### Overview

Rotational motion includes: non-static torque (pulleys, ball rolling down inclines), angular kinematics, **angular momentum**, and **rotational kinetic energy**. This post will cover the latter two.

The word rotational and angular are used interchangeably.

### Rotational Kinetic Energy

Kinetic energy can be translational, rotational, or both.

If a marbles *slides* down a ramp it is said to have translational (linear) kinetic energy equal to (1/2)mv^{2}.

If a marble *rolls* down a ramp is is said to have both translational and rotational kinetic energy. See this detailed chart to learn more about the variable used in the formula below.

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

Note all other types of energy (Potential, elastic, etc) don’t have a rotational equation counterpart. It’s just for kinetic energy.

#### Solving Problems

- Use the law of conservation of energy → E_i = E_f
- If something is rolling and moving linear, be sure to include both rotational and translational energy in the equation.

#### Rotational Energy Example

A marble starts from rest and rolls down an incline. What was the speed at the bottom of the incline.

**Solution:** Energy is conserved. Specifically, potential energy turns into linear and rotational kinetic energy. Mathematically this should look like:

- Initial and final energy equality: E_i = E_f
- Energy components: mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2
- Substituting moment of inertia for a sphere: mgh = \frac{1}{2}mv^2 + \frac{1}{2} \left[ \frac{2}{5}mr^2 \right] \left( \frac{v^2}{r^2} \right)
- Simplifying: mgh = \frac{1}{2}mv^2 + \frac{1}{5}mv^2
- Combining like terms: mgh = \frac{7}{10}mv^2
- Solving for v: v = \sqrt{\frac{10}{7}gh}

### Angular Momentum

This is arguably the most commonly missed topic on the AP exam.

Linear momentum is p = mv, thus angular momentum is **L = Iω**.

\text{Change in angular momentum} = \text{Angular Impulse} = \Delta L = I\Delta \omega = \tau \cdot t

Just like how a linear impulse is caused by a net external force, an angular impulse is caused by a net external torque.

#### Important Concepts

Linear and angular momentum are conserved *separately*. This means you should NOT combine the conservation of linear momentum (P_{i} = P_{f}) with the conservation of angular momentum (**L _{f} = L_{i}**).

You can, however use linear variables in angular momentum. Makes senses?

Let’s make a formula for the angular momentum of a particle traveling at velocity v \rightarrow L = I\omega \rightarrow L = (mr^2) \cdot \left(\frac{v}{r}\right). This is still angular momentum, despite substituting in linear variables. Note that the mr^2 is the moment of inertia of a point mass.

#### Example

A disk of mass 2 kg and radius 1 meter, is rotating at 6 rad/s. Another disk of mass 5 kg is dropped on top of the rotating disk. What is the new angular velocity of the two disk system? What is the ratio of energy before and after the collision?

**Solution:**

Part 1: Apply conservation of angular momentum.

- L_f = L_i \rightarrow 0.5m_1r^2\omega_i = (0.5m_1r^2 + 0.5m_2r^2)\omega_f.
- Substitute in values and solve for \omega_f \rightarrow \omega_f = 1.09, \text{rad/s}.
- Does this makes sense? Yes! Since angular momentum is conserved velocity of the system should decrease as the moment of inertia increases.

Part 2: The energy is purely rotational:

- \frac{KE_i}{KE_f} \rightarrow \frac{0.5I_i\omega_i^2}{0.5I_f\omega_f^2} \rightarrow \frac{m_i\omega_i^2}{m_f\omega_f^2}
- Substitute numbers and solve the ratio: \frac{ 2 \cdot 6^2}{7 \cdot 1.09^2} = 8.7

### For the AP Exam…

Angular momentum is arguably the most misunderstood concept on the AP Physics 1 Exam. Here are a couple of tips:

- Use conservation of angular momentum separately from conservation of linear momentum
- Angular momentum is always conserved in a collision
- Total angular momentum of a system should
*not change*if there is no external force. - Angular momentum is
*not*conserved if there is an external force on the system. This is an impulse. - It is okay to substitute linear variables into the formulas for angular momentum. For example you can substitute ω with v/r.
- Practicing the questions below will help you increase your understanding.

### Practice Questions for Mastery

The child walks towards the center of the platform.

The child walks towards the edge of the platform.

The child walks in a circle, opposite to the rotational direction of the platform.

The child walks in a circle, along with the rotational direction of the platform.

none of the above will change the total angular momentum of the system.

_{i}. You can approximate her shape to be a cylinder. She is spinning with velocity ω

_{i}. As she extends her arms she her rotational inertia changes by a factor of x and her angular velocity changes by a factor of y. Which one of the following options best describe x and y.

x = 1, y < 1

x = 2; y = 1/2

x > 1; y < 1

x < 1; y > 1

(e) x >1; y > 1

_{i}about P. The system is gradually brought to rest by friction. Derive an expressions for the following quantities in terms of µ, m, x, g, and ω

_{i}.

*M*and radius

*r*, and rotational inertia

*I*is released from the top of a inclined plane of height

*h*. The surface has considerable friction. Using only the variable mentioned, derive an expression for the sphere's center of mass velocity.