Question

Three thin uniform rods each of mass \( M \) and length \( L \) are placed along the three axes of a Cartesian coordinate system with one end of all the rods at origin. The moment of inertia of the system of the rods about z-axis is

• A. \(\dfrac{ML^2}{3}\)
• B. \(\dfrac{2ML^2}{3}\)
• C. \(\dfrac{ML^2}{2}\)
• D. \(ML^2\)

Hint:

When a rod is perpendicular to the axis of rotation and lies in the plane of rotation, use \(\int r^2 dm\) from 0 to \(L\) to compute its contribution.

Answer 1

The Correct Option is B

Step 1: Understand rod placement and axis of rotation 
There are three rods placed along the x, y, and z axes with one end at the origin. We need the moment of inertia about the z-axis. 
Step 2: Moment of inertia contribution from each rod 
- Rod along the z-axis: lies along the axis of rotation → contributes zero to moment of inertia about the z-axis. 
- Rod along x-axis: moment of inertia about z-axis is \(\int_0^L M/L \cdot x^2 dx = \dfrac{ML^2}{3}\) 
- Rod along y-axis: similar to the x-axis rod → contributes \(\dfrac{ML^2}{3}\) 
Step 3: Add contributions 
\[ I_z = \dfrac{ML^2}{3} + \dfrac{ML^2}{3} = \dfrac{2ML^2}{3} \]