A 3.00 kg mass is located at x = 2.0 cm and y = 0.0 cm. A 3.00 kg mass is located at x = 0.0 cm and y = 2.0 cm. A 4.00 kg mass is located at x = 3.0 cm and y = −3.0 cm. Where is the location of the center of mass?
Question
A 3.00 kg mass is located at x = 2.0 cm and y = 0.0 cm. A 3.00 kg mass is located at x = 0.0 cm and y = 2.0 cm. A 4.00 kg mass is located at x = 3.0 cm and y = −3.0 cm. Where is the location of the center of mass?
Solution
The center of mass (COM) of a system of particles can be found using the formula:
COM = (Σ mi*ri) / Σ mi
where mi is the mass of the ith particle and ri is the position of the ith particle. The sum is over all particles in the system.
In this case, we have three particles. We can find the x and y coordinates of the center of mass separately.
For the x-coordinate:
COM_x = (m1x1 + m2x2 + m3*x3) / (m1 + m2 + m3)
Substituting the given values:
COM_x = [(3 kg * 2 cm) + (3 kg * 0 cm) + (4 kg * 3 cm)] / (3 kg + 3 kg + 4 kg)
COM_x = (6 cmkg + 0 cmkg + 12 cm*kg) / 10 kg
COM_x = 18 cm*kg / 10 kg
COM_x = 1.8 cm
For the y-coordinate:
COM_y = (m1y1 + m2y2 + m3*y3) / (m1 + m2 + m3)
Substituting the given values:
COM_y = [(3 kg * 0 cm) + (3 kg * 2 cm) + (4 kg * -3 cm)] / (3 kg + 3 kg + 4 kg)
COM_y = (0 cmkg + 6 cmkg - 12 cm*kg) / 10 kg
COM_y = -6 cm*kg / 10 kg
COM_y = -0.6 cm
So, the center of mass is located at (1.8 cm, -0.6 cm).
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