A uniform disc of radius `R` lies in xy-plane with its centre at origin. Its moments of inertia about the axis `X=2R` and `Y=0` is equal to the moment of inertia about the axis `Y=d` and `Z=0`. What is the value of `d`?

Options

A. `(sqrt(17))/2R`
B. `sqrt(13)R`
C. `sqrt(15/2)R`
D. `4/3R`

Explanation

To solve the problem, we need to find the value of \( d \) such that the moment of inertia of a uniform disc about two different axes is equal. We will use the parallel axis theorem to compute the moments of inertia about the specified axes.

### Step-by-Step Solution:

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

2. **Moment of Inertia about Axis 1 (X = 2R, Y = 0)**:
- We will use the parallel axis theorem to find the moment of inertia about the axis at \( x = 2R \).
- According to the parallel axis theorem:
\[ I_1 = I_{CM} + m d^2 \]
where \( d \) is the distance from the center of mass to the new axis. Here, the distance \( d \) is \( 2R \).
- Therefore:
\[ I_1 = \frac{1}{2} m R^2 + m (2R)^2 \]
\[ I_1 = \frac{1}{2} m R^2 + 4mR^2 = \frac{1}{2} m R^2 + 8mR^2 = \frac{17}{2} m R^2 \]

3. **Moment of Inertia about Axis 2 (Y = d, Z = 0)**:
- Now, we find the moment of inertia about the axis at \( y = d \).
- Again, using the parallel axis theorem:
\[ I_2 = I_{CM} + m d^2 \]
- Here, the distance \( d \) is \( d \) itself (the distance from the center of mass to the new axis):
\[ I_2 = \frac{1}{2} m R^2 + m d^2 \]

4. **Setting the Moments of Inertia Equal**:
- According to the problem, \( I_1 = I_2 \):
\[ \frac{17}{2} m R^2 = \frac{1}{2} m R^2 + m d^2 \]
- We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{17}{2} R^2 = \frac{1}{2} R^2 + d^2 \]

5. **Solving for \( d^2 \)**:
- Rearranging the equation gives:
\[ d^2 = \frac{17}{2} R^2 - \frac{1}{2} R^2 \]
\[ d^2 = \frac{17 - 1}{2} R^2 = \frac{16}{2} R^2 = 8 R^2 \]

6. **Finding \( d \)**:
- Taking the square root of both sides:
\[ d = \sqrt{8} R = 2\sqrt{2} R \]

### Final Answer:
Thus, the value of \( d \) is:
\[ d = 2\sqrt{2} R \]