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The problem involves finding the center of mass of four particles arranged at the vertices of a square. Let me summarize the setup:

- The particles are located at the corners of a square with side length \( 2.0 \, \text{m} \).
- The masses are as follows:
  - \( m_1 = 1.0 \, \text{kg} \)
  - \( m_2 = 2.0 \, \text{kg} \)
  - \( m_3 = 3.0 \, \text{kg} \)
  - \( m_4 = 4.0 \, \text{kg} \)

We are tasked with determining the center of mass (in meters) of the system.

### Step 1: Coordinates of Each Particle

From the diagram, we can deduce the coordinates of the particles based on their positions in a 2D coordinate system.

- \( m_1 \) is located at \( (0, 0) \)
- \( m_2 \) is located at \( (0, 2) \)
- \( m_3 \) is located at \( (2, 2) \)
- \( m_4 \) is located at \( (2, 0) \)

### Step 2: Formula for the Center of Mass

The coordinates \( (x_{\text{cm}}, y_{\text{cm}}) \) of the center of mass are given by:

\[
x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i}
\]

Where:
- \( m_i \) is the mass of each particle,
- \( x_i \) and \( y_i \) are the x- and y-coordinates of each particle.

### Step 3: Calculating the x-Coordinate of the Center of Mass

\[
x_{\text{cm}} = \frac{(m_1 \cdot x_1) + (m_2 \cdot x_2) + (m_3 \cdot x_3) + (m_4 \cdot x_4)}{m_1 + m_2 + m_3 + m_4}
\]

Substitute the values:

\[
x_{\text{cm}} = \frac{(1.0 \cdot 0) + (2.0 \cdot 0) + (3.0 \cdot 2) + (4.0 \cdot 2)}{1.0 + 2.0 + 3.0 + 4.0}
\]

\[
x_{\text{cm}} = \frac{0 + 0 + 6.0 + 8.0}{10.0} = \frac{14.0}{10.0} = 1.4 \, \text{m}
\]

### Step 4: Calculating the y-Coordinate of the Center of Mass

\[
y_{\text{cm}} = \frac{(m_1 \cdot y_1) + (m_2 \cdot y_2) + (m_3 \cdot y_3) + (m_4 \cdot y_4)}{m_1 + m_2 + m_3 + m_4}
\]

Substitute the values:

\[
y_{\text{cm}} = \frac{(1.0 \cdot 0) + (2.0 \cdot 2) + (3.0 \cdot 2) + (4.0 \cdot 0)}{10.0}
\]

\[
y_{\text{cm}} = \frac{0 + 4.0 + 6.0 + 0}{10.0} = \frac{10.0}{10.0} = 1.0 \, \text{m}
\]

### Final Answer

The coordinates of the center of mass are:

\[
(x_{\text{cm}}, y_{\text{cm}}) = (1.4 \, \text{m}, 1.0 \, \text{m})
\]

Would you like more details on the calculations? Let me know if you have any further questions!

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Here are five questions related to this topic:
1. How does the mass of a particle affect the center of mass?
2. What would happen to the center of mass if one of the particles had zero mass?
3. Can you explain how the center of mass is affected by changing the position of one particle?
4. How would you calculate the center of mass for a system of three particles arranged in a triangle?
5. What is the significance of the center of mass in physical systems?

**Tip**: When working with multiple particles, always pay close attention to both their masses and positions since both factors equally influence the center of mass.

Determine the position of the center of mass (in meters) of a system of four particles (m1 = 1.0 kg, m2 = 2.0 kg, m3 = 3.0 kg, m4 = 4.0 kg) located at the vertices of a square with a side length of 2.0 m.

Learn how to calculate the center of mass of a system of four particles with masses 1.0 kg, 2.0 kg, 3.0 kg, and 4.0 kg, located at the vertices of a 2.0 m square. This problem covers key concepts in physics and coordinate geometry, perfect for high school students.

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