The moment of inertia l of a solid sphere having fixed volume depends upon its volume V as

Options

A. `I prop V`
B. `I prop V^(2//3)`
C. `I prop V^(5//3)`
D. `I prop V^(3//2)`

Answer

To determine how the moment of inertia \( I \) of a solid sphere with fixed volume \( V \) depends on its volume, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Moment of Inertia Formula**:
The moment of inertia \( I \) of a solid sphere is given by the formula:
\[ I = \frac{2}{5} m r^2 \]
where \( m \) is the mass and \( r \) is the radius of the sphere.

2. **Relate Mass to Volume**:
Since the volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
we can express the mass \( m \) in terms of volume and density \( \rho \):
\[ m = \rho V \]

3. **Substitute Mass in the Moment of Inertia Formula**:
Substitute \( m \) into the moment of inertia formula:
\[ I = \frac{2}{5} (\rho V) r^2 \]

4. **Express Radius in Terms of Volume**:
From the volume formula, we can express \( r \) in terms of \( V \):
\[ r = \left(\frac{3V}{4\pi}\right)^{1/3} \]

5. **Substitute Radius into the Moment of Inertia**:
Now substitute \( r \) into the moment of inertia equation:
\[ I = \frac{2}{5} (\rho V) \left(\left(\frac{3V}{4\pi}\right)^{1/3}\right)^2 \]
Simplifying this gives:
\[ I = \frac{2}{5} \rho V \left(\frac{3V}{4\pi}\right)^{2/3} \]

6. **Combine the Terms**:
Combine the terms to express \( I \) in terms of \( V \):
\[ I \propto V \cdot V^{2/3} = V^{5/3} \]

7. **Conclusion**:
Therefore, the moment of inertia \( I \) of a solid sphere with fixed volume \( V \) is directly proportional to \( V^{5/3} \):
\[ I \propto V^{5/3} \]

### Final Answer:
The moment of inertia \( I \) of a solid sphere with fixed volume \( V \) depends on its volume as:
\[ I \propto V^{5/3} \]