Two bodies of masses 1 kg and 2 kg are located in x-y plane at (0,2) and (2,0). Find the position of their centre of mass.

Answer

To find the position of the center of mass of two bodies located in the x-y plane, we can follow these steps:

### Step 1: Identify the masses and their coordinates
- Mass \( m_1 = 1 \, \text{kg} \) is located at coordinates \( (x_1, y_1) = (0, 2) \).
- Mass \( m_2 = 2 \, \text{kg} \) is located at coordinates \( (x_2, y_2) = (2, 0) \).

### Step 2: Write the formula for the center of mass
The coordinates of the center of mass \( (x_{cm}, y_{cm}) \) can be calculated using the formula:
\[ x_{cm} = \frac{m_1 \cdot x_1 + m_2 \cdot x_2}{m_1 + m_2} \]
\[ y_{cm} = \frac{m_1 \cdot y_1 + m_2 \cdot y_2}{m_1 + m_2} \]

### Step 3: Substitute the values into the formula
- For \( x_{cm} \):
\[ x_{cm} = \frac{(1 \, \text{kg} \cdot 0) + (2 \, \text{kg} \cdot 2)}{1 \, \text{kg} + 2 \, \text{kg}} = \frac{0 + 4}{3} = \frac{4}{3} \]

- For \( y_{cm} \):
\[ y_{cm} = \frac{(1 \, \text{kg} \cdot 2) + (2 \, \text{kg} \cdot 0)}{1 \, \text{kg} + 2 \, \text{kg}} = \frac{2 + 0}{3} = \frac{2}{3} \]

### Step 4: Write the final coordinates of the center of mass
Thus, the coordinates of the center of mass are:
\[ (x_{cm}, y_{cm}) = \left(\frac{4}{3}, \frac{2}{3}\right) \]

### Final Answer:
The position of the center of mass is \( \left(\frac{4}{3}, \frac{2}{3}\right) \).

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