Angular Momentum and Its Conservation

Jane Doe

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Study Guide Overview

This study guide covers the conservation of angular momentum, a crucial topic in AP Physics C: Mechanics. It explains the principle, calculations for point particles and extended objects, and key principles like the vector sum of angular momenta and conservation in isolated systems. It also explores common scenarios like colliding disks and the ballistic pendulum, combining concepts like rotational inertia, conservation of momentum, and conservation of energy. Finally, it provides practice questions and a final exam focus with high-priority topics, common question types, and last-minute tips.

#AP Physics C: Mechanics - Conservation of Angular Momentum 🚀

This topic is crucial, often appearing in both multiple-choice and free-response questions. Expect to see it combined with rotational motion and energy conservation.

# Conservation of Angular Momentum

Conservation of Angular Momentum: The total angular momentum of a system remains constant unless acted upon by a net external torque. 🔄

Angular momentum (L) is a measure of an object's rotational motion. It's calculated differently for point particles and extended objects:

Point particle: $L = r \times p = r m v \sin(\theta)$ , where $r$ is the position vector from the axis of rotation to the particle, $p$ is the linear momentum, and $\theta$ is the angle between $r$ and $p$ .
Extended object: $L = I \omega$ , where $I$ is the moment of inertia and $\omega$ is the angular velocity.

#Key Principles

The total angular momentum of a system is the vector sum of the angular momenta of all its parts.
In an isolated system (no net external torque), total angular momentum is conserved ( $L_{initial} = L_{final}$ ).
If a torque is applied, the angular momentum will change. The relationship is given by: $\tau = \frac{dL}{dt}$ , where $\tau$ is the net torque and $\frac{dL}{dt}$ is the rate of change of angular momentum.
If the moment of inertia changes, the angular velocity must change to conserve angular momentum.

Key Concept

Conservation of angular momentum is a direct consequence of Newton's second law for rotational motion, and it's a powerful tool for solving problems involving rotational motion.

#Visualizing Angular Momentum

Angular Momentum

Torque is the slope of an angular momentum vs. time graph!

Common Mistake

Don't confuse linear momentum ( $p = mv$ ) with angular momentum ( $L = I\omega$ ). You can't simply multiply linear momentum by a radius to get angular momentum unless you are dealing with a point particle!

#Common Situations of Conservation of Angular Momentum

# Disks Colliding

Similar to linear momentum problems, but now we must consider rotational inertia. When disks collide, they exert torques on each other, but these are internal to the system, so angular momentum is conserved.

Disk Collision

When disks collide, angular momentum is conserved because the torques are internal to the system.

# Ballistic Pendulum

This classic problem involves a projectile colliding with and embedding itself in a pendulum. It combines conservation of momentum (during the collision) and conservation of energy (as the pendulum swings).

Collision (Inelastic): Linear momentum is conserved, but kinetic energy is not. You can use the formula: $m_1v_1 = (m_1 + m_2)v_f$ where $m_1$ is the mass of the projectile, $v_1$ is the initial velocity of the projectile, $m_2$ is the mass of the pendulum, and $v_f$ is the final velocity of the combined mass just after the collision.
Swing: After the collision, the pendulum swings upward, converting kinetic energy into gravitational potential energy. You can use the formula: $\frac{1}{2}(m_1 + m_2)v_f^2 = (m_1 + m_2)gh$ where $h$ is the height of the swing.

Exam Tip

For a ballistic pendulum, remember to use linear momentum during the collision and then switch to energy conservation for the swing. If the pendulum has rotational inertia, use conservation of angular momentum during the collision.

Ballistic Pendulum

A ballistic pendulum combines conservation of momentum and energy.

#Satellites

Quick Fact

Angular momentum is conserved for satellites in orbit. As a satellite moves closer to a planet, its speed increases, and vice-versa. This is why planets move faster when they are closer to the sun.

We'll delve into this more in Unit 7, but remember that angular momentum is conserved in satellite motion. This means that as a satellite's distance from the planet changes, its orbital speed also changes to keep angular momentum constant.

# Final Exam Focus

High-Priority Topics:
- Conservation of Angular Momentum
- Rotational Kinematics and Dynamics
- Energy Conservation
- Combining Rotational and Translational Motion
Common Question Types:
- Disk collisions
- Ballistic pendulum problems (with and without rotational inertia)
- Satellite motion (qualitative)
- Problems involving changing moment of inertia
Last-Minute Tips:
- Time Management: Quickly identify the type of problem (e.g., collision, swing) and apply the correct conservation laws.
- Common Pitfalls: Be careful not to confuse linear and angular quantities. Always check units and ensure your answers make physical sense.
- Strategies: Draw free-body diagrams, clearly label variables, and show all your work. If you get stuck, move on and come back later.

Memory Aid

I Love Rotational Motion (ILRM): Use this to remember the key concepts: Inertia, Loss of Energy (in inelastic collisions), Rotational motion, Momentum (angular and linear).

# Practice Questions

Practice Question

Multiple Choice Questions:

A uniform disk rotates about a fixed axis. A second identical disk, initially at rest, is dropped onto the first disk. They stick together and rotate with a new angular velocity. Which of the following is true about the system's angular momentum and kinetic energy during this process?
- (A) Both angular momentum and kinetic energy are conserved.
- (B) Angular momentum is conserved, but kinetic energy is not.
- (C) Kinetic energy is conserved, but angular momentum is not.
- (D) Neither angular momentum nor kinetic energy is conserved.
Answer: B
A figure skater spins with her arms outstretched. As she pulls her arms in close to her body, what happens to her angular speed and moment of inertia?
- (A) Her angular speed increases, and her moment of inertia increases.
- (B) Her angular speed increases, and her moment of inertia decreases.
- (C) Her angular speed decreases, and her moment of inertia increases.
- (D) Her angular speed decreases, and her moment of inertia decreases.
Answer: B
A solid sphere rolls down an inclined plane without slipping. Which of the following quantities is conserved during its motion?
- (A) Linear momentum
- (B) Angular momentum about the center of mass
- (C) Total mechanical energy
- (D) Both linear and angular momentum
Answer: C

Free Response Question:

A bullet of mass $m$ is fired horizontally with a speed $v_0$ at a wooden block of mass $M$ that is initially at rest on a frictionless horizontal surface. The bullet embeds itself in the block. The block is attached to a massless rod of length $L$ that is pivoted at the other end. The system then swings upward. Assume the bullet is a point mass and the wooden block is a point mass.

(a) Determine the speed of the block and bullet immediately after the collision.
(b) Determine the angular speed of the block and bullet immediately after the collision.
(c) Determine the maximum angle <math-inline>\theta</math-inline> that the rod makes with the vertical as the block swings upward.
(d) If the wooden block is not a point mass but a uniform solid sphere of radius <math-inline>R</math-inline>, how will the maximum angle <math-inline>\theta</math-inline> change? Explain your reasoning.

Scoring Rubric:

(a) **2 points**
    *   1 point for using conservation of linear momentum: <math-inline>mv\_0 = (m+M)v\_f</math-inline>
    *   1 point for solving for <math-inline>v\_f</math-inline>: <math-inline>v\_f = \frac{m}{m+M}v\_0</math-inline>

(b) **2 points**
    *   1 point for relating linear speed to angular speed: <math-inline>v\_f = \omega L</math-inline>
    *   1 point for solving for <math-inline>\omega</math-inline>: <math-inline>\omega = \frac{v\_f}{L} = \frac{m}{L(m+M)}v\_0</math-inline>

(c) **3 points**
    *   1 point for using conservation of energy: <math-inline>\frac{1}{2}(m+M)v\_f^2 = (m+M)gh</math-inline>
    *   1 point for relating <math-inline>h</math-inline> to <math-inline>\theta</math-inline>: <math-inline>h = L(1 - \cos\theta)</math-inline>
    *   1 point for solving for <math-inline>\theta</math-inline>: <math-inline>\theta = \cos^{-1}(1 - \frac{v\_f^2}{2gL})</math-inline>

(d) **3 points**
    *   1 point for stating that the moment of inertia of the sphere is greater than that of a point mass.
    *   1 point for stating that the total kinetic energy of the sphere is greater than that of a point mass.
    *   1 point for concluding that the maximum angle <math-inline>\theta</math-inline> will be smaller.

Answers to Multiple Choice Questions:

Answers to Free Response Questions:

(a) $v_f = \frac{m}{m+M}v_0$

(b) $\omega = \frac{m}{L(m+M)}v_0$

(d) The maximum angle $\theta$ will be smaller because the sphere has a larger moment of inertia and will have a larger kinetic energy, which means less energy will be converted to potential energy resulting in a smaller height and angle.