A system consists of 3 particles each of mass 'm' are located at (1, 1) (2, 2) and (3, 3). The co-ordinates of the centre of mass are

Options

A. (6, 6)
B. (2, 2)
C. (3, 3)
D. (1, 1)

Explanation

To find the coordinates of the center of mass of a system consisting of three particles, we can use the formula for the center of mass in two dimensions. The coordinates of the center of mass (CM) can be calculated using the following steps:

### Step-by-Step Solution:

1. **Identify the masses and coordinates of the particles:**
- Each particle has a mass \( m \).
- The coordinates of the particles are:
- Particle 1: \( (x_1, y_1) = (1, 1) \)
- Particle 2: \( (x_2, y_2) = (2, 2) \)
- Particle 3: \( (x_3, y_3) = (3, 3) \)

2. **Use the formula for the center of mass:**
The coordinates of the center of mass \( (x_{cm}, y_{cm}) \) are given by the formulas:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \]
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3} \]

3. **Substituting the values:**
Since all the masses are equal \( m_1 = m_2 = m_3 = m \), we can simplify the equations:
\[ x_{cm} = \frac{m(1) + m(2) + m(3)}{m + m + m} = \frac{m(1 + 2 + 3)}{3m} = \frac{6m}{3m} = 2 \]
\[ y_{cm} = \frac{m(1) + m(2) + m(3)}{m + m + m} = \frac{m(1 + 2 + 3)}{3m} = \frac{6m}{3m} = 2 \]

4. **Final coordinates of the center of mass:**
Therefore, the coordinates of the center of mass are:
\[ (x_{cm}, y_{cm}) = (2, 2) \]

### Final Answer:
The coordinates of the center of mass are \( (2, 2) \).

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