Angular Momentum and Torque in Rotational Motion
- When dealing with extended rigid bodies, forces can cause both linear and rotational motion.
- To understand this, we need to explore angular momentum, torque, and their real-world applications.
Angular Momentum: A Measure of Rotational Motion
Angular momentum
Angular momentum is the rotational equivalent of linear momentum. More precisely, it is the product of its moment of inertia and its angular velocity.
For a rigid body rotating about a fixed axis, it is defined as:
$$
L = I \omega
$$
where:
- $L$ is the angular momentum.
- $I$ is the moment of inertia.
- $\omega$ is the angular velocity.
Angular momentum is a vector quantity, but in this course, we focus on its magnitude.
Conservation of Angular Momentum
Conservation of angular momentum
Angular momentum is conserved unless an external torque acts on the system.
- Imagine a figure skater spinning with her arms extended.
- As she pulls her arms in, her moment of inertia decreases.
- To conserve angular momentum, her angular velocity increases, causing her to spin faster.
A figure skater is spinning with her arms extended. Her moment of inertia with arms extended is $I_1 = 5.0 \, \mathrm{kg \cdot m^2}$, and her initial angular velocity is $\omega_1 = 2.0 \, \text{rad s}^{-1}$. She pulls her arms in, reducing her moment of inertia to $I_2 = 2.0 \, \mathrm{kg \cdot m^2}$.
What is her final angular velocity $\omega_2$?
Solution
- The angular momentum of the system is conserved, so: $$L_1 = L_2$$
- Where angular momentum $L$ is: $$L = I \cdot \omega$$
- Thus: $$I_1 \cdot \omega_1 = I_2 \cdot \omega_2$$
- Solving for $\omega_2$: $$\omega_2 = \frac{I_1 \cdot \omega_1}{I_2}$$
- Substitute the values: $$\omega_2 = \frac{5.0 \cdot 2.0}{2.0}$$ $$\omega_2 = 5.0 \, \text{rad s}^{-1}$$
- Result: Final angular velocity is $\omega_2 = 5.0 \, \text{rad s}^{-1}$
Torque and Angular Acceleration
Torque
Torque is the rotational equivalent of force. It measures the ability of a force to cause an object to rotate.
Torque and Angular Acceleration
Torque is directly related to angular acceleration through the equation:
$$
\tau = I \alpha
$$
