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

Options

A. (a)Ring
B. (b)Disc
C. (c)Hollow sphere
D. (d)Solid sphere

Answer

To determine which body of the same mass and radius has the minimum moment of inertia, we need to calculate the moment of inertia for each of the given shapes. Let's analyze each shape step by step.

### Step 1: Understand the Moment of Inertia
The moment of inertia (I) of a body depends on its mass distribution relative to the axis of rotation. The formula for the moment of inertia varies for different shapes.

### Step 2: Calculate the Moment of Inertia for Each Shape
1. **Ring**:
\[ I_{\text{ring}} = M R^2 \]
where \(M\) is the mass and \(R\) is the radius.

2. **Disc**:
\[ I_{\text{disc}} = \frac{1}{2} M R^2 \]

3. **Hollow Sphere**:
\[ I_{\text{hollow sphere}} = \frac{2}{3} M R^2 \]

4. **Solid Sphere**:
\[ I_{\text{solid sphere}} = \frac{2}{5} M R^2 \]

### Step 3: Compare the Moments of Inertia
Now that we have the formulas for the moment of inertia for each shape, we can compare them:

- \(I_{\text{ring}} = M R^2\)
- \(I_{\text{disc}} = \frac{1}{2} M R^2\)
- \(I_{\text{hollow sphere}} = \frac{2}{3} M R^2\)
- \(I_{\text{solid sphere}} = \frac{2}{5} M R^2\)

### Step 4: Determine the Minimum Moment of Inertia
To find the minimum moment of inertia, we can compare the coefficients of \(M R^2\):

- Ring: \(1\)
- Disc: \(\frac{1}{2}\)
- Hollow Sphere: \(\frac{2}{3}\)
- Solid Sphere: \(\frac{2}{5}\)

Among these values, \(\frac{2}{5}\) (for the solid sphere) is the smallest.

### Conclusion
Thus, the body with the minimum moment of inertia is the **solid sphere**.

### Final Answer
The correct option is **D: Solid Sphere**.

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