Four particles, each of mass `1kg` are placed at the corners of a square `OABC` of side `1m`. `O` is at the origin of the coordinate system. `OA` and `OC` are aligned along positive X-axis and positive Y-axis respectively. The position vector of the centre of mass is (in `m`)

Options

A. `hati+hatj`
B. `1/2(hati+hatj)`
C. `(hati-hatj)`
D. `1/2(hati-hatj)`

Explanation

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

### Step 1: Identify the coordinates of the particles
The four particles are located at the corners of square OABC with side length 1m. The coordinates of the corners are:
- O (0, 0)
- A (1, 0)
- B (1, 1)
- C (0, 1)

### Step 2: Assign masses to the particles
Each particle has a mass of 1 kg. Therefore:
- Mass at O (m1 = 1 kg)
- Mass at A (m2 = 1 kg)
- Mass at B (m3 = 1 kg)
- Mass at C (m4 = 1 kg)

### Step 3: Calculate the total mass
The total mass (M) of the system is the sum of the masses of all particles:
\[ M = m_1 + m_2 + m_3 + m_4 = 1 + 1 + 1 + 1 = 4 \text{ kg} \]

### Step 4: 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:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{M} \]
Substituting the values:
- \( x_1 = 0 \) (for O)
- \( x_2 = 1 \) (for A)
- \( x_3 = 1 \) (for B)
- \( x_4 = 0 \) (for C)

Thus,
\[ x_{cm} = \frac{(1 \cdot 0) + (1 \cdot 1) + (1 \cdot 1) + (1 \cdot 0)}{4} = \frac{0 + 1 + 1 + 0}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 5: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass (y_cm) can be calculated similarly:
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{M} \]
Substituting the values:
- \( y_1 = 0 \) (for O)
- \( y_2 = 0 \) (for A)
- \( y_3 = 1 \) (for B)
- \( y_4 = 1 \) (for C)

Thus,
\[ y_{cm} = \frac{(1 \cdot 0) + (1 \cdot 0) + (1 \cdot 1) + (1 \cdot 1)}{4} = \frac{0 + 0 + 1 + 1}{4} = \frac{2}{4} = \frac{1}{2} \]

### Step 6: Write the position vector of the center of mass
The position vector of the center of mass (R_cm) can be expressed as:
\[ R_{cm} = x_{cm} \hat{i} + y_{cm} \hat{j} = \frac{1}{2} \hat{i} + \frac{1}{2} \hat{j} \]

### Final Answer
Thus, the position vector of the center of mass is:
\[ R_{cm} = \frac{1}{2} \hat{i} + \frac{1}{2} \hat{j} \]