The radius of gyration of a uniform disc of radius R, about an axis passing through a point `(R )/(2)` away from the centre of disc, and perpendicular to the plane of disc is:

Options

A. `sqrt((1)/(2)) R`
B. `sqrt(2) R`
C. `(sqrt(3))/(2)R`
D. `(sqrt(3)/(4)) R`

Explanation

To find the radius of gyration of a uniform disc of radius \( R \) about an axis passing through a point \( \frac{R}{2} \) away from the center of the disc and perpendicular to the plane of the disc, we can follow these steps:

### Step 1: Understand the Problem
We need to calculate the radius of gyration \( k \) of a uniform disc about a specific axis (CD) that is located at a distance of \( \frac{R}{2} \) from the center of the disc.

### Step 2: Identify the Moment of Inertia
The moment of inertia \( I \) of a uniform disc about its central axis (AB) is given by:
\[ I_{AB} = \frac{1}{2} m R^2 \]
where \( m \) is the mass of the disc and \( R \) is the radius of the disc.

### Step 3: Apply the Parallel Axis Theorem
To find the moment of inertia about the axis CD, we use the parallel axis theorem, which states:
\[ I_{CD} = I_{AB} + m d^2 \]
where \( d \) is the distance between the two axes. Here, \( d = \frac{R}{2} \).

### Step 4: Calculate the Moment of Inertia about Axis CD
Substituting the values into the equation:
\[ I_{CD} = \frac{1}{2} m R^2 + m \left(\frac{R}{2}\right)^2 \]
Calculating \( m \left(\frac{R}{2}\right)^2 \):
\[ m \left(\frac{R}{2}\right)^2 = m \frac{R^2}{4} \]
Now substituting this back into the equation for \( I_{CD} \):
\[ I_{CD} = \frac{1}{2} m R^2 + m \frac{R^2}{4} = \frac{2}{4} m R^2 + \frac{1}{4} m R^2 = \frac{3}{4} m R^2 \]

### Step 5: Calculate the Radius of Gyration
The radius of gyration \( k \) is defined as:
\[ k = \sqrt{\frac{I_{CD}}{m}} \]
Substituting \( I_{CD} \) into this formula:
\[ k = \sqrt{\frac{\frac{3}{4} m R^2}{m}} = \sqrt{\frac{3}{4} R^2} = \sqrt{\frac{3}{4}} R = \frac{\sqrt{3}}{2} R \]

### Final Answer
Thus, the radius of gyration of the uniform disc about the specified axis is:
\[ k = \frac{\sqrt{3}}{2} R \]