Let's answer part (a) step by step as you requested: --- ## (a) Given: - Inertia of solid sphere: \( I_1 = \frac{2}{5}MR_1^2 \) - Inertia of hollow sphere: \( I_2 = \frac{2}{3}MR_2^2 \) - Both spheres have same mass \( M \) and \( R_1 = 2R_2 \). - Both are rotating about their own axis with same kinetic energy. We are to find the angular speed ratio \( \frac{\omega_1}{\omega_2} \). --- ### **Step 1: Set up the kinetic energy equation** Rotational kinetic energy for each: \[ K = \frac{1}{2}I\omega^2 \] Since both have the same kinetic energy: \[ \frac{1}{2} I_1 \omega_1^2 = \frac{1}{2} I_2 \omega_2^2 \] \[ I_1 \omega_1^2 = I_2 \omega_2^2 \] __Explanation__: We use the formula for rotational kinetic energy and set them equal since both have the same energy. --- ### **Step 2: Substitute values and solve for \( \frac{\omega_1}{\omega_2} \)** Substitute \( I_1 = \frac{2}{5}MR_1^2 \), \( I_2 = \frac{2}{3}MR_2^2 \), and \( R_1 = 2R_2 \): \[ \frac{2}{5}M(2R_2)^2 \omega_1^2 = \frac{2}{3}M(R_2^2) \omega_2^2 \] \[ \frac{2}{5}M(4R_2^2) \omega_1^2 = \frac{2}{3}M(R_2^2) \omega_2^2 \] \[ \frac{8}{5}MR_2^2 \omega_1^2 = \frac{2}{3}MR_2^2 \omega_2^2 \] Divide both sides by \( MR_2^2 \): \[ \frac{8}{5} \omega_1^2 = \frac{2}{3} \omega_2^2 \] \[ \frac{\omega_1^2}{\omega_2^2} = \frac{2}{3} \times \frac{5}{8} \] \[ \frac{\omega_1^2}{\omega_2^2} = \frac{10}{24} = \frac{5}{12} \] \[ \frac{\omega_1}{\omega_2} = \sqrt{\frac{5}{12}} \] **Explanation:** We substituted the expressions for moment of inertia, simplified using the given relationship between radii, and solved for the angular speed ratio. --- ## **Final Answer:** \[ \boxed{ \frac{\omega_1}{\omega_2} = \sqrt{\frac{5}{12}} } \]