Question

A solid cylinder of mass ' \( M \) ' and radius ' \( R \) ' is rotating about its geometrical axis. A solid sphere of same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be}

• A. 2 : 3
• B. 3 : 2
• C. 1 : 5
• D. 5 : 1

Hint:

Ratio $= \frac{I_1 \omega_1^2}{I_2 \omega_2^2}$. Don't forget to square the angular velocity!

Answer 1

The Correct Option is C

Step 1: Parameters
Cylinder: $I_c = \frac{1}{2}MR^2$, $\omega_c = \omega$.
Sphere: $I_s = \frac{2}{5}MR^2$, $\omega_s = \frac{\omega}{2}$.
Step 2: Kinetic Energy Formula
$K.E. = \frac{1}{2} I \omega^2$.
Step 3: Ratio Calculation
$\frac{K_s}{K_c} = \frac{\frac{1}{2} I_s \omega_s^2}{\frac{1}{2} I_c \omega_c^2} = \frac{\frac{2}{5}MR^2 \cdot (\frac{\omega}{2})^2}{\frac{1}{2}MR^2 \cdot \omega^2} = \frac{\frac{2}{5} \cdot \frac{1}{4}}{\frac{1}{2}} = \frac{1/10}{1/2} = \frac{1}{5}$.
Final Answer: (C)