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 1: Understand the coordinates of the particle
The particle has a mass \( m = 1 \, \text{kg} \) and is located at the coordinates \( (1 \, \text{m}, 1 \, \text{m}, 1 \, \text{m}) \).

### Step 2: Identify the formula for moment of inertia about the Z-axis
The moment of inertia \( I \) about the Z-axis for a point mass is given by the formula:
\[ I = m \cdot r^2 \]
where \( r \) is the perpendicular distance from the Z-axis to the point mass.

### Step 3: Calculate the distance \( r \) from the Z-axis
For a point located at coordinates \( (x, y, z) \), the distance \( r \) from the Z-axis (which is defined by \( x = 0 \) and \( y = 0 \)) can be calculated using the formula:
\[ r = \sqrt{x^2 + y^2} \]
In this case, the coordinates are \( (1, 1, 1) \):
\[ r = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \, \text{m} \]

### Step 4: Substitute the values into the moment of inertia formula
Now, substituting the values of \( m \) and \( r \) into the moment of inertia formula:
\[ I = m \cdot r^2 = 1 \, \text{kg} \cdot (\sqrt{2})^2 \]
Calculating \( r^2 \):
\[ r^2 = (\sqrt{2})^2 = 2 \]
Thus,
\[ I = 1 \, \text{kg} \cdot 2 = 2 \, \text{kg} \cdot \text{m}^2 \]

### Step 5: Conclusion
The moment of inertia of the particle about the Z-axis is:
\[ I = 2 \, \text{kg} \cdot \text{m}^2 \]