To solve the problem of finding the position of a fourth particle of mass 4g so that the center of mass of the four-particle system is at the origin, we can follow these steps:
### Step 1: Understand the given data
We have three particles with the following masses and coordinates:
- Mass \( m_1 = 1g \) at coordinates \( (2, 2, 2) \)
- Mass \( m_2 = 2g \) at coordinates \( (2, 2, 2) \)
- Mass \( m_3 = 3g \) at coordinates \( (2, 2, 2) \)
The total mass of these three particles is:
\[
m_1 + m_2 + m_3 = 1g + 2g + 3g = 6g
\]
### Step 2: Calculate the center of mass of the three particles
The center of mass \( (x_{cm}, y_{cm}, z_{cm}) \) for the three particles can be calculated using the formula:
\[
x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3}
\]
\[
y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3}
\]
\[
z_{cm} = \frac{m_1 z_1 + m_2 z_2 + m_3 z_3}{m_1 + m_2 + m_3}
\]
Substituting the values:
\[
x_{cm} = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2}{6} = \frac{12}{6} = 2
\]
\[
y_{cm} = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2}{6} = \frac{12}{6} = 2
\]
\[
z_{cm} = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2}{6} = \frac{12}{6} = 2
\]
Thus, the center of mass of the three particles is at \( (2, 2, 2) \).
### Step 3: Introduce the fourth particle
Let the fourth particle have mass \( m_4 = 4g \) and its coordinates be \( (x_4, y_4, z_4) \). We want the center of mass of the four particles to be at the origin \( (0, 0, 0) \).
### Step 4: Set up the equations for the center of mass of four particles
The center of mass for the four particles can be expressed as:
\[
x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
\[
y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
\[
z_{cm} = \frac{m_1 z_1 + m_2 z_2 + m_3 z_3 + m_4 z_4}{m_1 + m_2 + m_3 + m_4} = 0
\]
### Step 5: Substitute the known values into the equations
Substituting the known values into the x-coordinate equation:
\[
0 = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2 + 4 x_4}{6 + 4}
\]
\[
0 = \frac{12 + 4 x_4}{10}
\]
Multiplying both sides by 10:
\[
0 = 12 + 4 x_4
\]
Solving for \( x_4 \):
\[
4 x_4 = -12 \implies x_4 = -3
\]
### Step 6: Repeat for y and z coordinates
Using the same approach for the y-coordinate:
\[
0 = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2 + 4 y_4}{10}
\]
\[
0 = 12 + 4 y_4 \implies y_4 = -3
\]
For the z-coordinate:
\[
0 = \frac{1 \cdot 2 + 2 \cdot 2 + 3 \cdot 2 + 4 z_4}{10}
\]
\[
0 = 12 + 4 z_4 \implies z_4 = -3
\]
### Step 7: Conclusion
The coordinates of the fourth particle should be \( (-3, -3, -3) \).
### Final Answer
The fourth particle of mass 4g should be positioned at \( (-3, -3, -3) \).
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