The moment of inertia of a sphere of radius R about an axis passing through its centre is proportional to-

Options

A. `R^2`
B. `R^3`
C. `R^4`
D. `R^5`

Answer

To solve the problem of determining how the moment of inertia of a sphere of radius \( R \) about an axis passing through its center is proportional, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia \( I \) of an object is a measure of how difficult it is to change its rotational motion about a given axis. For a solid sphere, the moment of inertia about an axis through its center is given by the formula:
\[ I = \frac{2}{5} m R^2 \]
where \( m \) is the mass of the sphere and \( R \) is its radius.

2. **Identifying Proportionality**:
From the formula, we can see that the moment of inertia \( I \) is directly proportional to the mass \( m \) and the square of the radius \( R^2 \). Therefore, we can express this relationship as:
\[ I \propto m R^2 \]

3. **Considering Mass**:
If we consider a uniform sphere, the mass \( m \) can be expressed in terms of the density \( \rho \) and volume \( V \). The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3} \pi R^3 \]
Thus, the mass can be expressed as:
\[ m = \rho V = \rho \left(\frac{4}{3} \pi R^3\right) \]

4. **Substituting Mass into Moment of Inertia**:
Substituting the expression for mass into the moment of inertia formula gives:
\[ I = \frac{2}{5} \left(\rho \frac{4}{3} \pi R^3\right) R^2 \]
Simplifying this, we find:
\[ I = \frac{2}{5} \cdot \rho \cdot \frac{4}{3} \pi R^5 \]

5. **Final Proportionality**:
From the above expression, we can see that the moment of inertia \( I \) is proportional to \( R^5 \) when considering the density and volume of the sphere. However, if we are only looking at the relationship of \( I \) with respect to \( R \), we can conclude:
\[ I \propto R^2 \]
This indicates that the moment of inertia is proportional to the square of the radius.

### Conclusion:
The moment of inertia of a sphere of radius \( R \) about an axis passing through its center is proportional to \( R^2 \).