The radius of gyration of a solid shapere of radius r about a certain axis is r. The distance of this axis from the centre of the shpere is

Options

A. remain stationary
B. 0.5r
C. `sqrt(0.6)r`
D. `sqrt(0.4)r`

Answer

To solve the problem, we need to find the distance of the axis from the center of the sphere given that the radius of gyration about that axis is equal to the radius of the sphere \( r \).

### Step-by-Step Solution:

1. **Understanding Radius of Gyration**:
The radius of gyration \( k \) is defined as:
\[ k = \sqrt{\frac{I}{m}} \]
where \( I \) is the moment of inertia and \( m \) is the mass. Given that \( k = r \), we can express this as:
\[ r = \sqrt{\frac{I}{m}} \implies I = m r^2 \]

2. **Moment of Inertia of a Sphere**:
The moment of inertia of a solid sphere about its own center (axis through the center) is given by:
\[ I_{center} = \frac{2}{5} m r^2 \]

3. **Using the Parallel Axis Theorem**:
If we have an axis parallel to the axis through the center of the sphere, the moment of inertia about this new axis (let's call it \( O' \)) is given by:
\[ I_{O'} = I_{center} + m d^2 \]
where \( d \) is the distance from the center of the sphere to the new axis.

4. **Setting Up the Equation**:
From the previous steps, we can write:
\[ m r^2 = \frac{2}{5} m r^2 + m d^2 \]

5. **Simplifying the Equation**:
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ r^2 = \frac{2}{5} r^2 + d^2 \]
Rearranging gives:
\[ d^2 = r^2 - \frac{2}{5} r^2 \]
\[ d^2 = \left(1 - \frac{2}{5}\right) r^2 = \frac{3}{5} r^2 \]

6. **Finding \( d \)**:
Taking the square root of both sides gives:
\[ d = \sqrt{\frac{3}{5}} r \]

### Final Answer:
The distance of the axis from the center of the sphere is:
\[ d = \sqrt{0.6} r \]