The radius of gyration of a uniform circular disc of radius `R`, about any diameter of the disc is

Options

A. `K = R`
B. `K = R//2`
C. `K = 2 R`
D. none of these

Answer

To find the radius of gyration \( k \) of a uniform circular disc of radius \( R \) about any diameter of the disc, we can follow these steps:

### Step-by-Step Solution:

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

2. **Use the Perpendicular Axis Theorem**:
The perpendicular axis theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes lying in the plane. For a disc, if we consider two diameters, we can denote their moments of inertia as \( I_1 \) and \( I_2 \):
\[ I = I_1 + I_2 \]
Since the disc is symmetrical, \( I_1 = I_2 \). Therefore, we can write:
\[ I = 2I_1 \]

3. **Relate the Moments of Inertia**:
From the moment of inertia about the center (perpendicular to the plane), we have:
\[ I = \frac{1}{2} m R^2 \]
Thus, substituting into the equation from the perpendicular axis theorem:
\[ \frac{1}{2} m R^2 = 2I_1 \]
This simplifies to:
\[ I_1 = \frac{1}{4} m R^2 \]

4. **Relate Moment of Inertia to Radius of Gyration**:
The moment of inertia about the diameter can also be expressed in terms of the radius of gyration \( k \):
\[ I_1 = m k^2 \]
Setting the two expressions for \( I_1 \) equal gives:
\[ m k^2 = \frac{1}{4} m R^2 \]

5. **Solve for the Radius of Gyration**:
Dividing both sides by \( m \) (assuming \( m \neq 0 \)):
\[ k^2 = \frac{1}{4} R^2 \]
Taking the square root of both sides, we find:
\[ k = \frac{R}{2} \]

### Final Result:
The radius of gyration of a uniform circular disc of radius \( R \) about any diameter of the disc is:
\[ k = \frac{R}{2} \]

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