A ring and a solid sphere of same mass and radius are rotating with the same angular velocity about their diameteric axes then :-

Options

A. it is easier to stop the ring
B. it is easier to stop the solid sphere
C. it is equally difficult to stop both of them
D. it is not possible to stop a rotating body

Answer

To solve the problem, we need to analyze the moment of inertia of both the ring and the solid sphere, as well as the implications of their angular velocities. Here’s a step-by-step breakdown:

### Step 1: Understand the Moment of Inertia
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation.

- For a **ring** rotating about its diameter, the moment of inertia is given by:
\[ I_{\text{ring}} = m r^2 \]
where \( m \) is the mass and \( r \) is the radius.

- For a **solid sphere** rotating about its diameter, the moment of inertia is given by:
\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \]

### Step 2: Compare the Moments of Inertia
Since both the ring and the solid sphere have the same mass \( m \) and radius \( r \), we can compare their moments of inertia directly.

- The moment of inertia of the ring is:
\[ I_{\text{ring}} = m r^2 \]

- The moment of inertia of the solid sphere is:
\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \]

### Step 3: Determine Which Has Greater Moment of Inertia
Now we can see that:
\[ I_{\text{ring}} > I_{\text{sphere}} \]
This means the ring has a greater moment of inertia than the solid sphere.

### Step 4: Analyze the Effect of Angular Velocity
Both objects are rotating with the same angular velocity (\( \omega \)). The torque (\( \tau \)) required to change the angular velocity is related to the moment of inertia and angular acceleration (\( \alpha \)) by the equation:
\[ \tau = I \alpha \]

### Step 5: Conclusion on Stopping the Objects
Since the ring has a greater moment of inertia, it will require more torque to stop it compared to the solid sphere. Therefore, it will be easier to stop the solid sphere than the ring.

### Final Answer
The correct option is:
**B: It is easier to stop the solid sphere.**