Three particles each of 'm' are kept at the three vertices of an equilateral triangle of side 'a' . Moment of inertia of the system about an axis passing through the centroid and perpendicular to its plane is

Options

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

Explanation

To find the moment of inertia of a system of three particles each of mass 'm' located at the vertices of an equilateral triangle of side 'a', we follow these steps:

### Step 1: Identify the Centroid of the Triangle
The centroid of an equilateral triangle is located at a point where the medians intersect. For an equilateral triangle, the centroid divides each median in a 2:1 ratio.

### Step 2: Calculate the Distance from the Centroid to Each Vertex
For an equilateral triangle with vertices A, B, and C, the distance from the centroid (G) to any vertex (say A) can be calculated using the formula:
\[ d = \frac{a}{\sqrt{3}} \]
This is derived from the geometry of the triangle, where the median length is \(\frac{\sqrt{3}}{2}a\) and the distance from the centroid to a vertex is \(\frac{2}{3}\) of the median.

### Step 3: Write the Moment of Inertia Formula
The moment of inertia \(I\) of a system of point masses is given by:
\[ I = \sum m_i r_i^2 \]
where \(m_i\) is the mass of each particle and \(r_i\) is the distance from the axis of rotation (in this case, the centroid).

### Step 4: Substitute Values into the Moment of Inertia Formula
Since all three masses are equal and the distances from the centroid to each vertex are the same, we can simplify the moment of inertia calculation:
\[ I = 3 \cdot m \cdot d^2 \]
Substituting \(d = \frac{a}{\sqrt{3}}\):
\[ I = 3 \cdot m \cdot \left(\frac{a}{\sqrt{3}}\right)^2 \]

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

### Final Result
Thus, the moment of inertia of the system about an axis passing through the centroid and perpendicular to its plane is:
\[ I = m a^2 \]