Rotational Energy and Angular Momentum Speed Review

Angular Momentum

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

Linear momentum is \(p = mv\), thus angular momentum is \(L = I\omega\).

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

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. Make sense?

Let’s make a formula for the angular momentum of a particle traveling at velocity [katex]v \rightarrow L = I\omega \rightarrow L = (mr^2) \cdot \left(\frac{v}{r}\right)[/katex]. This is still angular momentum, despite substituting in linear variables. Note that the [katex] mr^2 [/katex] 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 and the same 1 meter radius 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.

1. [katex]L_f = L_i \rightarrow 0.5m_1r^2\omega_i = (0.5m_1r^2 + 0.5m_2r^2)\omega_f[/katex].
2. Substitute in values and solve for [katex]\omega_f \rightarrow \omega_f = 1.71\,\text{rad/s}[/katex].
3. Does this make 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:

1. [katex]\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}[/katex]
2. Substitute numbers and solve the ratio: [katex]\frac{ 2 \cdot 6^2}{7 \cdot 1.71^2} = 3.5[/katex]

📌 Key Takeaways (a.k.a. Your Cheat Sheet)

• Rotational kinetic energy: \( KE_{\text{rot}} = \frac{1}{2} I \omega^2 \)
• Rolling motion includes both linear & rotational kinetic energy
• Angular momentum is conserved unless an external torque acts
• Momentum formulas: \( L = I \omega \) and \( \Delta L = \tau t \)
• If inertia increases, angular velocity decreases (spinning slower)
• Rotational collisions often lose energy

For the AP Exam…

Angular momentum is one of the most missed concepts on the AP Physics 1 Exam. Here are a couple of things to look out for:

1. Use conservation of angular momentum separately from conservation of linear momentum
2. Angular momentum is conserved in a collision only if there is no net external torque
3. Total angular momentum of a system should not change if there is no external torque.
4. Angular momentum is not conserved if there is a net external torque on the system. This is an angular impulse.
5. It is okay to substitute linear variables into the formulas for angular momentum. For example you can substitute \(\omega\) with \(v/r\).
6. Practicing the questions below will help you increase your understanding.

Practice Questions for Mastery