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 of determining which object (a ring or a solid sphere) is easier to stop when both are rotating with the same angular velocity about their diametric axes, we will analyze the moment of inertia for both shapes.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
- The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. The greater the moment of inertia, the more torque is required to change the object's rotational state.

2. **Moment of Inertia of a Solid Sphere**:
- The formula for the moment of inertia of a solid sphere about its diameter is given by:
\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \]
- Here, \( m \) is the mass and \( r \) is the radius of the sphere.

3. **Moment of Inertia of a Ring**:
- The moment of inertia of a ring about its diameter can be derived using the perpendicular axis theorem. The moment of inertia about an axis perpendicular to the plane of the ring is:
\[ I_{\text{ring, perpendicular}} = m r^2 \]
- According to the perpendicular axis theorem, the moment of inertia about any diameter of the ring is:
\[ I_{\text{ring}} = \frac{1}{2} I_{\text{ring, perpendicular}} = \frac{1}{2} m r^2 \]

4. **Comparing the Moments of Inertia**:
- Now we compare the moments of inertia:
\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \quad \text{and} \quad I_{\text{ring}} = \frac{1}{2} m r^2 \]
- To compare, we can convert both to a common denominator:
- For the sphere: \( \frac{2}{5} m r^2 = \frac{4}{10} m r^2 \)
- For the ring: \( \frac{1}{2} m r^2 = \frac{5}{10} m r^2 \)
- Thus, we find:
\[ I_{\text{ring}} > I_{\text{sphere}} \]

5. **Conclusion**:
- Since the ring has a greater moment of inertia than the solid sphere, it will be more difficult to stop the ring compared to the solid sphere.
- Therefore, it is easier to stop the solid sphere.

### Final Answer:
- It is easier to stop the solid sphere, and it is more difficult to stop the ring.