Angular momentum describes how much rotational motion an object has, and it is a key idea for understanding spinning systems. A figure skater, a rotating planet, and a bicycle wheel all have angular momentum. When no external torque acts on a system, its angular momentum stays constant.
This conservation law helps explain why changing body shape can change spin rate without adding a new push.
For a rotating object, angular momentum is given by L = Iω, where I is moment of inertia and ω is angular velocity. Moment of inertia depends on how mass is distributed around the rotation axis, so spreading mass outward increases I and pulling mass inward decreases I. If L stays constant, a smaller I must be matched by a larger ω.
This is why a skater spins faster when pulling in their arms and slower when extending them.
Key Facts
- Angular momentum is rotational motion quantity: L = Iω.
- Angular momentum is conserved when net external torque is zero: τnet = 0 means L is constant.
- Moment of inertia measures resistance to changes in rotation and depends on mass distribution.
- For the same angular momentum, decreasing I increases ω: I1ω1 = I2ω2.
- A skater pulling arms inward decreases moment of inertia and spins faster.
- External torque changes angular momentum according to τnet = ΔL/Δt.
Vocabulary
- Angular momentum
- Angular momentum is the quantity of rotational motion an object has, equal to moment of inertia times angular velocity for a rigid rotating object.
- Moment of inertia
- Moment of inertia is a measure of how hard it is to change an object's rotation based on how its mass is spread around the axis.
- Angular velocity
- Angular velocity is the rate at which an object rotates, usually measured in radians per second.
- Torque
- Torque is a twisting effect that can change an object's rotational motion.
- Conservation law
- A conservation law states that a physical quantity stays constant in a system when no outside influence changes it.
Common Mistakes to Avoid
- Confusing angular momentum with angular velocity is wrong because L depends on both spin rate and mass distribution through L = Iω.
- Assuming a skater speeds up because they create angular momentum is wrong because the skater mainly changes I while total L stays constant if external torque is negligible.
- Ignoring external torque is wrong because angular momentum is only conserved when the net external torque is zero or small enough to neglect.
- Using ordinary speed instead of angular velocity is wrong because rotational equations use ω in radians per second, not linear speed in meters per second.
Practice Questions
- 1 A skater has a moment of inertia of 4.0 kg m^2 and spins at 2.0 rad/s with arms extended. If the skater pulls in their arms and the moment of inertia becomes 1.6 kg m^2, what is the new angular velocity?
- 2 A rotating stool system has angular momentum 18 kg m^2/s. If its moment of inertia is 3.0 kg m^2, what is its angular velocity? If the moment of inertia decreases to 2.0 kg m^2 with no external torque, what is the new angular velocity?
- 3 Explain why a figure skater spins faster when pulling their arms inward even though no one gives the skater an extra push.