To find the coordinates of the center of mass of a system of four particles with given masses and positions, we can follow these steps:
### Step 1: Identify the positions of the particles
We have four particles with the following masses and positions:
- Particle 1 (1 kg) at the origin: \( (0, 0) \)
- Particle 2 (2 kg) at \( (1, 0) \) (on the x-axis)
- Particle 3 (3 kg) at \( (0, 1) \) (on the y-axis)
- Particle 4 (4 kg) at \( (1, 1) \) (diagonal corner)
### Step 2: Write down the coordinates
The coordinates of the particles are:
- \( m_1 = 1 \, \text{kg} \) at \( (x_1, y_1) = (0, 0) \)
- \( m_2 = 2 \, \text{kg} \) at \( (x_2, y_2) = (1, 0) \)
- \( m_3 = 3 \, \text{kg} \) at \( (x_3, y_3) = (0, 1) \)
- \( m_4 = 4 \, \text{kg} \) at \( (x_4, y_4) = (1, 1) \)
### Step 3: Calculate the total mass
The total mass \( M \) of the system is:
\[
M = m_1 + m_2 + m_3 + m_4 = 1 + 2 + 3 + 4 = 10 \, \text{kg}
\]
### Step 4: Calculate the x-coordinate of the center of mass
The x-coordinate of the center of mass \( x_{cm} \) is given by:
\[
x_{cm} = \frac{1}{M} \sum_{i=1}^{n} m_i x_i = \frac{1}{10} \left( m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4 \right)
\]
Substituting the values:
\[
x_{cm} = \frac{1}{10} \left( 1 \cdot 0 + 2 \cdot 1 + 3 \cdot 0 + 4 \cdot 1 \right) = \frac{1}{10} \left( 0 + 2 + 0 + 4 \right) = \frac{6}{10} = 0.6
\]
### Step 5: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass \( y_{cm} \) is given by:
\[
y_{cm} = \frac{1}{M} \sum_{i=1}^{n} m_i y_i = \frac{1}{10} \left( m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4 \right)
\]
Substituting the values:
\[
y_{cm} = \frac{1}{10} \left( 1 \cdot 0 + 2 \cdot 0 + 3 \cdot 1 + 4 \cdot 1 \right) = \frac{1}{10} \left( 0 + 0 + 3 + 4 \right) = \frac{7}{10} = 0.7
\]
### Step 6: Final coordinates of the center of mass
Thus, the coordinates of the center of mass of the system are:
\[
(x_{cm}, y_{cm}) = (0.6, 0.7)
\]
