To find the center of mass of the system of three masses, we use the formula for the coordinates $(x_{cm}, y_{cm})$ of the center of mass:

$x_{cm} = \frac{\sum (m_i x_i)}{\sum m_i}$ $y_{cm} = \frac{\sum (m_i y_i)}{\sum m_i}$

where $m_i$ is the mass and $(x_i, y_i)$ are the coordinates of the masses.

Given:

• Mass $m_1 = 5 \, \text{kg}$ at coordinates $(-4, 3)$
• Mass $m_2 = 9 \, \text{kg}$ at coordinates $(2, 2)$
• Mass $m_3 = 12 \, \text{kg}$ at coordinates $(5, 0)$

Step 1: Find $x_{cm}$

$x_{cm} = \frac{(5)(-4) + (9)(2) + (12)(5)}{5 + 9 + 12}$ $x_{cm} = \frac{-20 + 18 + 60}{26}$ $x_{cm} = \frac{58}{26} \approx 2.23$

Step 2: Find $y_{cm}$

$y_{cm} = \frac{(5)(3) + (9)(2) + (12)(0)}{5 + 9 + 12}$ $y_{cm} = \frac{15 + 18 + 0}{26}$ $y_{cm} = \frac{33}{26} \approx 1.27$

Thus, the center of mass is approximately at $(2.23, 1.27)$.

Would you like more details or have any questions?

Related Questions:

1. How does changing the mass at a specific point affect the center of mass?
2. What if the mass distribution were in 3D? How would we calculate the center of mass?
3. How do we calculate the center of mass for continuous distributions?
4. What happens if one mass is at the origin?
5. Can the center of mass lie outside the system of particles?

Tip:

The center of mass is a balance point, so it can be useful in analyzing the motion of the entire system in physics problems.