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 can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Moment of Inertia**:
The moment of inertia (I) is a measure of an object's resistance to changes in its rotation. For a solid sphere, the moment of inertia depends on its mass and the distribution of that mass relative to the axis of rotation.

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

3. **Identify the Variables**:
In this formula, we see that the moment of inertia \( I \) is directly related to the mass \( m \) and the square of the radius \( R \).

4. **Consider the Proportionality**:
Since the question specifies that the sphere is made of a certain material, we can assume that the density of the material is constant. Therefore, if the radius \( R \) changes, the mass \( m \) will also change proportionally because:
\[ m = \text{density} \times \text{volume} = \text{density} \times \left(\frac{4}{3} \pi R^3\right) \]
Thus, the mass \( m \) is proportional to \( R^3 \).

5. **Substitute Mass into the Moment of Inertia Formula**:
Substituting \( m \) into the moment of inertia formula gives:
\[ I = \frac{2}{5} \left(\text{density} \times \frac{4}{3} \pi R^3\right) R^2 \]
Simplifying this, we get:
\[ I = \frac{8}{15} \pi \text{density} R^5 \]
This shows that the moment of inertia \( I \) is proportional to \( R^5 \).

6. **Conclusion**:
Therefore, the moment of inertia of the sphere is proportional to \( R^5 \).

### Final Answer:
The moment of inertia of the sphere is proportional to \( R^5 \).