Three masses of 5 kg, 9 kg, and 12 kg are placed at (-4, 3), (2, 2) , and (5, 0) respectively. Find the center of mass.

To find the center of mass \((x_{\text{cm}}, y_{\text{cm}})\) for a system of point masses, we use the formulas:

\[
x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{\text{cm}} = \frac{\sum m_i y_i}{\sum m_i}
\]
where \(m_i\) represents the mass at each point, and \((x_i, y_i)\) are the coordinates of each mass.

Given masses and coordinates:
- Mass \(m_1 = 5\) kg at \((-4, 3)\),
- Mass \(m_2 = 9\) kg at \((2, 2)\),
- Mass \(m_3 = 12\) kg at \((5, 0)\).

### Step 1: Find the total mass
\[
M = m_1 + m_2 + m_3 = 5 + 9 + 12 = 26 \text{ kg}
\]

### Step 2: Calculate \(x_{\text{cm}}\)
\[
x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{M} = \frac{(5 \times -4) + (9 \times 2) + (12 \times 5)}{26}
\]
\[
x_{\text{cm}} = \frac{-20 + 18 + 60}{26} = \frac{58}{26} \approx 2.23
\]

### Step 3: Calculate \(y_{\text{cm}}\)
\[
y_{\text{cm}} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{M} = \frac{(5 \times 3) + (9 \times 2) + (12 \times 0)}{26}
\]
\[
y_{\text{cm}} = \frac{15 + 18 + 0}{26} = \frac{33}{26} \approx 1.27
\]

### Final Result:
The center of mass is approximately \((x_{\text{cm}}, y_{\text{cm}}) \approx (2.23, 1.27)\).

Would you like more details or have any questions?

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Here are some related questions:
1. How does the location of the center of mass change if the masses are closer together?
2. What happens to the center of mass if one of the masses is moved to a different coordinate?
3. How can the concept of center of mass be applied to a continuous object?
4. What is the difference between center of mass and center of gravity?
5. How would the center of mass change if all masses were doubled?

**Tip**: The center of mass depends on both the positions and the masses of the objects involved.

Three masses of 5 kg, 9 kg, and 12 kg are placed at (-4, 3), (2, 2), and (5, 0) respectively. Find the center of mass.

This problem demonstrates how to calculate the center of mass for three point masses located at specific coordinates: (-4, 3), (2, 2), and (5, 0). Using the center of mass formulas, this example covers key concepts in coordinate geometry and weighted averages, making it ideal for high school students in Grades 10-12.

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