Four particles of masses `m, 2m, 3m, 4m` are placed at corners of a square of side 'a' as shown in fig. Find out coordinateds of centre of mass.

Explanation

To find the coordinates of the center of mass of the four particles placed at the corners of a square, we can follow these steps:

### Step 1: Define the positions of the particles
Let's place the square in the coordinate system. We can assign the following coordinates to the corners of the square:
- Particle 1 (mass = m) at (0, 0)
- Particle 2 (mass = 2m) at (a, 0)
- Particle 3 (mass = 3m) at (a, a)
- Particle 4 (mass = 4m) at (0, a)

### Step 2: Calculate the total mass
The total mass \( M \) of the system is the sum of the individual masses:
\[ M = m + 2m + 3m + 4m = 10m \]

### Step 3: Calculate the x-coordinate of the center of mass
The x-coordinate of the center of mass \( x_{cm} \) is given by the formula:
\[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \]
Substituting the values:
\[ x_{cm} = \frac{m \cdot 0 + 2m \cdot a + 3m \cdot a + 4m \cdot 0}{10m} \]
This simplifies to:
\[ x_{cm} = \frac{(0 + 2ma + 3ma + 0)}{10m} = \frac{5ma}{10m} = \frac{a}{2} \]

### Step 4: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass \( y_{cm} \) is given by:
\[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \]
Substituting the values:
\[ y_{cm} = \frac{m \cdot 0 + 2m \cdot 0 + 3m \cdot a + 4m \cdot a}{10m} \]
This simplifies to:
\[ y_{cm} = \frac{(0 + 0 + 3ma + 4ma)}{10m} = \frac{7ma}{10m} = \frac{7a}{10} \]

### Step 5: Final coordinates of the center of mass
Thus, the coordinates of the center of mass are:
\[ \left( \frac{a}{2}, \frac{7a}{10} \right) \]

### Summary
The coordinates of the center of mass of the four particles are:
\[ \left( \frac{a}{2}, \frac{7a}{10} \right) \]