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 find the square of the distance of the center of mass of four particles from point A, we can follow these steps:

### Step 1: Define the coordinates of the particles
Let's place the particles at the vertices of a square with side length 1 m. We can assign the following coordinates based on the vertices of the square:
- A (0, 0) for the 1 kg mass
- B (1, 0) for the 2 kg mass
- C (1, 1) for the 3 kg mass
- D (0, 1) for the 4 kg mass

### Step 2: Write down the masses and their coordinates
- Mass \( m_A = 1 \, \text{kg} \) at \( (0, 0) \)
- Mass \( m_B = 2 \, \text{kg} \) at \( (1, 0) \)
- Mass \( m_C = 3 \, \text{kg} \) at \( (1, 1) \)
- Mass \( m_D = 4 \, \text{kg} \) at \( (0, 1) \)

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

### Step 4: Calculate the coordinates of the center of mass
The coordinates of the center of mass \( (x_{cm}, y_{cm}) \) can be calculated using the formula:
\[ x_{cm} = \frac{m_A x_A + m_B x_B + m_C x_C + m_D x_D}{M} \]
\[ y_{cm} = \frac{m_A y_A + m_B y_B + m_C y_C + m_D y_D}{M} \]

Substituting the values:
\[ x_{cm} = \frac{(1 \cdot 0) + (2 \cdot 1) + (3 \cdot 1) + (4 \cdot 0)}{10} = \frac{0 + 2 + 3 + 0}{10} = \frac{5}{10} = 0.5 \]
\[ y_{cm} = \frac{(1 \cdot 0) + (2 \cdot 0) + (3 \cdot 1) + (4 \cdot 1)}{10} = \frac{0 + 0 + 3 + 4}{10} = \frac{7}{10} = 0.7 \]

### Step 5: Calculate the distance from point A to the center of mass
The distance \( d \) from point A to the center of mass can be calculated using the distance formula:
\[ d = \sqrt{(x_{cm} - x_A)^2 + (y_{cm} - y_A)^2} \]
Substituting the coordinates of A and the center of mass:
\[ d = \sqrt{(0.5 - 0)^2 + (0.7 - 0)^2} = \sqrt{(0.5)^2 + (0.7)^2} = \sqrt{0.25 + 0.49} = \sqrt{0.74} \]

### Step 6: Calculate the square of the distance
The square of the distance from A to the center of mass is:
\[ d^2 = 0.74 \, \text{m}^2 \]

### Final Answer
The square of the distance of the center of mass from point A is \( 0.74 \, \text{m}^2 \).