.

Assuming no torque is acting

Angular momentum is constant only for as long as the net external torque about that axis stays zero. Friction in a bearing, drag through the air and a hand on the rim are all torques, and each of them changes L. Working out how much takes a torque and a time, which is what the torque calculator is for.

Rounded figures in, rounded figures out

The working prints every operand to four significant figures while the answer is carried unrounded, so re-multiplying the printed operands can differ from the printed answer in the last digit. Type a value already cut to four figures and the answer inherits that rounding too.

Speeds approaching the speed of light

Everything here is classical. Near light speed the relation between momentum and speed changes and these formulae stop describing the motion. Nothing in the ranges this page uses goes anywhere near that, and no figure on the page should be read as covering it.

Where angular momentum is actually used

Flywheels and energy storage

A flywheel is chosen for its moment of inertia, and what it stores is the ½Iω2 this page reports as the rotational energy. Doubling the spin rate quadruples the stored energy while only doubling the angular momentum, which is why the spin rate is the design lever.

Gyroscopes and attitude control

A spinning rotor resists having its axis turned, because turning it means changing a vector angular momentum, which takes a torque. Reaction wheels on spacecraft use the same fact in reverse: driving a wheel one way turns the body the other way, and nothing outside has to push.

Rotating machinery run-up and run-down

How long a motor takes to bring a load to speed, and how long that load takes to coast to a stop, are set by the angular momentum it has to gain or lose divided by the torque doing the work. The moment of inertia behind it extra is the number a drive sizing calculation starts from; the torque available is the other half of the sum, and neither time follows from the angular momentum on its own.

Sport and human movement

A diver leaves the board with an angular momentum that nothing in the air can change much, and sets the rotation rate by tucking or opening out. Nothing on this page measures a diver; you supply the moment of inertia and the arithmetic does the rest.

Orbits and spinning astronomical bodies

A planet keeps the same angular momentum about its star all the way round, which is why it moves fastest where it is closest; at the nearest and furthest points, where its motion is square to the line to the star, that angular momentum is exactly mvr. The same idea applied to a collapsing cloud is why compact remnants spin quickly. Momentum p = mv obeys a rule of the same shape when no outside force acts, which the conservation of momentum guide works through in a straight line.

Laboratory turntable and rotating-stool experiments

The standard demonstration measures a spin rate before and after weights are pulled in, and checks that Iω is unchanged. Enter each state here and the two angular momenta should agree to within your measurement error; the rotational energies will not, and setting them beside the straight-line ½mv2 from the kinetic energy calculator is a good way to see why that is not a contradiction.

Where to go next

For what the quantity means, why it is conserved and a set of worked problems, read Angular Momentum Explained. Take a shape to the moment of inertia calculator, a spin rate to the angular velocity calculator, or a twist to the torque calculator. To watch the trade-off happen instead of typing it, the Angular Momentum Simulator moves the masses for you, and the whole physics lab library is open beside it.