AP® Physics 1: Algebra-Based · Unit 6 · Topic 6.4
Conservation of Angular Momentum: The Spinning Skater
When no outside torque acts on a spinning object, its angular momentum L = I omega stays constant, so pulling mass closer to the axis shrinks I and forces omega to rise to match.
A figure skater spinning with arms out has a certain rotational inertia I, built from how her mass is spread around the spin axis. Angular momentum is L = I omega, and as long as no outside torque acts on her, L stays fixed. Friction with the ice is tiny and acts right at the axis, so it contributes almost no torque. That means she is free to change I without changing L. When she pulls her arms in, she moves mass closer to the axis, so I drops. Since I omega must stay the same number, omega has to climb to make up the difference. Pull I down to a quarter of its starting value and omega quadruples.
This feels almost magical, but it is just bookkeeping on L, not on energy. Kinetic energy is (1/2) I omega2, and omega grows faster than I shrinks, so KE actually goes up. That energy is not free: the skater does real muscular work yanking her arms inward against the outward tug her spin creates, and that work becomes the extra kinetic energy. Conservation of angular momentum never promises conservation of kinetic energy; only an idealized, perfectly elastic process would do that, and muscles pulling in limbs are not that.
This scene is the rotational twin of a skater on an icy pond catching a heavy ball, or a planet whose orbital angular momentum stays fixed as it swings from far to near its star. It also sets up the next idea: a person on a frictionless stool holding a spinning bicycle wheel. Flip the wheel over and its angular momentum reverses, so to keep total L constant, the stool and student must start spinning to cancel the change.
Key terms
- Angular momentum (L)
- A spinning object's quantity of rotational motion, L = I omega, conserved when no net external torque acts.
- Rotational inertia (I)
- A measure of how spread out an object's mass is from its spin axis, resisting changes in angular velocity.
- Angular velocity (omega)
- The rate at which an object rotates, measured in radians per second.
- Rotational kinetic energy
- Energy of spinning motion, equal to one-half times rotational inertia times angular velocity squared.
Watch it explained
- How it worksHow does angular momentum staying fixed force omega up when rotational inertia goes down?
- Worked exampleFind the new spin rate of a skater at 2 rad/s whose inertia drops from 4 to 1.6 kg m squared.
- Common mistakeWhy isn't the skater's kinetic energy conserved even though her angular momentum is?
- Going furtherWhat does a student on a rotating stool feel when handed a spinning wheel to flip over?
Check yourself
End of topic.