Question

A uniform rod of length \( 2L \) has constant mass per unit length \( m \). The moment of inertia of the rod about an axis passing through its center and perpendicular to its length is

• A. \( \frac{mL^2}{4} \)
• B. \( \frac{mL^3}{3} \)
• C. \( \frac{2mL^3}{3} \)
• D. \( \frac{mL^2}{12} \)

Hint:

The moment of inertia of a rod about an axis through its center and perpendicular to its length is given by \( I = \frac{1}{12} m L^2 \).

Answer 1

The Correct Option is C

Step 1: Moment of inertia of a uniform rod.
The moment of inertia 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 \( m \) is the mass of the rod and \( L \) is its length. Step 2: Adjust for the rod’s mass per unit length.
The mass per unit length of the rod is \( m \), and its total mass is \( M = m \times 2L = 2mL \). The moment of inertia is then given by: \[ I = \frac{1}{12} \times 2mL \times L^2 = \frac{2mL^3}{3} \] Step 3: Conclusion.
Thus, the moment of inertia of the rod is \( \frac{2mL^3}{3} \), which corresponds to option (C).