AP Physics C: Mechanics questions on linear/angular relationship with answers

📚 Quick Study Guide

• 📏 Arc Length: The distance traveled along a circular path is given by $s = r\theta$, where $s$ is the arc length, $r$ is the radius, and $\theta$ is the angular displacement in radians.
• ⏱️ Angular Velocity: The rate of change of angular displacement is angular velocity, $\omega = \frac{d\theta}{dt}$.
• ⚙️ Angular Acceleration: The rate of change of angular velocity is angular acceleration, $\alpha = \frac{d\omega}{dt}$.
• 🔄 Tangential Velocity: The linear velocity of a point on a rotating object is related to the angular velocity by $v = r\omega$.
• 🎢 Tangential Acceleration: The linear acceleration of a point on a rotating object is related to the angular acceleration by $a = r\alpha$.
• 💪 Torque: The rotational equivalent of force, torque is given by $\tau = I\alpha$, where $I$ is the moment of inertia.
• 🤸 Moment of Inertia: A measure of an object's resistance to rotational acceleration. For a point mass, $I = mr^2$. For more complex objects, $I$ depends on the shape and axis of rotation.
• 🛢️ Kinetic Energy (Rotational): The kinetic energy due to rotation is given by $K = \frac{1}{2}I\omega^2$.
• 💫 Angular Momentum: The rotational equivalent of linear momentum, given by $L = I\omega$.

📝 Practice Quiz

1. A wheel with a radius of 0.5 m rotates with a constant angular acceleration of 2 rad/s². What is the tangential acceleration of a point on the edge of the wheel?
   1. 0.5 m/s²
   2. 1 m/s²
   3. 2 m/s²
   4. 4 m/s²
2. A disc is rotating at 4 rad/s. It slows down with a constant angular acceleration of -0.5 rad/s². How long does it take for the disc to come to rest?
   1. 2 s
   2. 4 s
   3. 8 s
   4. 16 s
3. A particle moves in a circle of radius 2 m with a constant speed of 6 m/s. What is its angular speed?
   1. 1 rad/s
   2. 2 rad/s
   3. 3 rad/s
   4. 6 rad/s
4. A bicycle wheel has a radius of 0.35 m. If the bicycle is traveling at 10 m/s, what is the angular speed of the wheel?
   1. 14.3 rad/s
   2. 28.6 rad/s
   3. 3.5 rad/s
   4. 7.0 rad/s
5. A rotating object has a moment of inertia of 5 kg·m² and an angular speed of 2 rad/s. What is its rotational kinetic energy?
   1. 5 J
   2. 10 J
   3. 15 J
   4. 20 J
6. A solid sphere rolls down an incline without slipping. What fraction of its total kinetic energy is rotational?
   1. 2/7
   2. 2/5
   3. 5/7
   4. 5/12
7. What is the relationship between torque ($\tau$), moment of inertia ($I$), and angular acceleration ($\alpha$)?
   1. $\tau = I + \alpha$
   2. $\tau = I - \alpha$
   3. $\tau = I\alpha$
   4. $\tau = I/\alpha$