Rotational Dynamics Quiz: Pulleys and Rotating Objects with Answer Key

📚 Quick Study Guide

• 📐 Moment of Inertia (I): A body's resistance to angular acceleration. For a point mass, $I = mr^2$. For different shapes, use standard formulas (e.g., for a solid cylinder, $I = \frac{1}{2}MR^2$).
• 💪 Torque ($\tau$): The rotational equivalent of force. $\tau = rF\sin(\theta)$, where $r$ is the distance from the axis of rotation to the point where the force is applied, $F$ is the magnitude of the force, and $\theta$ is the angle between $r$ and $F$. Also, $\tau = I\alpha$, where $\alpha$ is the angular acceleration.
• 🔄 Angular Velocity ($\omega$): The rate of change of angular displacement. Measured in rad/s.
• 🎢 Angular Acceleration ($\alpha$): The rate of change of angular velocity. Measured in rad/s².
• ⚡ Rotational Kinetic Energy (KE): The kinetic energy due to rotational motion. $KE = \frac{1}{2}I\omega^2$.
• 🧱 Work-Energy Theorem: The net work done is equal to the change in kinetic energy. $W = \Delta KE$. In rotational terms, $W = \tau \Delta \theta$.
• 🔗 Pulleys: When dealing with pulleys, consider the tension in the string and the radius of the pulley. The torque exerted by the tension is $\tau = rT$, where $T$ is the tension.

🧪 Practice Quiz

1. A solid cylinder (mass $M$, radius $R$) rolls down an incline without slipping. What is its linear acceleration?
   1. $a = \frac{1}{2}g\sin(\theta)$
   2. $a = \frac{2}{3}g\sin(\theta)$
   3. $a = g\sin(\theta)$
   4. $a = \frac{3}{4}g\sin(\theta)$
2. A block of mass $m$ is attached to a string wrapped around a pulley of mass $M$ and radius $R$. The block is released from rest. What is the tension in the string?
   1. $T = mg$
   2. $T = \frac{mg}{2}$
   3. $T = \frac{mg}{3}$
   4. $T = 2mg$
3. A rotating wheel has an angular velocity of 10 rad/s and an angular acceleration of 2 rad/s². After 5 seconds, what is its angular velocity?
   1. 10 rad/s
   2. 20 rad/s
   3. 30 rad/s
   4. 40 rad/s
4. A uniform rod of length $L$ and mass $M$ is pivoted at one end. What is the moment of inertia about the pivot point?
   1. $\frac{1}{12}ML^2$
   2. $\frac{1}{3}ML^2$
   3. $\frac{1}{2}ML^2$
   4. $ML^2$
5. A yo-yo is released from rest. What is the acceleration of its center of mass?
   1. $g$
   2. $\frac{g}{2}$
   3. $\frac{g}{3}$
   4. $\frac{2g}{3}$
6. A sphere of mass $M$ and radius $R$ rolls without slipping along a horizontal surface with a velocity $v$. What is its total kinetic energy?
   1. $\frac{1}{2}Mv^2$
   2. $\frac{3}{4}Mv^2$
   3. $\frac{5}{7}Mv^2$
   4. $\frac{7}{10}Mv^2$
7. What force should be applied on the edge of a 2 kg, 0.5m radius disc to produce an angular acceleration of 4 rad/$s^2$?
   1. 2 N
   2. 4 N
   3. 6 N
   4. 8 N