Angular Momentum Conservation Reference Cheat Sheet

Key Facts

• Angular momentum for a particle is $\vec{L} = \vec{r} \times \vec{p}$, and its magnitude is $L = r p \sin\theta$.
• For a rigid object rotating about a fixed axis, angular momentum is $L = I\omega$.
• Angular momentum is conserved when the net external torque is zero, so $L_i = L_f$.
• For changing rotational inertia with no external torque, conservation gives $I_i\omega_i = I_f\omega_f$.
• Net torque changes angular momentum according to $\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$.
• For constant torque over a time interval, angular impulse equals change in angular momentum: $\tau_{\text{net}}\Delta t = \Delta L$.
• Moment of inertia depends on mass distribution, so moving mass farther from the axis increases $I$ and usually decreases $\omega$ if $L$ is conserved.
• Angular momentum is a vector, and its direction is found using the right-hand rule.

Vocabulary

Angular Momentum

A vector measure of rotational motion, given by $\vec{L} = \vec{r} \times \vec{p}$ for a particle or $L = I\omega$ for a rigid body about a fixed axis.

Torque

A rotational effect of a force, given by $\vec{\tau} = \vec{r} \times \vec{F}$ with magnitude $\tau = rF\sin\theta$.

Moment of Inertia

A measure of how difficult it is to change an object's rotational motion, depending on mass and how far the mass is from the axis.

Conservation Law

A rule stating that a quantity stays constant in a closed system when no external interaction changes it.

External Torque

A torque caused by forces from outside the chosen system, which can change the system's total angular momentum.

Right-Hand Rule

A method for finding the direction of angular velocity, torque, or angular momentum by curling the fingers of the right hand in the rotation direction.