Question

A solid sphere of mass $M$ and radius $R$ is rotating about its diameter. A disc of same mass and radius is also rotating about an axis passing through its centre and perpendicular to the plane but angular speed is twice that of the sphere. The ratio of kinetic energy of disc to that of sphere is

• A. $5:1$
• B. $6:1$
• C. $4:1$
• D. $3:1$

Hint:

Always check if angular speeds are same or different before comparing rotational kinetic energies.

Answer 1

The Correct Option is A

Step 1: Moment of inertia expressions.
For solid sphere:
\[ I_s = \frac{2}{5}MR^2 \] For disc:
\[ I_d = \frac{1}{2}MR^2 \]
Step 2: Rotational kinetic energy formula.
\[ K = \frac{1}{2}I\omega^2 \]
Step 3: Kinetic energies.
Sphere:
\[ K_s = \frac{1}{2}\cdot\frac{2}{5}MR^2\omega^2 = \frac{1}{5}MR^2\omega^2 \] Disc (angular speed $=2\omega$):
\[ K_d = \frac{1}{2}\cdot\frac{1}{2}MR^2(2\omega)^2 = MR^2\omega^2 \]
Step 4: Ratio of kinetic energies.
\[ \frac{K_d}{K_s} = \frac{MR^2\omega^2}{\frac{1}{5}MR^2\omega^2} = 5 \]
Step 5: Conclusion.
The ratio of kinetic energy of disc to sphere is $5:1$.