A $200\mathrm{g}$ mass is at (0, 0), a $300\mathrm{g}$ mass is at ($10\mathrm{cm}$, 0), and a $400\mathrm{g}$ mass is at ($13\mathrm{cm}$, $15\mathrm{cm}$). What are the coordinates of the center of mass?

• A. $9.1\mathrm{cm}$, $4.5\mathrm{cm}$
• B. $8.6\mathrm{cm}$, $4.5\mathrm{cm}$
• C. $4.1\mathrm{cm}$, $6.7\mathrm{cm}$
• D. $7.0\mathrm{cm}$, $4.5\mathrm{cm}$
• E. $9.1\mathrm{cm}$, $6.7\mathrm{cm}$

Answer: (E)

Verified step by step guidance

1. Identify the masses and their coordinates: m1 = 200g at (0, 0), m2 = 300g at (10cm, 0), m3 = 400g at (13cm, 15cm).
2. Convert the masses to kilograms for consistency in units: m1 = 0.2kg, m2 = 0.3kg, m3 = 0.4kg.
3. Use the formula for the x-coordinate of the center of mass: x_cm = (m1 * x1 + m2 * x2 + m3 * x3) / (m1 + m2 + m3). Substitute the values: x_cm = (0.2 * 0 + 0.3 * 10 + 0.4 * 13) / (0.2 + 0.3 + 0.4).
4. Use the formula for the y-coordinate of the center of mass: y_cm = (m1 * y1 + m2 * y2 + m3 * y3) / (m1 + m2 + m3). Substitute the values: y_cm = (0.2 * 0 + 0.3 * 0 + 0.4 * 15) / (0.2 + 0.3 + 0.4).
5. Calculate the x_cm and y_cm using the substituted values to find the coordinates of the center of mass.