11.1.5 Angular momentum

Definition of Angular Momentum

Angular momentum is the product of an object's moment of inertia and its angular velocity. It is a measure of how much rotational motion an object possesses and is given in units of Nms.

The formula for angular momentum $( L)$ is:

$$L = I \omega$$

Where:

• $I$ is the moment of inertia of the object (a measure of how resistant it is to changes in rotational speed),
• $\omega$ is the angular velocity (how quickly the object rotates).

Law of Conservation of Angular Momentum

The law of conservation of angular momentum states that when there is no external torque acting on a system, the total angular momentum of that system remains constant.

This principle is often illustrated by an ice skater spinning with their arms extended. When the skater pulls their arms in close to their body:

• Their moment of inertia decreases (mass is closer to the axis of rotation),
• To conserve angular momentum, their angular velocity must increase, so they spin faster. In mathematical terms:

$$L_{\text{initial}} = L_{\text{final}}$$

or,

$$I_1 \omega_1 = I_2 \omega_2$$

where:

• $I_1$ and $\omega_1$ are the initial moment of inertia and angular velocity,
• $I_2$ and $\omega_2$ are the final moment of inertia and angular velocity.

Angular Impulse and Change in Angular Momentum

Angular impulse represents the change in angular momentum due to an applied torque over a time interval. It is calculated as:

$$\text{Angular Impulse} = T \Delta t = \Delta (I \omega)$$

Where:

• $T$ is the applied torque,
• $\Delta t$ is the time over which the torque is applied. To find the angular impulse graphically, one can calculate the area under a torque-time graph. This area corresponds to the change in angular momentum of the object.