Centre of mass of 3 particles 10 kg, 20 kg and 30 kg is at (0, 0, 0). Where should a particle of mass 40 kg be placed so that the combination centre of mass will be at (3,3,3)

Options

A. `(0,0,0)`
B. `(7.5,7.5,7.5)`
C. `(1,2,3)`
D. `(4,4,4)`

Answer

To solve the problem of finding the position of a 40 kg particle such that the center of mass of the system (10 kg, 20 kg, 30 kg, and 40 kg) is at (3, 3, 3), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the masses and their positions:**
- Let the masses be:
- \( m_1 = 10 \, \text{kg} \) at position \( \mathbf{r_1} = (0, 0, 0) \)
- \( m_2 = 20 \, \text{kg} \) at position \( \mathbf{r_2} = (0, 0, 0) \)
- \( m_3 = 30 \, \text{kg} \) at position \( \mathbf{r_3} = (0, 0, 0) \)
- The center of mass of these three particles is at the origin (0, 0, 0).

2. **Introduce the new mass:**
- Let the new mass \( m_4 = 40 \, \text{kg} \) be at position \( \mathbf{r_4} = (x, y, z) \).

3. **Calculate the total mass:**
- The total mass \( M \) of the system is:
\[ M = m_1 + m_2 + m_3 + m_4 = 10 + 20 + 30 + 40 = 100 \, \text{kg} \]

4. **Set up the center of mass equation:**
- The center of mass \( \mathbf{R} \) of the system is given by:
\[ \mathbf{R} = \frac{m_1 \mathbf{r_1} + m_2 \mathbf{r_2} + m_3 \mathbf{r_3} + m_4 \mathbf{r_4}}{M} \]
- Substituting the known values:
\[ \mathbf{R} = \frac{10(0, 0, 0) + 20(0, 0, 0) + 30(0, 0, 0) + 40(x, y, z)}{100} \]
- This simplifies to:
\[ \mathbf{R} = \frac{40(x, y, z)}{100} = \frac{2}{5}(x, y, z) \]

5. **Set the center of mass to the desired position:**
- We want the center of mass to be at \( (3, 3, 3) \):
\[ \frac{2}{5}(x, y, z) = (3, 3, 3) \]

6. **Solve for \( x, y, z \):**
- From the equation, we can equate each component:
\[ \frac{2}{5}x = 3 \implies x = 3 \cdot \frac{5}{2} = 7.5 \]
\[ \frac{2}{5}y = 3 \implies y = 3 \cdot \frac{5}{2} = 7.5 \]
\[ \frac{2}{5}z = 3 \implies z = 3 \cdot \frac{5}{2} = 7.5 \]

7. **Conclusion:**
- The position of the 40 kg particle should be \( (7.5, 7.5, 7.5) \).

### Final Answer:
The 40 kg particle should be placed at the coordinates \( (7.5, 7.5, 7.5) \).