Two particles of masses 2 kg and 3 kg are separated by 5 m. Then moment of inertia of the system about an axis passing through the centre of mass of the system and perpendicular to the line joining them is

Options

A. `10 kg m^(2)`
B. `20 kg m^(2)`
C. `30 kg m^(2)`
D. `40 kg m^(2)`

Explanation

To find the moment of inertia of the system about an axis passing through the center of mass and perpendicular to the line joining the two particles, we can follow these steps:

### Step 1: Identify the masses and the distance between them
We have two particles:
- Mass \( m_1 = 2 \, \text{kg} \)
- Mass \( m_2 = 3 \, \text{kg} \)
- Distance \( d = 5 \, \text{m} \)

### Step 2: Calculate the position of the center of mass (CM)
The position of the center of mass \( x_{CM} \) for two particles can be calculated using the formula:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
Assuming \( m_1 \) is at position \( x_1 = 0 \) and \( m_2 \) is at position \( x_2 = 5 \, \text{m} \):
\[ x_{CM} = \frac{(2 \, \text{kg} \cdot 0) + (3 \, \text{kg} \cdot 5 \, \text{m})}{2 \, \text{kg} + 3 \, \text{kg}} = \frac{15 \, \text{kg m}}{5 \, \text{kg}} = 3 \, \text{m} \]

### Step 3: Calculate the distances from the center of mass to each mass
Now, we need to find the distances from the center of mass to each mass:
- Distance from \( m_1 \) to \( x_{CM} \):
\[ r_1 = x_{CM} - x_1 = 3 \, \text{m} - 0 \, \text{m} = 3 \, \text{m} \]
- Distance from \( m_2 \) to \( x_{CM} \):
\[ r_2 = x_2 - x_{CM} = 5 \, \text{m} - 3 \, \text{m} = 2 \, \text{m} \]

### Step 4: Use the formula for moment of inertia about the center of mass
The moment of inertia \( I \) about the center of mass is given by:
\[ I = m_1 r_1^2 + m_2 r_2^2 \]
Substituting the values:
\[ I = (2 \, \text{kg} \cdot (3 \, \text{m})^2) + (3 \, \text{kg} \cdot (2 \, \text{m})^2) \]
Calculating each term:
\[ I = (2 \, \text{kg} \cdot 9 \, \text{m}^2) + (3 \, \text{kg} \cdot 4 \, \text{m}^2) \]
\[ I = 18 \, \text{kg m}^2 + 12 \, \text{kg m}^2 = 30 \, \text{kg m}^2 \]

### Final Answer
The moment of inertia of the system about the axis passing through the center of mass and perpendicular to the line joining them is:
\[ \boxed{30 \, \text{kg m}^2} \]