Three thin rods each of length Land mass M are placed along x, y and z-axes such that one end of each rod is at origin. The moment of inertia of this system about z-axis is

Options

A. `(2)/(3) ML^(2)`
B. `(4ML^(2))/(3)`
C. `(5ML^(2))/(3)`
D. `(ML^(2))/(3)`

Explanation

To find the moment of inertia of the system of three thin rods about the z-axis, we can follow these steps:

### Step 1: Understand the Configuration
We have three thin rods, each of length \( L \) and mass \( M \). The rods are aligned along the x, y, and z-axes, with one end of each rod at the origin (0, 0, 0).

### Step 2: Identify the Moment of Inertia for Each Rod
The moment of inertia \( I \) of a thin rod of mass \( M \) and length \( L \) about an axis through one end and perpendicular to its length is given by the formula:
\[ I = \frac{1}{3} M L^2 \]

### Step 3: Determine Which Rods Contribute to the Moment of Inertia About the z-axis
- The rod along the z-axis does not contribute to the moment of inertia about the z-axis because all its mass is located on the axis of rotation.
- The rods along the x-axis and y-axis will contribute to the moment of inertia about the z-axis.

### Step 4: Calculate the Moment of Inertia for the Contributing Rods
- For the rod along the x-axis (let's call it \( I_1 \)):
\[ I_1 = \frac{1}{3} M L^2 \]
- For the rod along the y-axis (let's call it \( I_2 \)):
\[ I_2 = \frac{1}{3} M L^2 \]

### Step 5: Sum the Contributions
The total moment of inertia \( I_{total} \) about the z-axis is the sum of the moments of inertia of the rods along the x and y axes:
\[ I_{total} = I_1 + I_2 = \frac{1}{3} M L^2 + \frac{1}{3} M L^2 = \frac{2}{3} M L^2 \]

### Step 6: Conclusion
Thus, the moment of inertia of the system about the z-axis is:
\[ I_{total} = \frac{2}{3} M L^2 \]

### Final Answer
The correct option is \( \frac{2}{3} M L^2 \).

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