A particle of mass 1 kg is kept at (1m,1m,1m). The moment of inertia of this particle about Z-axis would be

Options

A. `1 "kg-m"^(2)`
B. `2 "kg-m"^(2)`
C. `3 "kg-m"^(2)`
D. None of these

Explanation

To find the moment of inertia of a particle about the Z-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the mass and position of the particle**:
The mass \( m \) of the particle is given as 1 kg, and its position is at coordinates \( (1 \, \text{m}, 1 \, \text{m}, 1 \, \text{m}) \).

2. **Understand the moment of inertia formula**:
The moment of inertia \( I \) about the Z-axis is given by the formula:
\[ I_z = m \cdot R_z^2 \]
where \( R_z \) is the perpendicular distance from the Z-axis to the particle.

3. **Calculate the distance \( R_z \)**:
Since the Z-axis is perpendicular to the XY-plane, the distance \( R_z \) can be calculated using the coordinates of the particle in the XY-plane. The coordinates of the particle are \( (1, 1) \) in the XY-plane. The distance \( R_z \) is given by:
\[ R_z = \sqrt{x^2 + y^2} \]
Substituting the values:
\[ R_z = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \, \text{m} \]

4. **Substitute \( R_z \) into the moment of inertia formula**:
Now substituting \( R_z \) back into the moment of inertia formula:
\[ I_z = m \cdot R_z^2 = 1 \, \text{kg} \cdot (\sqrt{2} \, \text{m})^2 \]

5. **Calculate \( I_z \)**:
\[ I_z = 1 \cdot 2 = 2 \, \text{kg m}^2 \]

6. **Conclusion**:
Therefore, the moment of inertia of the particle about the Z-axis is:
\[ I_z = 2 \, \text{kg m}^2 \]

### Final Answer:
The moment of inertia of the particle about the Z-axis is **2 kg m²**.

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