The coordinates of centre of mass of three particles of masses 1 kg, 2 kg, 3 kg are (2m, 2m, 2m). Where should a fourth particle of mass 4 kg be placed so that the coordinates of centre of mass are (4m, 4m, 4m)?

Answer

To solve the problem, we need to find the coordinates of the fourth particle such that the center of mass (CM) of all four particles is at the desired location (4m, 4m, 4m).

### Step-by-Step Solution:

1. **Identify the Masses and Their Coordinates:**
- We have three particles with the following masses and coordinates:
- Particle 1: mass \( m_1 = 1 \, \text{kg} \), coordinates \( (x_1, y_1, z_1) = (2, 2, 2) \)
- Particle 2: mass \( m_2 = 2 \, \text{kg} \), coordinates \( (x_2, y_2, z_2) = (2, 2, 2) \)
- Particle 3: mass \( m_3 = 3 \, \text{kg} \), coordinates \( (x_3, y_3, z_3) = (2, 2, 2) \)

2. **Calculate the Total Mass of the First Three Particles:**
\[ M = m_1 + m_2 + m_3 = 1 + 2 + 3 = 6 \, \text{kg} \]

3. **Determine the Current Center of Mass:**
The center of mass \( (x_{cm}, y_{cm}, z_{cm}) \) of the three particles is given by:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{M} = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2}{6} = \frac{2 + 4 + 6}{6} = \frac{12}{6} = 2 \]
Similarly, \( y_{cm} \) and \( z_{cm} \) will also be 2.

4. **Introduce the Fourth Particle:**
Let the coordinates of the fourth particle (mass \( m_4 = 4 \, \text{kg} \)) be \( (x_4, y_4, z_4) \).

5. **Set Up the Equation for the New Center of Mass:**
We want the new center of mass to be \( (4, 4, 4) \). The total mass now becomes:
\[ M' = M + m_4 = 6 + 4 = 10 \, \text{kg} \]
The new center of mass equations become:
\[ x_{cm}' = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{M'} = 4 \]
Substituting the known values:
\[ \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2 + 4 x_4}{10} = 4 \]
Simplifying gives:
\[ \frac{12 + 4 x_4}{10} = 4 \]

6. **Solve for \( x_4 \):**
Multiply both sides by 10:
\[ 12 + 4 x_4 = 40 \]
Rearranging gives:
\[ 4 x_4 = 40 - 12 = 28 \]
Thus,
\[ x_4 = \frac{28}{4} = 7 \]

7. **Repeat for \( y_4 \) and \( z_4 \):**
Since the coordinates of the center of mass are the same in all three dimensions, we find:
\[ y_4 = 7 \quad \text{and} \quad z_4 = 7 \]

8. **Final Coordinates of the Fourth Particle:**
Therefore, the coordinates of the fourth particle should be \( (7, 7, 7) \).

### Final Answer:
The fourth particle of mass 4 kg should be placed at the coordinates \( (7m, 7m, 7m) \).