# Angular Momentum and Conservation

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/angular-momentum-and-conservation

## Angular Momentum Basics

A spinning skater pulls her arms in and speeds up with no one pushing her. With no external torque, the product $L = I\omega$ is locked. Shrink $I$ and $\omega$ must rise.

**Angular momentum** is the rotational analogue of linear momentum:

$$\vec{L} = \vec{r}\times\vec{p} = I\vec{\omega}$$

Here, $I$ is rotational inertia and $\omega$ is angular velocity.

**Newton's second law (rotational form)** connects torque and momentum:

$$\vec{\tau}_{\text{net}} = \frac{d\vec{L}}{dt}$$

If net external torque is zero, **conservation of angular momentum** applies:

$$\vec{L}_i = \vec{L}_f \quad\Longrightarrow\quad I_i\omega_i = I_f\omega_f$$

**Classic example**: An ice skater pulls their arms in. $I$ decreases as mass moves closer to the axis, forcing $\omega$ to increase to keep $L$ constant.

## Energy and Analogues

**Rotational kinetic energy** is given by:

$$KE_{\text{rot}} = \frac{1}{2}I\omega^2$$

For **rolling without slipping**, $v_{\text{cm}} = R\omega$ and total KE combines translation and rotation:

$$KE_{\text{total}} = \frac{1}{2}mv_{\text{cm}}^2 + \frac{1}{2}I\omega^2$$

| Translation | Rotation |
| --- | --- |
| $\vec{F}$ | $\vec{\tau}$ |
| $\vec{p}=m\vec{v}$ | $\vec{L}=I\vec{\omega}$ |
| $KE=\tfrac{1}{2}mv^{2}$ | $KE=\tfrac{1}{2}I\omega^{2}$ |

**Common pitfall**: Conserving $L$ does not conserve rotational kinetic energy. The skater’s $KE = L^{2}/2I$ increases as she pulls her arms in because her muscles supply extra work. One conservation law never implies another.

## Practice questions

3 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. A spinning figure skater pulls her arms in and speeds up (angular momentum is conserved). What happens to her **rotational kinetic energy**?

A. It decreases, because her moment of inertia decreases
B. It stays the same, because no external torque acts
C. It increases, her muscles do work pulling her arms inward
D. It stays the same, because angular momentum is conserved

**Answer:** C. It increases, her muscles do work pulling her arms inward

**Why:** With $L = I\omega$ constant, $KE = L^{2}/2I$ rises as $I$ falls. Conservation of one quantity does not imply conservation of another: the extra energy is supplied by the skater’s muscles.

Page: https://tryals.app/practice/physics-i/angular-momentum-and-conservation/a-spinning-figure-skater-pulls-her-arms-in-and-speeds-up-angular

### 2. A skater spins at **2.0 rev/s** with arms out. Pulling her arms in halves her moment of inertia. Predict her new spin rate, in rev/s.

**Answer:** 4 (within ±0.4)

**Why:** Halving $I$ doubles $\omega$: she now spins at 4 rev/s. Neutron stars do the same trick on a cosmic scale, a collapsing stellar core spins up to hundreds of revolutions per second.

Page: https://tryals.app/practice/physics-i/angular-momentum-and-conservation/a-skater-spins-at-2-0-rev-s-with-arms-out-pulling-her-arms-in-halves

### 3. Sort: is angular momentum conserved or not in each scenario?

**Answer:**

- Conserved: Spinning top with no friction, Planet orbiting the Sun, Ice skater pulling arms in, Two disks coupling on a shared axle
- Not conserved: Wheel slowed by brake friction, Gyroscope on a stand (torque from gravity)

**Why:** Angular momentum is conserved when net external torque is zero.

Page: https://tryals.app/practice/physics-i/angular-momentum-and-conservation/sort-is-angular-momentum-conserved-or-not-in-each-scenario