There are two identical spherical balls of same material one being solid and the other being hollow. Then they can be distinguished by

Options

A. spinning them by applying rqual torques
B. rolling them down on the same inclined plane
C. determining their moments of inertia
D. any one of the three method mentioned above

Answer

To distinguish between the two identical spherical balls, one solid and one hollow, we can use the following methods based on their physical properties, particularly their moment of inertia and behavior under rolling and torque.

### Step-by-Step Solution:

1. **Understanding the Properties**:
- Both balls are identical in mass (m) and radius (r).
- The solid sphere has a moment of inertia given by \( I_s = \frac{2}{5} m r^2 \).
- The hollow sphere has a moment of inertia given by \( I_h = \frac{2}{3} m r^2 \).
- Since \( I_h > I_s \), the hollow sphere has a greater moment of inertia than the solid sphere.

2. **Method 1: Spinning by Applying Equal Torque**:
- If we apply the same torque (τ) to both spheres, we can analyze their angular accelerations.
- The relationship is given by \( \tau = I \alpha \), where \( \alpha \) is the angular acceleration.
- Since the hollow sphere has a larger moment of inertia, it will have a smaller angular acceleration compared to the solid sphere when the same torque is applied.
- Therefore, the sphere with the greater angular acceleration is the solid sphere.

3. **Method 2: Rolling Down an Inclined Plane**:
- When both spheres are released from the same height on an inclined plane, they will roll down.
- The acceleration of a rolling object is given by \( a = \frac{g \sin \theta}{1 + \frac{I}{m r^2}} \).
- Since the moment of inertia of the hollow sphere is greater, its acceleration will be less than that of the solid sphere.
- Thus, the solid sphere will reach the bottom of the incline first.

4. **Method 3: Determining Moment of Inertia**:
- We can directly measure the moment of inertia of both spheres using a rotational apparatus.
- The sphere with the lower moment of inertia will be the solid sphere, while the one with the higher moment of inertia will be the hollow sphere.

5. **Conclusion**:
- All three methods (spinning with equal torque, rolling down an incline, and measuring moment of inertia) can effectively distinguish between the solid and hollow spheres.
- Therefore, the correct answer is option 4: "Any one of the three methods mentioned above."