Angular Momentum Calculator
Work out the moment of inertia and angular momentum of a spinning body, and the new spin rate when it changes shape.
What it does
Angular momentum is the measure of how much rotation something carries. For a rigid body turning about a fixed axis it is the moment of inertia times the angular velocity, and the OpenStax physics text puts it plainly: the angular momentum along the axis is L equals I times omega, while for a single particle it is the cross product of the position and the momentum, which reduces to m v r when the velocity is at right angles to the radius.
The moment of inertia is the part that changes with shape. It measures how far the mass sits from the axis, so the same mass at the same radius gives very different numbers depending on how it is arranged. A solid disc has half the value of a ring of the same mass and radius, because every part of the disc is closer to the centre.
The consequence everyone meets is the spinning skater. With no outside torque the total angular momentum is conserved, so pulling the arms in reduces the moment of inertia and the spin rate has to rise to match. OpenStax works through exactly this, writing the conservation as I omega equals I prime omega prime, and the same relation explains the diver, the spinning basketball and the contracting cloud that forms a star.
This page does the arithmetic in two modes. Enter the shape, mass, size and spin rate and it gives the moment of inertia, the angular momentum and the rotational kinetic energy. Add a second size and it also gives the new spin rate that keeps the angular momentum the same.
The models are ideal. A real skater is not a uniform disc, and a real body is not rigid, so the numbers are a clean way to see how the relationship works rather than a prediction of a spin rate on the ice.
How to use
- Pick metric or imperial units.
- Choose the shape of the body and the axis it turns about.
- Enter the mass.
- Enter the radius, the length of a rod, or the distance from the axis for a point mass.
- Enter the spin rate in revolutions per minute.
- Add a second size to find the new spin rate after the body changes shape.
Formula
- I
- Moment of inertia, in kg m2.
- omega
- Angular velocity in radians per second.
- L
- Angular momentum in kg m2 per second.
- m
- Mass of the body.
- r
- Radius, rod length or perpendicular distance to the axis.
- k
- Shape factor for the moment of inertia.
The formulas in the table of shapes are for the axis named there, and the rod value of 1/12 applies to a rod turned about its centre, not about one end, where it would be 1/3. Rotational kinetic energy is half of I omega squared. The conversion used for the imperial display is 1 kg m2 = 7.06268 lb ft2 and 1 joule = 0.737562 ft lbf.
Worked example
Example: a 60 kg disc pulled in from 0.5 m to 0.2 m
Defaults: metric units, a solid disc, 60 kg, a radius of 0.5 m, 120 rpm, and a new radius of 0.2 m.
- Moment of inertia at 0.5 m: 0.5 x 60 x 0.5 squared = 7.5 kg m2.
- Angular velocity: 120 rpm x 2 pi / 60 = 12.566 rad/s.
- Angular momentum: 7.5 x 12.566 = 94.248 kg m2/s.
- Rotational kinetic energy: 0.5 x 7.5 x 12.566 squared = 592.2 J.
- Moment of inertia at 0.2 m: 0.5 x 60 x 0.2 squared = 1.2 kg m2.
- New spin rate: 94.248 / 1.2 = 78.54 rad/s, which is 750.0 rpm.
- The tool shows 7.5 kg m2, 94.248 kg m2/s, 592.2 J and 750.0 rpm.
Reference table
| Shape and axis | Factor | Moment of inertia |
|---|---|---|
| Solid disc, through the centre | 0.5 | 7.500 kg m2 |
| Solid sphere, about any diameter | 0.4 | 6.000 kg m2 |
| Thin ring, through the centre | 1 | 15.000 kg m2 |
| Thin rod of 1 m, about its centre | 1/12 | 1.250 kg m2 |
| Point mass 0.5 m from the axis | 1 | 15.000 kg m2 |
Limits
- Models the body as rigid, so it cannot show a wobble or a change of shape in the middle of the turn.
- Assumes the axis of rotation is fixed and passes through the centre of mass for the shape values given.
- Ignores any outside torque, so the new spin rate only holds when the body is free to rotate.
- Treats the shape as uniform, which is a good model for a disc and a poor one for a person.
Common mistakes
- Leaving the spin rate in revolutions per second instead of converting to radians per second.
- Using the point mass formula for a solid body, which assumes all the mass sits at one distance.
- Forgetting that the rod formula changes when the rod turns about an end instead of its centre.
- Applying conservation of angular momentum while a torque is acting, for instance while a motor drives the shaft.
- Reading the rotational kinetic energy as the total energy, which ignores the linear motion as well.
Try it yourself
Quick checks. Answers sit tucked under each question.
What is the angular momentum of a 2 kg point mass 0.3 m from the axis at 5 rad/s?
The moment of inertia is 2 x 0.3 squared = 0.18 kg m2 and the angular momentum is 0.18 x 5 = 0.90 kg m2/s. Enter 5 rad/s as 47.7465 rpm.
How much faster does the example skater spin after pulling in from 0.5 m to 0.2 m?
The moment of inertia falls from 7.5 to 1.2 kg m2, so the spin rate rises by a factor of 6.25, from 120 rpm to 750.0 rpm.
What is the angular momentum of a 1 kg mass on a 2 m radius turning at 60 rpm?
The moment of inertia is 1 x 2 squared = 4 kg m2 and the angular velocity is 60 x 2 pi / 60 = 6.283 rad/s, so the angular momentum is 4 x 6.283 = 25.133 kg m2/s.
Questions
What is angular momentum?
The measure of rotation. For a body turning about a fixed axis it is the moment of inertia times the angular velocity, and it is conserved when no outside torque acts on the body.
Why does a skater speed up when pulling in the arms?
Pulling the arms in lowers the moment of inertia, and the angular momentum has to stay the same, so the spin rate rises. The extra rotational energy comes from the work the skater does pulling the arms in.
How do I convert rpm to radians per second?
Multiply by 2 pi and divide by 60, so 120 rpm is 12.566 rad/s. The angular momentum formula needs radians per second.
Does the shape of the body really matter that much?
Yes. A solid disc of 60 kg at 0.5 m has a moment of inertia of 7.5 kg m2 while a ring of the same mass and radius has 15, twice as much, and a solid sphere 6.
What about a point mass on a string?
Use the point mass shape with the distance to the axis. That is the case of a stone on a string, where the string tension supplies the centripetal force and the angular momentum is m r squared omega.
Is angular momentum conserved in a collision?
In a collision the total angular momentum of the system is conserved, even though the kinetic energy usually is not. That is why a spinning cue ball can pass spin to another ball.
Sources
- OpenStax University Physics Volume 1: 11.2, angular momentum
- OpenStax University Physics Volume 1: 11.3, conservation of angular momentum
- Kyoto University: moment of inertia formula sheet for standard shapes