Three uniform thin rods, each of mass 1 kg and length `sqrt3`\ m, are placed along three co-ordinate axes with one end at the origin. The moment of inertia of the system about X-axis is

Options

A. `2 kg m^(2)`
B. `3 kg m^(2)`
C. `0..75 kg m^(2)`
D. `1 kg m^(2)`

Answer

To find the moment of inertia of the system about the X-axis, we will follow these steps:

### Step 1: Understand the Configuration
We have three uniform thin rods, each of mass 1 kg and length \( \sqrt{3} \) m, placed along the coordinate axes (X, Y, Z) with one end at the origin (0, 0, 0).

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

### Step 3: Calculate Moment of Inertia for Each Rod
1. **Rod along the X-axis**:
- Since the axis of rotation (X-axis) passes through the rod, its moment of inertia about the X-axis is:
\[ I_1 = 0 \]

2. **Rod along the Y-axis**:
- For the rod along the Y-axis:
\[ I_2 = \frac{1}{3} m L^2 = \frac{1}{3} \times 1 \, \text{kg} \times (\sqrt{3} \, \text{m})^2 = \frac{1}{3} \times 1 \times 3 = 1 \, \text{kg m}^2 \]

3. **Rod along the Z-axis**:
- For the rod along the Z-axis:
\[ I_3 = \frac{1}{3} m L^2 = \frac{1}{3} \times 1 \, \text{kg} \times (\sqrt{3} \, \text{m})^2 = \frac{1}{3} \times 1 \times 3 = 1 \, \text{kg m}^2 \]

### Step 4: Sum the Moments of Inertia
The total moment of inertia \( I \) about the X-axis is the sum of the moments of inertia of the rods along the Y and Z axes:
\[ I = I_1 + I_2 + I_3 = 0 + 1 + 1 = 2 \, \text{kg m}^2 \]

### Final Answer
The moment of inertia of the system about the X-axis is:
\[ \boxed{2 \, \text{kg m}^2} \]