The radius of gyration of a uniform disc about a line perpendicular to the disc equals to its radius. Find the distance of the line from the centre.

Explanation

To solve the problem, we need to find the distance \( d \) from the center of a uniform disc to a line perpendicular to the disc, given that the radius of gyration \( k \) about this line equals the radius \( R \) of the disc.

### Step-by-Step Solution:

1. **Understand the Moment of Inertia of the Disc**:
The moment of inertia \( I \) of a uniform disc about an axis through its center and perpendicular to the plane of the disc is given by:
\[ I = \frac{1}{2} m R^2 \]
where \( m \) is the mass of the disc and \( R \) is its radius.

2. **Moment of Inertia about the New Axis**:
We are given that the radius of gyration \( k \) about the new axis is equal to the radius \( R \). The moment of inertia \( I' \) about this new axis can be expressed as:
\[ I' = m k^2 = m R^2 \]

3. **Apply the Parallel Axis Theorem**:
The parallel axis theorem states that if you know the moment of inertia about a parallel axis through the center of mass, you can find the moment of inertia about another parallel axis by:
\[ I' = I + m d^2 \]
where \( d \) is the distance between the two axes.

4. **Set Up the Equation**:
Substituting the known values into the equation:
\[ m R^2 = \frac{1}{2} m R^2 + m d^2 \]

5. **Simplify the Equation**:
Dividing through by \( m \) (assuming \( m \neq 0 \)):
\[ R^2 = \frac{1}{2} R^2 + d^2 \]
Rearranging gives:
\[ d^2 = R^2 - \frac{1}{2} R^2 = \frac{1}{2} R^2 \]

6. **Solve for \( d \)**:
Taking the square root of both sides:
\[ d = \sqrt{\frac{1}{2} R^2} = \frac{R}{\sqrt{2}} \]

### Final Answer:
The distance of the line from the center of the disc is:
\[ d = \frac{R}{\sqrt{2}} \]