Three bodies of equal masses are placed at (0,0) ,(a,0) and `(a/2.(asqrt(3))/2)` . Find out the cordinates of centre of mass .

Options

A. `(a/2,(asqrt(3))/6)`
B. `(a,a/2)`
C. `((sqrt(3a))/6,a)`
D. `(a/2,a/2)`

Answer

To find the coordinates of the center of mass of three bodies placed at the given coordinates, we can follow these steps:

### Step 1: Identify the coordinates and masses
We have three bodies with equal masses \( m \) located at the following coordinates:
1. Body 1: \( (0, 0) \)
2. Body 2: \( (a, 0) \)
3. Body 3: \( \left( \frac{a}{2}, \frac{a \sqrt{3}}{2} \right) \)

### Step 2: Use the center of mass formula
The formula for the center of mass \( (x_{cm}, y_{cm}) \) for a system of particles is given by:
\[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \]
\[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \]
Since all masses are equal, we can denote the mass of each body as \( m \).

### Step 3: Calculate \( x_{cm} \)
Substituting the coordinates into the formula for \( x_{cm} \):
\[ x_{cm} = \frac{m \cdot 0 + m \cdot a + m \cdot \frac{a}{2}}{m + m + m} \]
\[ = \frac{0 + a + \frac{a}{2}}{3m} \]
\[ = \frac{a + \frac{a}{2}}{3} \]
\[ = \frac{\frac{2a}{2} + \frac{a}{2}}{3} = \frac{\frac{3a}{2}}{3} = \frac{a}{2} \]

### Step 4: Calculate \( y_{cm} \)
Now, substituting the coordinates into the formula for \( y_{cm} \):
\[ y_{cm} = \frac{m \cdot 0 + m \cdot 0 + m \cdot \frac{a \sqrt{3}}{2}}{m + m + m} \]
\[ = \frac{0 + 0 + \frac{a \sqrt{3}}{2}}{3m} \]
\[ = \frac{\frac{a \sqrt{3}}{2}}{3} = \frac{a \sqrt{3}}{6} \]

### Step 5: Final coordinates of the center of mass
Thus, the coordinates of the center of mass are:
\[ \left( \frac{a}{2}, \frac{a \sqrt{3}}{6} \right) \]

### Summary
The center of mass of the three bodies is located at:
\[ \left( \frac{a}{2}, \frac{a \sqrt{3}}{6} \right) \]