Four particles of masses 1kg, 2kg, 3kg and 4kg are placed at the four vertices A, B, C and D of a square of side 1m. Find the position of centre of mass of the particles.

Explanation

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

### Step 1: Define the coordinates of the vertices
Let’s assign the coordinates of the vertices of the square:
- Vertex A (0, 0) for mass \( m_A = 1 \, \text{kg} \)
- Vertex B (1, 0) for mass \( m_B = 2 \, \text{kg} \)
- Vertex C (1, 1) for mass \( m_C = 3 \, \text{kg} \)
- Vertex D (0, 1) for mass \( m_D = 4 \, \text{kg} \)

### Step 2: Calculate the total mass
The total mass \( M \) of the system is the sum of the individual masses:
\[ M = m_A + m_B + m_C + m_D = 1 + 2 + 3 + 4 = 10 \, \text{kg} \]

### 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:
\[ x_{cm} = \frac{m_A x_A + m_B x_B + m_C x_C + m_D x_D}{M} \]
Substituting the values:
\[ x_{cm} = \frac{(1 \times 0) + (2 \times 1) + (3 \times 1) + (4 \times 0)}{10} \]
Calculating the numerator:
\[ = \frac{0 + 2 + 3 + 0}{10} = \frac{5}{10} = 0.5 \, \text{m} \]

### Step 4: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass \( y_{cm} \) is given by:
\[ y_{cm} = \frac{m_A y_A + m_B y_B + m_C y_C + m_D y_D}{M} \]
Substituting the values:
\[ y_{cm} = \frac{(1 \times 0) + (2 \times 0) + (3 \times 1) + (4 \times 1)}{10} \]
Calculating the numerator:
\[ = \frac{0 + 0 + 3 + 4}{10} = \frac{7}{10} = 0.7 \, \text{m} \]

### Step 5: Conclusion
The position of the center of mass of the four particles is:
\[ \text{Center of Mass} = (x_{cm}, y_{cm}) = (0.5 \, \text{m}, 0.7 \, \text{m}) \]

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