To find the coordinates of the center of mass of the system, we use the following formulas for the center of mass in the x- and y-directions:

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

Where $m_i$ are the masses and $x_i$, $y_i$ are their corresponding coordinates.

From the figure:

• $m_1 = 3.3 \, \text{kg}$, at $(x_1, y_1) = (1, 0)$
• $m_2 = 4.9 \, \text{kg}$, at $(x_2, y_2) = (2, 1)$
• $m_3 = 6.4 \, \text{kg}$, at $(x_3, y_3) = (0, 3)$

Step 1: Calculate $x_{cm}$

$x_{cm} = \frac{(3.3 \times 1) + (4.9 \times 2) + (6.4 \times 0)}{3.3 + 4.9 + 6.4} = \frac{3.3 + 9.8 + 0}{14.6} = \frac{13.1}{14.6} \approx 0.90 \, \text{m}$

Step 2: Calculate $y_{cm}$

$y_{cm} = \frac{(3.3 \times 0) + (4.9 \times 1) + (6.4 \times 3)}{3.3 + 4.9 + 6.4} = \frac{0 + 4.9 + 19.2}{14.6} = \frac{24.1}{14.6} \approx 1.65 \, \text{m}$

Thus, the coordinates of the center of mass are:

• $x_{cm} \approx 0.90 \, \text{m}$
• $y_{cm} \approx 1.65 \, \text{m}$

Would you like further clarification on any of these steps?

Related Questions:

1. What happens to the center of mass if all the particles have the same mass?
2. How would you find the center of mass for a system with more than three particles?
3. How do the coordinates of the individual masses influence the center of mass position?
4. Can the center of mass ever lie outside the physical system of particles?
5. How does the center of mass change if the coordinate system is shifted?

Tip: The center of mass represents the average position of mass distribution in a system and can simplify the analysis of motion in complex systems.