Centre of mass of two particle with masses 2 kg and 1 kg locsted at (1, 0, 1) and (2, 2, 0) has the co - ordinates of

Options

A. `(4)/(3),(2)/(3),(2)/(3)`
B. `(2)/(3),(4)/(3),(2)/(3)`
C. `(2)/(3),(2)/(3),(4)/(3)`
D. `(1)/(3),(2)/(3),(2)/(3)`

Answer

To find the center of mass of two particles with given masses and coordinates, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the masses and their coordinates:**
- Mass \( m_1 = 2 \, \text{kg} \) located at coordinates \( (x_1, y_1, z_1) = (1, 0, 1) \).
- Mass \( m_2 = 1 \, \text{kg} \) located at coordinates \( (x_2, y_2, z_2) = (2, 2, 0) \).

2. **Use the formula for the center of mass (COM):**
The coordinates of the center of mass \( (X_{cm}, Y_{cm}, Z_{cm}) \) can be calculated using the formula:
\[ X_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]
\[ Y_{cm} = \frac{m_1 y_1 + m_2 y_2}{m_1 + m_2} \]
\[ Z_{cm} = \frac{m_1 z_1 + m_2 z_2}{m_1 + m_2} \]

3. **Calculate the total mass:**
\[ m_{total} = m_1 + m_2 = 2 \, \text{kg} + 1 \, \text{kg} = 3 \, \text{kg} \]

4. **Calculate the \( X_{cm} \):**
\[ X_{cm} = \frac{(2 \, \text{kg} \cdot 1) + (1 \, \text{kg} \cdot 2)}{3 \, \text{kg}} = \frac{2 + 2}{3} = \frac{4}{3} \]

5. **Calculate the \( Y_{cm} \):**
\[ Y_{cm} = \frac{(2 \, \text{kg} \cdot 0) + (1 \, \text{kg} \cdot 2)}{3 \, \text{kg}} = \frac{0 + 2}{3} = \frac{2}{3} \]

6. **Calculate the \( Z_{cm} \):**
\[ Z_{cm} = \frac{(2 \, \text{kg} \cdot 1) + (1 \, \text{kg} \cdot 0)}{3 \, \text{kg}} = \frac{2 + 0}{3} = \frac{2}{3} \]

7. **Write the final coordinates of the center of mass:**
The coordinates of the center of mass are:
\[ (X_{cm}, Y_{cm}, Z_{cm}) = \left(\frac{4}{3}, \frac{2}{3}, \frac{2}{3}\right) \]

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