The radius of gyration of a uniform rod of length `l` about an axis passing through one of its ends and perpendicular to its length is

Options

A. `(l)/(sqrt(2))`
B. `(l)/(3)`
C. `(l)/(sqrt(3))`
D. `(l)/(2)`

Explanation

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

### Step 1: Understand the moment of inertia
The moment of inertia \( I \) of a uniform rod of length \( l \) about an axis passing through one end and perpendicular to its length is given by the formula:
\[ I = \frac{1}{3} m l^2 \]
where \( m \) is the mass of the rod.

### Step 2: Relate moment of inertia to radius of gyration
The radius of gyration \( K \) is defined by the relationship:
\[ I = m K^2 \]
where \( K \) is the radius of gyration.

### Step 3: Substitute the moment of inertia into the equation
We can substitute the expression for \( I \) into the equation:
\[ \frac{1}{3} m l^2 = m K^2 \]

### Step 4: Simplify the equation
We can cancel \( m \) from both sides (assuming \( m \neq 0 \)):
\[ \frac{1}{3} l^2 = K^2 \]

### Step 5: Solve for the radius of gyration \( K \)
To find \( K \), we take the square root of both sides:
\[ K = \sqrt{\frac{1}{3} l^2} = \frac{l}{\sqrt{3}} \]

### Conclusion
Thus, the radius of gyration of the uniform rod about the specified axis is:
\[ K = \frac{l}{\sqrt{3}} \]

### Final Answer
The radius of gyration of the uniform rod about the axis passing through one of its ends and perpendicular to its length is \( \frac{l}{\sqrt{3}} \).

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