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

Hey there! Let's dive into the fascinating world of angular momentum. This principle is super important, and understanding it will definitely boost your confidence on the exam. Think of it as the rotational version of linear momentum conservation. Let's get started!

#Conservation of Angular Momentum

#What is it?

At its core, the conservation of angular momentum states that the total angular momentum of a closed system remains constant unless acted upon by an external torque. It's like a spinning top that keeps spinning unless something stops it. This principle is used everywhere, from planets to figure skaters! 🌌

#Sum of Angular Momenta 🔄

• Total Angular Momentum: To find the total angular momentum of a system, you simply add up the angular momenta of all its parts. Think of it like adding up all the individual spins to get the overall spin.
• Superposition: This principle applies the concept of superposition. Just add the individual contributions together. Easy peasy!
• Versatile: Works for point particles and extended objects. Just calculate each component's angular momentum relative to the axis of rotation.

#Changes in Angular Momentum

• External Torques: To change a system's total angular momentum, you need an external torque. This is the only way to speed up or slow down a system's rotation.
• Newton's Third Law: Angular impulse between two objects is equal in magnitude but opposite in direction. Just like regular forces, but for rotation.
• Closed Systems: If there are no external torques, angular momentum is conserved. This is when things get really interesting!
• Non-Rigid Systems: In non-rigid systems, you can change angular speed by altering mass distribution without changing the total angular momentum. Think of a figure skater pulling their arms in.
• Angular Impulse: The change in angular momentum is equal to the net angular impulse. We'll get to that in a sec.

#Angular Impulse

• Definition: Angular impulse is the product of the net torque and the time interval over which it acts. Units are $\text{kg} \cdot \text{m}^2/\text{s}$.
• Effect: It changes an object's angular momentum, just like linear impulse changes linear momentum.
• Calculation: You can calculate it using the integral $\int \vec{\tau} dt$ or $\vec{\tau} \Delta t$ for a constant torque.
• Impulse-Momentum Theorem: Change in angular momentum $\Delta \vec{L}$ is equal to the angular impulse. This is a crucial connection!

#Constant Angular Momentum 🌀

• Closed Systems: When there's no net external torque, the total angular momentum is conserved. This is the heart of the principle.
• Redistribution: Angular momentum can be redistributed within the system, but the total remains constant. Think of a spinning figure skater.
• Counterintuitive Behavior: This conservation leads to some cool effects, like a skater spinning faster when they pull their arms in.

#Angular Speed in Non-Rigid Systems

• Changing Speed: Angular speed can change without altering the total angular momentum if the mass distribution changes.
• Moment of Inertia: If the moment of inertia (I) decreases, the angular speed (ω) increases, and vice versa. Remember $L = Iω$!
• Mass Distribution: When components move closer to the rotational axis, the moment of inertia decreases, and the angular speed increases. 📏

#Angular Impulse Effects

• Change in Angular Momentum: $\Delta \vec{L} = \vec{\tau} \Delta t$ for a constant torque. This equation is your best friend!
• Calculations: You can use this to find the final angular momentum or angular velocity given the initial values and the applied angular impulse.
• Collisions: It's super useful for analyzing collisions and interactions involving rotating objects, like a ball hitting a spinning wheel.

#Final Exam Focus 🎯

Okay, let's get down to brass tacks. Here's what you should really focus on for the exam:

• High-Value Topics:
  • Conservation of angular momentum in closed systems.
  • Angular impulse and its relationship to changes in angular momentum.
  • How changes in the moment of inertia affect angular speed.
• Common Question Types:
  • Problems involving collisions of rotating objects.
  • Scenarios where a system's mass distribution changes (e.g., figure skater, rotating rod).
  • Conceptual questions about the conditions for conservation of angular momentum.
• Time Management:
  • Quickly assess if angular momentum is conserved before diving into calculations.
  • Use the impulse-momentum theorem for rotational motion to relate torques and changes in angular momentum.
• Common Pitfalls:
  • Forgetting to account for external torques.
  • Not considering the vector nature of angular momentum.
  • Incorrectly calculating the moment of inertia.

#

Practice Question

Practice Questions

Okay, let's put your knowledge to the test with some practice questions. These are designed to mimic what you'll see on the AP exam.

#Multiple Choice Questions

1. A spinning disk has a rotational inertia $I$ and an angular speed $\omega$. If the disk's rotational inertia is doubled while its angular momentum remains constant, what is the new angular speed of the disk? (A) $4\omega$ (B) $2\omega$ (C) $\omega/2$ (D) $\omega/4$

   Answer: (C) $\omega/2$ Since $L = I\omega$ and $L$ is constant, if $I$ doubles, $\omega$ must halve.

2. A figure skater is spinning with her arms outstretched. She then pulls her arms in close to her body. Which of the following statements is correct regarding the skater's angular momentum and rotational kinetic energy? (A) Both angular momentum and rotational kinetic energy increase. (B) Angular momentum increases, and rotational kinetic energy remains constant. (C) Angular momentum remains constant, and rotational kinetic energy increases. (D) Both angular momentum and rotational kinetic energy remain constant.

   Answer: (C) Angular momentum remains constant, and rotational kinetic energy increases. Angular momentum is conserved in the absence of external torques. Rotational kinetic energy increases because the skater does work to pull her arms in.

3. A uniform rod of mass M and length L is rotating about an axis perpendicular to the rod and passing through its center. If the rod is then rotated about a parallel axis at one end, how does the angular momentum change if the angular velocity remains the same?

   (A) Angular momentum increases (B) Angular momentum decreases (C) Angular momentum remains the same (D) Cannot be determined

   Answer: (A) Angular momentum increases. The moment of inertia about the end is greater than that about the center of the rod. Since L = Iω, if I increases and ω is constant, L increases.

#Free Response Question

A uniform disk of mass M and radius R is initially at rest on a frictionless horizontal surface. A small object of mass m is moving with a velocity v towards the disk, perpendicular to its radius, and collides with the edge of the disk. After the collision, the object sticks to the disk.

(a) Determine the angular momentum of the system before the collision about the center of the disk.

(b) Determine the moment of inertia of the disk about its center.

(c) Determine the moment of inertia of the system after the collision.

(d) Determine the angular velocity of the disk-object system immediately after the collision.

(e) Determine the change in kinetic energy of the system due to the collision.

Solution:

(a) Angular Momentum Before Collision

The angular momentum of the object before the collision is given by: $$L_{object} = mvr$$

The disk is initially at rest, so its angular momentum is zero. Thus, the total initial angular momentum of the system is: $$L_{initial} = mvr$$

(1 point) for correct expression of angular momentum of the object

(b) Moment of Inertia of the Disk

The moment of inertia of a uniform disk about its center is: $$I_{disk} = \frac{1}{2}MR^2$$

(1 point) for correct moment of inertia of the disk

(c) Moment of Inertia of the System After Collision

The moment of inertia of the object about the center of the disk is $mR^2$. The moment of inertia of the system after the collision is the sum of the moment of inertia of the disk and the object: $$I_{final} = I_{disk} + mR^2 = \frac{1}{2}MR^2 + mR^2$$

(1 point) for correct moment of inertia of the object (1 point) for correct moment of inertia of the system

(d) Angular Velocity After Collision

Since there are no external torques, the angular momentum of the system is conserved. Therefore, $L_{initial} = L_{final}$: $$mvr = I_{final} \omega_{final}$$ $$mvr = (\frac{1}{2}MR^2 + mR^2) \omega_{final}$$ $$\omega_{final} = \frac{mvr}{\frac{1}{2}MR^2 + mR^2} = \frac{mv}{\frac{1}{2}MR + mR}$$

(1 point) for correct application of conservation of angular momentum (1 point) for correct expression of final angular velocity

(e) Change in Kinetic Energy

The initial kinetic energy of the system is the kinetic energy of the object: $$KE_{initial} = \frac{1}{2}mv^2$$

The final kinetic energy of the system is the rotational kinetic energy of the disk-object system: $$KE_{final} = \frac{1}{2}I_{final}\omega_{final}^2 = \frac{1}{2}(\frac{1}{2}MR^2 + mR^2) \left(\frac{mv}{\frac{1}{2}MR + mR}\right)^2$$

The change in kinetic energy is: $$\Delta KE = KE_{final} - KE_{initial}$$

(1 point) for correct initial kinetic energy (1 point) for correct final kinetic energy (1 point) for correct change in kinetic energy