Three point sized bodies each of mass `M` are fixed at three corners of light triangular frame of side length `L`. About an axis perpendicular to the plane of frame and passing through centre of frame the moment of inertia of three bodies is

Options

A. `ML^(2)`
B. `(3ML^(2))/2`
C. `sqrt(3)ML^(2)`
D. `3ML^(2)`

Explanation

To find the moment of inertia of three point-sized bodies, each of mass \( M \), fixed at the corners of a triangular frame of side length \( L \), about an axis perpendicular to the plane of the frame and passing through the center of the frame, we can follow these steps:

### Step 1: Understand the Geometry of the Triangle
The three masses are located at the corners of an equilateral triangle. The centroid (center of mass) of an equilateral triangle is located at a distance of \( \frac{L}{\sqrt{3}} \) from each vertex.

### Step 2: Calculate the Distance from the Centroid to Each Mass
For an equilateral triangle with side length \( L \), the distance \( R \) from the centroid to each vertex (where the masses are located) can be calculated using the formula:
\[ R = \frac{L}{\sqrt{3}} \]

### Step 3: Use the Moment of Inertia Formula
The moment of inertia \( I \) about an axis is given by the formula:
\[ I = \sum m_i r_i^2 \]
where \( m_i \) is the mass and \( r_i \) is the distance from the axis of rotation to the mass.

### Step 4: Substitute the Values
Since all three masses are equal and located at the same distance \( R \) from the centroid, we can express the moment of inertia as:
\[ I = 3 \cdot M \cdot R^2 \]
Substituting \( R = \frac{L}{\sqrt{3}} \):
\[ I = 3 \cdot M \cdot \left(\frac{L}{\sqrt{3}}\right)^2 \]

### Step 5: Simplify the Expression
Calculating \( R^2 \):
\[ R^2 = \left(\frac{L}{\sqrt{3}}\right)^2 = \frac{L^2}{3} \]
Now substituting this back into the moment of inertia formula:
\[ I = 3 \cdot M \cdot \frac{L^2}{3} = M \cdot L^2 \]

### Conclusion
Thus, the moment of inertia of the three bodies about the specified axis is:
\[ I = M L^2 \]

### Final Answer
The correct answer is \( I = M L^2 \).

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