Find the coordinates of centre of mass of a square of side 1 m in which four particles of masses 2m, 2m, 3m, 5m are placed at four corners.

Answer

To find the coordinates of the center of mass of a square with side 1 m, where four particles of masses 2m, 2m, 3m, and 5m are placed at the corners, we can follow these steps:

### Step 1: Define the positions of the particles
Let's place the square in the Cartesian coordinate system. Assume the corners of the square are at the following coordinates:
- Particle 1 (mass = 2m) at (0, 0)
- Particle 2 (mass = 2m) at (1, 0)
- Particle 3 (mass = 3m) at (1, 1)
- Particle 4 (mass = 5m) at (0, 1)

### Step 2: Calculate the total mass
The total mass \(M\) of the system is the sum of the individual masses:
\[ M = 2m + 2m + 3m + 5m = 12m \]

### 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:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{M} \]
Substituting the values:
\[ x_{cm} = \frac{(2m \cdot 0) + (2m \cdot 1) + (3m \cdot 1) + (5m \cdot 0)}{12m} \]
Calculating the numerator:
\[ = \frac{0 + 2m + 3m + 0}{12m} = \frac{5m}{12m} \]
Thus,
\[ x_{cm} = \frac{5}{12} \text{ m} \]

### Step 4: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass \(y_{cm}\) is given by:
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{M} \]
Substituting the values:
\[ y_{cm} = \frac{(2m \cdot 0) + (2m \cdot 0) + (3m \cdot 1) + (5m \cdot 1)}{12m} \]
Calculating the numerator:
\[ = \frac{0 + 0 + 3m + 5m}{12m} = \frac{8m}{12m} \]
Thus,
\[ y_{cm} = \frac{8}{12} = \frac{2}{3} \text{ m} \]

### Final Answer
The coordinates of the center of mass of the system are:
\[ \left( \frac{5}{12}, \frac{2}{3} \right) \text{ m} \]

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