Which of the following bodies of same mass and same radius has minimum moment of inertia?

Options

A. Ring
B. Disc
C. Hollow sphere
D. Solid sphere

Answer

To determine which of the given bodies of the same mass and same radius has the minimum moment of inertia, we need to calculate the moment of inertia for each body. The moment of inertia (I) is a measure of an object's resistance to changes in its rotation and depends on the mass distribution relative to the axis of rotation.

### Step-by-Step Solution:

1. **Identify the Bodies**: We are considering four bodies: a ring, a disk, a hollow sphere, and a solid sphere. All have the same mass (m) and radius (r).

2. **Write the Moment of Inertia Formulas**:
- Moment of inertia of a **ring** about its central axis:
\[ I_{\text{ring}} = m r^2 \]
- Moment of inertia of a **disk** about its central axis:
\[ I_{\text{disk}} = \frac{1}{2} m r^2 \]
- Moment of inertia of a **hollow sphere** about its diameter:
\[ I_{\text{hollow sphere}} = \frac{2}{3} m r^2 \]
- Moment of inertia of a **solid sphere** about its diameter:
\[ I_{\text{solid sphere}} = \frac{2}{5} m r^2 \]

3. **Compare the Coefficients**:
- For the ring: \( I_{\text{ring}} = m r^2 \) (coefficient = 1)
- For the disk: \( I_{\text{disk}} = \frac{1}{2} m r^2 \) (coefficient = 0.5)
- For the hollow sphere: \( I_{\text{hollow sphere}} = \frac{2}{3} m r^2 \) (coefficient ≈ 0.67)
- For the solid sphere: \( I_{\text{solid sphere}} = \frac{2}{5} m r^2 \) (coefficient = 0.4)

4. **Identify the Minimum Moment of Inertia**:
- Now we can compare the coefficients:
- Ring: 1
- Disk: 0.5
- Hollow Sphere: 0.67
- Solid Sphere: 0.4
- The smallest coefficient is 0.4, which corresponds to the solid sphere.

5. **Conclusion**:
- Therefore, the body with the minimum moment of inertia is the **solid sphere**.

### Final Answer:
The solid sphere has the minimum moment of inertia among the given bodies.

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