#AP Physics 1: Angular Momentum & Impulse - The Night Before 🚀

Hey! Let's get you feeling super confident about angular momentum and impulse for tomorrow. This is your high-impact, last-minute guide, designed to make everything click. We'll break it down, connect the dots, and get you ready to ace this!

#Angular Momentum: The Spin of Things

Angular momentum is all about how much 'rotational oomph' an object has. It's like linear momentum, but for spinning stuff. Think of it as the tendency of a rotating object to keep rotating.

#Angular Momentum Equation 🌀

- For a rigid object: $L = I\omega$ - $L$ is **angular momentum** - $I$ is the **moment of inertia** (how hard it is to change rotation) - $\omega$ is the **angular velocity** (how fast it's spinning) -

#Angular Momentum About a Point

- For a point object: $L = rmv \sin \theta$ - $r$ is the distance from the reference point - $m$ is the mass - $v$ is the velocity - $\theta$ is the angle between $r$ and $v$ - The axis of rotation matters! A spinning top has different angular momentum if you rotate it around its center versus an off-center axis. - - Think of a ball thrown: - At 90°: Max angular momentum - At 45°: Less angular momentum
*Caption: A spinning top demonstrating angular momentum. The axis of rotation greatly impacts the calculation.*

#Angular Impulse: Changing the Spin

Angular impulse is what changes an object's angular momentum. It's like a 'rotational push' over time.

#Definition of Angular Impulse

- Angular impulse is the product of **torque** and **time**: $\text{Angular Impulse} = \tau \Delta t$ - $\tau$ is the **torque** (rotational force) - $\Delta t$ is the **time interval** -

#Direction of Angular Impulse

- Angular impulse has the **same direction** as the torque. - Clockwise torque = clockwise angular impulse, and vice versa.

#Graphical Representation of Impulse

- The **area under a torque vs. time graph** gives you the angular impulse. 📈 - If the torque is constant, it's just a rectangle; if it varies, you may need to calculate the area under the curve (e.g., with integration).
*Caption: The area under the curve of a torque vs. time graph represents the angular impulse.*

#Change in Angular Momentum: The Result

How much did the rotation change? That's what we're looking at here.

#Magnitude of Angular Momentum Change

- Change in angular momentum: $\Delta L = L\_f - L\_i$ - $L\_f$ is the **final angular momentum** - $L\_i$ is the **initial angular momentum** -

#Impulse-Momentum Theorem for Rotation

- The big connection: **Angular impulse = Change in angular momentum** - $\tau \Delta t = \Delta L$ - From Newton's second law: $\tau\_{net} = \frac{\Delta L}{\Delta t} = I\alpha$ - $\tau\_{net}$ is the **net torque** - $I$ is the **moment of inertia** - $\alpha$ is the **angular acceleration** -

#Torque and Angular Momentum Graphs

- The **slope of an angular momentum vs. time graph** is the **net torque**. - A constant slope means constant torque. 📊

#Graphical Representation of Impulse

- The **area under a net torque vs. time graph** is the **angular impulse**.
*Caption: The slope of an angular momentum vs. time graph represents the net torque.*

#🚫 Boundary Statement

- AP Physics 1 focuses on the *magnitude* of angular momentum, using 1D vector conventions. You don't need to worry about the direction of angular momentum and angular impulse.

#

- **High-Priority Topics:** - Angular momentum calculations (both $L = I\omega$ and $L = rmv \sin \theta$) - Angular impulse and its relation to torque and time - The impulse-momentum theorem for rotation - Interpreting graphs of torque vs. time and angular momentum vs. time - **Common Question Types:** - Multiple-choice questions involving calculations of angular momentum or impulse - Free-response questions requiring application of the impulse-momentum theorem - Questions asking you to interpret graphs and relate them to physical quantities - **Last-Minute Tips:** - **Time Management:** Quickly identify the core concepts in each question. - **Common Pitfalls:** Pay attention to units, directions, and the correct formula. - **Challenging Questions:** Break down complex problems into smaller parts. - **Stay Calm:** You've got this! Focus on what you know, and use your time wisely.

#

Practice Question

Practice Questions

### Multiple Choice Questions
1. A spinning skater pulls their arms inward. What happens to their angular momentum and angular velocity? (A) Angular momentum increases, angular velocity decreases. (B) Angular momentum decreases, angular velocity increases. (C) Angular momentum remains constant, angular velocity increases. (D) Angular momentum remains constant, angular velocity decreases.
2. A torque of 10 Nm is applied to a wheel for 2 seconds. What is the angular impulse? (A) 5 Nms (B) 10 Nms (C) 20 Nms (D) 40 Nms
3. A ball is thrown at an angle towards a reference point. Which of the following factors does NOT affect the angular momentum of the ball with respect to the reference point? (A) Mass of the ball (B) Speed of the ball (C) Distance between the reference point and the ball (D) Acceleration due to gravity

#Free Response Question

A solid disk with a mass of 2 kg and a radius of 0.5 m is initially rotating at 10 rad/s. A frictional force applies a constant torque of 2 Nm to the disk, slowing it down.
(a) Calculate the initial angular momentum of the disk. (The moment of inertia of a solid disk is $I = \frac{1}{2}mr^2$) (2 points)
(b) Calculate the angular acceleration of the disk. (2 points)
(c) How long does it take for the disk to come to a complete stop? (3 points)
(d) What is the angular impulse exerted by the frictional force to stop the disk? (2 points)
(e) Sketch a graph of angular momentum vs. time for the disk. (2 points)

### Answer Key
**Multiple Choice:**
1. (C) 2. (C) 3. (D)
**Free Response:**
(a) Calculate the initial angular momentum of the disk. (2 points) - $I = \frac{1}{2}mr^2 = \frac{1}{2}(2 kg)(0.5 m)^2 = 0.25 kgm^2$ - $L = I\omega = (0.25 kgm^2)(10 rad/s) = 2.5 kgm^2/s$
(b) Calculate the angular acceleration of the disk. (2 points) - $\tau = I\alpha$ - $\alpha = \frac{\tau}{I} = \frac{-2 Nm}{0.25 kgm^2} = -8 rad/s^2$ (negative because it's slowing down)
(c) How long does it take for the disk to come to a complete stop? (3 points) - $\omega\_f = \omega\_i + \alpha t$ - $0 = 10 rad/s + (-8 rad/s^2)t$ - $t = \frac{10}{8} = 1.25 s$
(d) What is the angular impulse exerted by the frictional force to stop the disk? (2 points) - Angular impulse = change in angular momentum - $\Delta L = L\_f - L\_i = 0 - 2.5 kgm^2/s = -2.5 kgm^2/s$
(e) Sketch a graph of angular momentum vs. time for the disk. (2 points) - The graph should be a straight line with a negative slope, starting at 2.5 $kgm^2/s$ and going to 0 $kgm^2/s$ at 1.25 seconds.

You've got this! Go crush that exam! 💪