Two particles of masses `m` and `2m` are placed at separation `L`. Find the `M.I.` about an axis passing through the center of mass and perpendicular to the line joining the point masses.

Options

A. `(2mL^(2))/(3)`
B. `(mr^(2))/(2)`
C. `(mr^(2))/(4)`
D. `(3mr^(2))/(4)`

Answer

To find the moment of inertia (M.I.) of two particles of masses \( m \) and \( 2m \) separated by a distance \( L \) about an axis passing through the center of mass and perpendicular to the line joining the point masses, we can follow these steps:

### Step 1: Determine the position of the center of mass (CM)
The position of the center of mass \( x_{CM} \) for two point masses can be calculated using the formula:

\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

Here, let:
- \( m_1 = m \) (mass of the first particle)
- \( m_2 = 2m \) (mass of the second particle)
- \( x_1 = 0 \) (position of mass \( m \))
- \( x_2 = L \) (position of mass \( 2m \))

Substituting these values:

\[ x_{CM} = \frac{m \cdot 0 + 2m \cdot L}{m + 2m} = \frac{2mL}{3m} = \frac{2L}{3} \]

### Step 2: Calculate the distances from the center of mass
Now, we need to find the distances of each mass from the center of mass:

- Distance \( d_1 \) from mass \( m \) to the center of mass:

\[ d_1 = x_{CM} - x_1 = \frac{2L}{3} - 0 = \frac{2L}{3} \]

- Distance \( d_2 \) from mass \( 2m \) to the center of mass:

\[ d_2 = x_2 - x_{CM} = L - \frac{2L}{3} = \frac{L}{3} \]

### Step 3: Calculate the moment of inertia
The moment of inertia \( I \) about the center of mass is given by:

\[ I = m d_1^2 + 2m d_2^2 \]

Substituting the values of \( d_1 \) and \( d_2 \):

\[ I = m \left(\frac{2L}{3}\right)^2 + 2m \left(\frac{L}{3}\right)^2 \]

Calculating each term:

\[ I = m \cdot \frac{4L^2}{9} + 2m \cdot \frac{L^2}{9} \]

\[ I = \frac{4mL^2}{9} + \frac{2mL^2}{9} = \frac{(4m + 2m)L^2}{9} = \frac{6mL^2}{9} = \frac{2mL^2}{3} \]

### Final Answer
Thus, the moment of inertia about the axis passing through the center of mass and perpendicular to the line joining the point masses is:

\[ I = \frac{2mL^2}{3} \]

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