To solve the problem, we need to find the mass \( M \) given the coordinates of four masses and the center of mass (COM). The masses and their coordinates are as follows:
- Mass \( m_1 = 1 \, \text{kg} \) at \( (2, 1, 1) \)
- Mass \( m_2 = 1.5 \, \text{kg} \) at \( (1, 2, 1) \)
- Mass \( m_3 = 2 \, \text{kg} \) at \( (2, -2, 1) \)
- Mass \( m_4 = M \, \text{kg} \) at \( (-1, 4, 3) \)
The center of mass is given as \( (1, 1, \frac{3}{2}) \).
### Step 1: Write the formula for the center of mass
The coordinates of the center of mass are given by:
\[
x_{COM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{m_1 + m_2 + m_3 + m_4}
\]
\[
y_{COM} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{m_1 + m_2 + m_3 + m_4}
\]
\[
z_{COM} = \frac{m_1 z_1 + m_2 z_2 + m_3 z_3 + m_4 z_4}{m_1 + m_2 + m_3 + m_4}
\]
### Step 2: Substitute the known values into the equations
Using the given values:
1. For \( x_{COM} \):
\[
1 = \frac{1 \cdot 2 + 1.5 \cdot 1 + 2 \cdot 2 + M \cdot (-1)}{1 + 1.5 + 2 + M}
\]
2. For \( y_{COM} \):
\[
1 = \frac{1 \cdot 1 + 1.5 \cdot 2 + 2 \cdot (-2) + M \cdot 4}{1 + 1.5 + 2 + M}
\]
3. For \( z_{COM} \):
\[
\frac{3}{2} = \frac{1 \cdot 1 + 1.5 \cdot 1 + 2 \cdot 1 + M \cdot 3}{1 + 1.5 + 2 + M}
\]
### Step 3: Simplify each equation
1. **For \( x_{COM} \)**:
\[
1 = \frac{2 + 1.5 + 4 - M}{4.5 + M}
\]
\[
1(4.5 + M) = 7.5 - M
\]
\[
4.5 + M = 7.5 - M
\]
\[
2M = 3 \implies M = \frac{3}{2} = 1.5 \, \text{kg}
\]
2. **For \( y_{COM} \)**:
\[
1 = \frac{1 + 3 - 4 + 4M}{4.5 + M}
\]
\[
1(4.5 + M) = 0 + 4M
\]
\[
4.5 + M = 4M
\]
\[
3M = 4.5 \implies M = \frac{4.5}{3} = 1.5 \, \text{kg}
\]
3. **For \( z_{COM} \)**:
\[
\frac{3}{2} = \frac{1 + 1.5 + 2 + 3M}{4.5 + M}
\]
\[
\frac{3}{2}(4.5 + M) = 4.5 + 3M
\]
\[
6.75 + \frac{3}{2}M = 4.5 + 3M
\]
\[
2.25 = \frac{3}{2}M - 3M
\]
\[
2.25 = -\frac{3}{2}M \implies M = 1.5 \, \text{kg}
\]
### Final Answer
Thus, the value of \( M \) is \( 1.5 \, \text{kg} \).
