Angular Momentum

A body rotating about a fixed axis has angular momentum about that axis. Its angular momentum depends on both how its mass is distributed around the axis and how quickly it rotates.

Here, $$L$$ is angular momentum, $$I$$ is the moment of inertia about the axis, and $$\omega$$ is the angular velocity.

Moment of inertia describes how mass is distributed relative to the axis. For the same total mass, placing more mass farther from the axis gives a larger moment of inertia. Angular velocity describes the rate and direction of rotation.

For a given angular velocity, a larger moment of inertia gives a larger angular momentum. For a given moment of inertia, a greater angular speed also gives a larger angular momentum. Angular momentum has SI units of $$\mathrm{kg,m^2,s^{-1}}$$.

Conservation of Angular Momentum

A rotating system can change how quickly it spins by changing how its mass is distributed. If there is no resultant external turning effect on the system, its total angular momentum stays constant. This is the principle of conservation of angular momentum.

A turning effect is called a torque. For rotation about a fixed axis, angular momentum about that axis is conserved when the resultant external torque about the axis is zero.

For rotation about a fixed axis, angular momentum $$L$$ is given by:

Here, $$I$$ is the moment of inertia, which depends on how mass is distributed relative to the axis, and $$\omega$$ is the angular velocity. Moving mass closer to the axis decreases the moment of inertia; moving it farther away increases it.

If angular momentum is conserved between two stages of the motion, the initial and final values are equal:

The product $$I\omega$$ must therefore remain constant. If $$I$$ decreases, the magnitude of $$\omega$$ increases: the system spins faster. If $$I$$ increases, the magnitude of $$\omega$$ decreases: it spins more slowly.

Angular momentum in sport

A figure skater spinning with arms outstretched can pull them closer to the axis of rotation. This decreases the skater’s moment of inertia, so the skater spins faster. Extending the arms again increases the moment of inertia and slows the spin. This analysis assumes that the resultant external torque about the spin axis is negligible during the movement.

Angular Impulse

A resultant torque changes an object’s angular momentum. The longer a constant torque acts, the greater the change it produces. This change in angular momentum is called angular impulse.

Here, $$T$$ is the constant resultant torque, $$\Delta t$$ is the time interval for which it acts, and $$\Delta L$$ is the change in angular momentum. For rotation about a fixed axis, angular momentum is $$L = I\omega$$, where $$I$$ is the moment of inertia and $$\omega$$ is the angular velocity.

Angular impulse depends on both torque and time. For example, the same constant torque acting for twice as long produces twice the change in angular momentum. This is twice the change, not necessarily twice the final angular momentum.

The change in angular momentum is in the direction of the resultant torque. If the moment of inertia remains constant, the change can be written as $$\Delta L = I\Delta\omega$$. If it changes, the change in the whole product $$I\omega$$ must be considered.