For a sphere made out of a certain material, the moment of inertia of the sphere is proportional to [ radius of the sphere = R ]

Options

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

Answer

To solve the question regarding the moment of inertia of a sphere made from a certain material, we need to analyze the relationship between the moment of inertia and the radius of the sphere.

### Step-by-Step Solution:

1. **Understanding Moment of Inertia**:
The moment of inertia (I) of an object is a measure of its resistance to rotational motion about an axis. For a sphere, it depends on both its mass (m) and its radius (R).

2. **Moment of Inertia for a Solid Sphere**:
For a solid sphere, the moment of inertia about an axis through its center is given by the formula:
\[ I = \frac{2}{5} m R^2 \]
Here, \(m\) is the mass of the sphere and \(R\) is its radius.

3. **Moment of Inertia for a Hollow Sphere**:
For a hollow sphere, the moment of inertia about an axis through its center is given by:
\[ I = \frac{2}{3} m R^2 \]
Again, \(m\) is the mass of the hollow sphere and \(R\) is its radius.

4. **Proportionality to Radius**:
In both cases (solid and hollow), we can observe that the moment of inertia is proportional to \(R^2\). This means that if we increase the radius of the sphere, the moment of inertia increases with the square of the radius.

5. **Conclusion**:
Therefore, we conclude that the moment of inertia \(I\) of a sphere is proportional to the square of its radius \(R\). Thus, we can express this relationship as:
\[ I \propto R^2 \]