Angular Momentum

Angular momentum

• Linear momentum is defined as $p = mv$. Angular momentum is the rotational equivalent: $$L = I\omega$$
• Where: $L$ = angular momentum (kg m$^2$ rad s$^{-1}$), $I$ = moment of inertia (kg m$^2$), $\omega$ = angular velocity (rad s$^{-1}$).

Angular momentum of a point mass

• For a point mass $m$ at distance $r$ from the axis, with $I = mr^2$ and $\omega = \frac{v}{r}$: $$L = I\omega = (mr^2) \times \frac{v}{r} = mvr$$
• This is useful when dealing with particles moving in straight lines that have angular momentum about an external axis.

Conservation of angular momentum

• As with linear momentum, angular momentum is always conserved.
• The principle states: the angular momentum of a system always remains constant, unless a net torque is acting on the system.
• This conservation law has many real-world applications:
• A person on a spinning chair spins faster when they pull their arms in and slower when they extend them.
• Objects in elliptical orbits travel faster nearer the body they orbit and slower when further away.
• Ice skaters change their rotational velocity by extending or contracting their arms.
• Tornadoes spin faster as their radius decreases.
• Problems involving a change in angular momentum use: $$I_i\omega_i = I_f\omega_f$$

Worked example: exploding star

• A spherical star of mass $M$ and radius $R$ rotates about its axis. It explodes, leaving a remnant with mass $\frac{M}{10}$ and radius $\frac{R}{50}$. The moment of inertia of a sphere is $I = \frac{2}{5}MR^2$.
• Applying conservation of angular momentum: $$\left(\frac{2}{5}MR^2\right)\omega_i = \left(\frac{2}{5} \times \frac{M}{10} \times \left(\frac{R}{50}\right)^2\right)\omega_f$$
• Simplifying: $\frac{\omega_f}{\omega_i} = 25\,000$
• The remnant star spins 25,000 times faster than the original.