Three thin uniform rods each of mass M and length L and placed along the three axis of a Cartesian coordinate system with one end of each rod at the origin. The M.I. of the system about z-axis is

Options

A. `(ML^2)/3`
B. `(2ML^2)/3`
C. `(ML^2)/6`
D. `ML^2`

Answer

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

### Step-by-Step Solution:

1. **Understand the Configuration**: We have three rods, each of mass \( M \) and length \( L \). They are positioned along the x-axis, y-axis, and z-axis of a Cartesian coordinate system, with one end of each rod at the origin (0,0,0).

2. **Identify the Moment of Inertia for Each Rod**:
- The moment of inertia \( I \) of a rod about an axis perpendicular to its length and passing through one end is given by the formula:
\[ I = \frac{1}{3}ML^2 \]

3. **Calculate 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 since its distance from the z-axis is zero.
- For the rod along the x-axis, its moment of inertia about the z-axis is:
\[ I_x = \frac{1}{3}ML^2 \]
- For the rod along the y-axis, its moment of inertia about the z-axis is also:
\[ I_y = \frac{1}{3}ML^2 \]

4. **Sum the Moments of Inertia**:
- 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-axis and y-axis:
\[ I_{total} = I_x + I_y = \frac{1}{3}ML^2 + \frac{1}{3}ML^2 = \frac{2}{3}ML^2 \]

5. **Final Result**:
- Therefore, the moment of inertia of the system about the z-axis is:
\[ I_{total} = \frac{2}{3}ML^2 \]