Which of the following has the highest moment of inertia when each of them has the same mass and the same radius ? a) A ring about any of its diameter. b) A disc about any of its diameter. c) A hollow sphere about any of its diameter. d) A solid sphere about any of its diameter

Options

A. A ring about any of its diameter.
B. A disc about any of its diameter.
C. A hollow sphere about any of its diameter.
D. A solid sphere about any of its diameter.

Explanation

To determine which of the given objects has the highest moment of inertia when they all have the same mass and radius, we need to calculate the moment of inertia for each object.

### Step-by-Step Solution:

1. **Identify the Objects**: We have four objects to consider:
- A ring about any of its diameter.
- A disc about any of its diameter.
- A hollow sphere about any of its diameter.
- A solid sphere about any of its diameter.

2. **Define Mass and Radius**: Let the mass of each object be \( m \) and the radius be \( R \).

3. **Calculate the Moment of Inertia for Each Object**:
- **For the Ring**: The moment of inertia \( I \) about any diameter is given by:
\[ I_{\text{ring}} = \frac{1}{2} m R^2 \]
- **For the Disc**: The moment of inertia \( I \) about any diameter is given by:
\[ I_{\text{disc}} = \frac{1}{4} m R^2 \]
- **For the Hollow Sphere**: The moment of inertia \( I \) about any diameter is given by:
\[ I_{\text{hollow sphere}} = \frac{2}{5} m R^2 \]
- **For the Solid Sphere**: The moment of inertia \( I \) about any diameter is given by:
\[ I_{\text{solid sphere}} = \frac{2}{5} m R^2 \]

4. **Compare the Moments of Inertia**:
- Ring: \( \frac{1}{2} m R^2 \)
- Disc: \( \frac{1}{4} m R^2 \)
- Hollow Sphere: \( \frac{2}{3} m R^2 \)
- Solid Sphere: \( \frac{2}{5} m R^2 \)

5. **Convert to a Common Form**:
- To compare these values, we can express them with a common denominator:
- Ring: \( \frac{1}{2} = \frac{3}{6} \)
- Disc: \( \frac{1}{4} = \frac{1.5}{6} \)
- Hollow Sphere: \( \frac{2}{3} = \frac{4}{6} \)
- Solid Sphere: \( \frac{2}{5} = \frac{2.4}{6} \)

6. **Determine the Highest Moment of Inertia**:
- From the comparisons, we see:
- Ring: \( \frac{3}{6} m R^2 \)
- Disc: \( \frac{1.5}{6} m R^2 \)
- Hollow Sphere: \( \frac{4}{6} m R^2 \)
- Solid Sphere: \( \frac{2.4}{6} m R^2 \)
- The hollow sphere has the highest moment of inertia.

### Conclusion:
The object with the highest moment of inertia is the **hollow sphere** about any of its diameter. Therefore, the correct answer is option **C**.