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 square of distance of their centre of mass from A.

Explanation

To solve the problem of finding the square of the distance of the center of mass from point A, we will follow these steps:

### Step 1: Define the positions of the particles
We have four particles located at the vertices of a square with side length 1 m. We can assign the following coordinates to the vertices:
- A (0, 0) for the mass of 1 kg
- B (1, 0) for the mass of 2 kg
- C (1, 1) for the mass of 3 kg
- D (0, 1) for the mass of 4 kg

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

### Step 3: Calculate the x-coordinate of the center of mass
The x-coordinate of the center of mass \( x_{cm} \) can be calculated using 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:
\[ x_{cm} = \frac{(1 \times 0) + (2 \times 1) + (3 \times 1) + (4 \times 0)}{10} = \frac{0 + 2 + 3 + 0}{10} = \frac{5}{10} = 0.5 \, \text{m} \]

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

### Step 5: Calculate the distance from A to the center of mass
The distance \( d \) from point A to the center of mass can be calculated using the Pythagorean theorem:
\[ d = \sqrt{x_{cm}^2 + y_{cm}^2} \]
Calculating \( d^2 \):
\[ d^2 = x_{cm}^2 + y_{cm}^2 = (0.5)^2 + (0.7)^2 = 0.25 + 0.49 = 0.74 \, \text{m}^2 \]

### Final Answer
The square of the distance of the center of mass from point A is:
\[ \boxed{0.74 \, \text{m}^2} \]