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.
Understanding Physics: Angular Momentum Conservation
Angular momentum has a direction as well as a size. The direction points along the rotation axis, following the right hand rule. Curl the fingers of your right hand in the direction of rotation.
Your thumb points in the angular momentum direction. This detail matters because rotations in different directions can cancel or combine. It explains why a spinning bicycle wheel resists being turned sideways.
When a rider steers a moving bicycle, forces acting on the wheel can change the direction of its angular momentum. The result helps the bicycle stay balanced while moving.
Torque is the rotational effect of a force. A force has a larger turning effect when it acts farther from the axis. Pushing a door near its handle is easier than pushing near its hinges for this reason.
For a spinning object, a torque can speed it up, slow it down, or tilt its axis. Friction in a wheel bearing produces torque that gradually reduces the wheel's rotation. Gravity can produce torque too.
A top begins to wobble and precess because gravity pulls on its center of mass while the support force acts at the tip. Its axis moves around rather than simply falling straight down.
Conservation depends on choosing the system carefully. A person on a swivel chair can change their body position, but their muscles are internal to the person and chair system. Internal forces can move mass closer to or farther from the axis.
They cannot create a net external torque for that whole system. In contrast, a foot pressing on the floor creates an external torque if the floor is outside the chosen system. This distinction is useful in sports.
A diver can change their rotation rate in the air by tucking their body. They cannot start a large rotation from rest in midair without an earlier push from a board or platform.
A common mistake is to think that faster spinning always means more rotational energy. A skater uses chemical energy in their muscles while pulling their arms inward. The spin rate rises, but the rotational energy rises too.
Angular momentum can remain constant while energy changes. Another useful habit is to identify the axis before making a prediction. Mass close to one axis may be far from another.
A long rod is much harder to spin around an axis through one end than around a parallel axis through its center. In problems, first decide what object belongs in the system, locate the axis, check for external torques, then compare how the mass distribution changes.
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.