To find the position of the fourth mass (4 kg) such that the new center of mass of the system is at (0, 0, 0), we can follow these steps:
### Step 1: Understand the Given Data
We have three masses:
- \( m_1 = 1 \, \text{kg} \)
- \( m_2 = 2 \, \text{kg} \)
- \( m_3 = 3 \, \text{kg} \)
The center of mass of these three particles is given as \( (2, 2, 2) \).
### Step 2: Calculate the Total Mass
The total mass of the three particles is:
\[
M = m_1 + m_2 + m_3 = 1 + 2 + 3 = 6 \, \text{kg}
\]
### Step 3: Use the Center of Mass Formula
The formula for the center of mass \( (x_{cm}, y_{cm}, z_{cm}) \) of a system of particles is given by:
\[
x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3}
\]
Similarly for \( y_{cm} \) and \( z_{cm} \).
Given that the center of mass is at \( (2, 2, 2) \), we can write:
\[
m_1 x_1 + m_2 x_2 + m_3 x_3 = 12 \quad \text{(for x-coordinate)}
\]
\[
m_1 y_1 + m_2 y_2 + m_3 y_3 = 12 \quad \text{(for y-coordinate)}
\]
\[
m_1 z_1 + m_2 z_2 + m_3 z_3 = 12 \quad \text{(for z-coordinate)}
\]
### Step 4: Introduce the Fourth Mass
Let the position of the fourth mass \( m_4 = 4 \, \text{kg} \) be \( (x_4, y_4, z_4) \).
The new center of mass after adding the fourth mass should be at \( (0, 0, 0) \). Thus, we can write:
\[
x_{cm}' = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
\[
y_{cm}' = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
\[
z_{cm}' = \frac{m_1 z_1 + m_2 z_2 + m_3 z_3 + m_4 z_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
### Step 5: Set Up the Equations
Substituting the known values into the equations:
\[
\frac{12 + 4x_4}{10} = 0 \quad \Rightarrow \quad 12 + 4x_4 = 0 \quad \Rightarrow \quad 4x_4 = -12 \quad \Rightarrow \quad x_4 = -3
\]
\[
\frac{12 + 4y_4}{10} = 0 \quad \Rightarrow \quad 12 + 4y_4 = 0 \quad \Rightarrow \quad 4y_4 = -12 \quad \Rightarrow \quad y_4 = -3
\]
\[
\frac{12 + 4z_4}{10} = 0 \quad \Rightarrow \quad 12 + 4z_4 = 0 \quad \Rightarrow \quad 4z_4 = -12 \quad \Rightarrow \quad z_4 = -3
\]
### Step 6: Conclusion
Thus, the position of the fourth mass should be:
\[
(x_4, y_4, z_4) = (-3, -3, -3)
\]
