A uniform rod of length '2L' has mass per unit length 'm' . The moment of inertia of the rod about an axis passing through its centre and perpendicular to its length is

Options

A. `(2)/(3)mL^(2)`
B. `(1)/(3)mL^(2)`
C. `(2)/(3)mL^(3)`
D. `(4)/(3)mL^(3)`

Explanation

To find the moment of inertia of a uniform rod of length \(2L\) and mass per unit length \(m\) about an axis passing through its center and perpendicular to its length, we can follow these steps:

### Step 1: Determine the total mass of the rod
The mass per unit length of the rod is given as \(m\). Since the length of the rod is \(2L\), the total mass \(M\) of the rod can be calculated as:
\[ M = m \times \text{length} = m \times 2L = 2mL \]

### Step 2: Use the formula for the moment of inertia of a rod
The moment of inertia \(I\) of a uniform rod about an axis passing through its center and perpendicular to its length is given by the formula:
\[ I = \frac{1}{12} M L^2 \]
where \(L\) is the length of the rod.

### Step 3: Substitute the total mass and length into the formula
In our case, the length of the rod is \(2L\). Therefore, substituting \(M = 2mL\) and \(L = 2L\) into the moment of inertia formula:
\[ I = \frac{1}{12} (2mL) (2L)^2 \]

### Step 4: Simplify the expression
Now we simplify the expression:
\[ I = \frac{1}{12} (2mL) (4L^2) = \frac{1}{12} \times 8mL^3 = \frac{2mL^3}{3} \]

### Conclusion
Thus, the moment of inertia of the rod about the specified axis is:
\[ I = \frac{2mL^3}{3} \]