Make numerical problems on rotational motion

Here are several numerical problems related to rotational motion, ranging from basic to advanced levels. Each problem is followed by its solution for clarity. ### Problem 1: Moment of Inertia **Problem:** A solid cylinder of radius \( R = 0.5 \, \text{m} \) and mass \( M = 10 \, \text{kg} \) is rotating about its central axis. Calculate the moment of inertia \( I \) of the cylinder. **Solution:** The moment of inertia \( I \) of a solid cylinder rotating about its central axis is given by: \[ I = \frac{1}{2} M R^2 \] Substituting the given values: \[ I = \frac{1}{2} (10 \, \text{kg}) (0.5 \, \text{m})^2 = \frac{1}{2} (10) (0.25) = 1.25 \, \text{kg m}^2 \] Thus, \( I = 1.25 \, \text{kg m}^2 \). --- ### Problem 2: Angular Kinematics **Problem:** A disk starts from rest and rotates with a constant angular acceleration of \( 2 \, \text{rad/s}^2 \). How much time (in seconds) will it take to reach an angular speed of \( 10 \, \text{rad/s} \)? **Solution:** Using the angular kinematics equation: \[ \omega_f = \omega_i + \alpha t \] Where: - \( \omega_f = 10 \, \text{rad/s} \) (final angular velocity) - \( \omega_i = 0 \, \text{rad/s} \) (initial angular velocity) - \( \alpha = 2 \, \text{rad/s}^2 \) (angular acceleration) Rearranging gives: \[ t = \frac{\omega_f - \omega_i}{\alpha} = \frac{10 - 0}{2} = 5 \, \text{s} \] --- ### Problem 3: Torque and Rotational Equilibrium **Problem:** A seesaw is in equilibrium with a 30 kg child sitting 2 m from the central pivot. If another child sits on the opposite side of the pivot at a distance of 1.5 m, what is the mass of the second child for the seesaw to be balanced? **Solution:** For equilibrium, the clockwise moment equals the counterclockwise moment. \[ \text{Torque from Child 1} = \text{Torque from Child 2} \] \[ M_1 \cdot d_1 = M_2 \cdot d_2 \] Where: - \( M_1 = 30 \, \text{kg} \) - \( d_1 = 2 \, \text{m} \) - \( d_2 = 1.5 \, \text{m} \) - \( M_2 \) is the mass of the second child. Substituting values: \[ 30 \cdot 2 = M_2 \cdot 1.5 \] \[ 60 = M_2 \cdot 1.5 \] \[ M_2 = \frac{60}{1.5} = 40 \, \text{kg} \] --- ### Problem 4: Conservation of Angular Momentum **Problem:** A rotating disk with a moment of inertia of \( 3 \, \text{kg m}^2 \) is spinning at an angular speed of \( 4 \, \text{rad/s} \). If a small mass of \( 1 \, \text{kg} \) is dropped on the edge of the disk, calculate the new angular speed of the disk if the moment of inertia of the mass with respect to the pivot is \( 0.5 \, \text{kg m}^2 \). **Solution:** Using the conservation of angular momentum: \[ L_i = L_f \] Where: \[ L_i = I_i \omega_i, \quad I_f = I_i + I_m, \quad L_f = I_f \omega_f \] Given: - \( I_i = 3 \, \text{kg m}^2 \) - \( \omega_i = 4 \, \text{rad/s} \) - \( I_m = 0.5 \, \text{kg m}^2 \) Calculating initial angular momentum: \[ L_i = 3 \cdot 4 = 12 \, \text{kg m}^2/\text{s} \] Calculating final moment of inertia: \[ I_f = 3 + 0.5 = 3.5 \, \text{kg m}^2 \] Setting angular momentum equal: \[ 12 = 3.5 \cdot \omega_f \] Solving for \( \omega_f \): \[ \omega_f = \frac{12}{3.5} \approx 3.43 \, \text{rad/s} \] --- These problems provide a comprehensive overview of key concepts in rotational motion, including moment of inertia, angular kinematics, torque, and angular momentum. Feel free to adjust the levels of complexity or create additional problems based on these concepts!