Rotational Motion and Angular Momentum Cheat Sheet

Key Facts

• Angular displacement, angular velocity, and angular acceleration are related by $\omega = \frac{\Delta \theta}{\Delta t}$ and $\alpha = \frac{\Delta \omega}{\Delta t}$.
• For constant angular acceleration, the rotational kinematics equations include $\omega_f = \omega_i + \alpha t$ and $\theta = \omega_i t + \frac{1}{2}\alpha t^2$.
• Tangential motion connects to angular motion using $v = r\omega$ and $a_t = r\alpha$.
• Centripetal acceleration for circular motion is $a_c = \frac{v^2}{r} = r\omega^2$ and points toward the center of the circle.
• Torque is calculated by $\tau = rF\sin\theta$, where $\theta$ is the angle between the lever arm and the force.
• Newton's second law for rotation is $\tau_{net} = I\alpha$, where $I$ is the moment of inertia.
• Rotational kinetic energy is $K_{rot} = \frac{1}{2}I\omega^2$, and total kinetic energy for rolling can be $K_{total} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$.
• Angular momentum is $L = I\omega$ for a rigid rotating object, and it is conserved when $\tau_{net,external} = 0$.

Vocabulary

Angular displacement

Angular displacement is the angle $\Delta \theta$ through which an object rotates, usually measured in radians.

Angular velocity

Angular velocity is the rate of change of angular position, given by $\omega = \frac{\Delta \theta}{\Delta t}$.

Torque

Torque is the turning effect of a force about an axis, calculated by $\tau = rF\sin\theta$.

Moment of inertia

Moment of inertia is a measure of how strongly an object resists angular acceleration, represented by $I$.

Angular momentum

Angular momentum is the rotational version of momentum, given by $L = I\omega$ for a rigid rotating object.

Conservation of angular momentum

Conservation of angular momentum means total angular momentum stays constant when the net external torque is zero.