The radius of gyration of an uniform rod of length `L` about an axis passing through its centre of mass and perpendicular to its length is.

Options

A. `(L)/(sqrt(2))`
B. `(L^2)/(sqrt(12))`
C. `(L)/(sqrt(3))`
D. `(L)/(sqrt(2))`

Explanation

To find the radius of gyration of a uniform rod of length \( L \) about an axis passing through its center of mass and perpendicular to its length, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Radius of Gyration**:
The radius of gyration \( k \) is defined as the distance from the axis of rotation at which the entire mass of the body can be assumed to be concentrated for the purpose of calculating the moment of inertia.

2. **Identify the Moment of Inertia**:
For a uniform rod of length \( L \) rotating about an axis through its center and perpendicular to its length, the moment of inertia \( I \) is given by the formula:
\[ I = \frac{1}{12} m L^2 \]
where \( m \) is the mass of the rod.

3. **Relate Moment of Inertia to Radius of Gyration**:
The relationship between the moment of inertia \( I \) and the radius of gyration \( k \) is given by:
\[ I = m k^2 \]
We can substitute the expression for \( I \) from step 2 into this equation.

4. **Set Up the Equation**:
Substituting the moment of inertia into the equation gives:
\[ \frac{1}{12} m L^2 = m k^2 \]

5. **Simplify the Equation**:
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{12} L^2 = k^2 \]

6. **Solve for \( k \)**:
Taking the square root of both sides, we find:
\[ k = \sqrt{\frac{1}{12}} L = \frac{L}{\sqrt{12}} \]

7. **Final Result**:
Thus, the radius of gyration \( k \) of the uniform rod about the specified axis is:
\[ k = \frac{L}{\sqrt{12}} \]