A 3 kg mass is positioned at (0, 8), and a 1 kg mass is positioned at (12, 0). What are the coordinates of a 4 kg mass which will result in the center of mass of the system of three masses being located at the origin (0, 0)?
A) (-3, -6)
B) (-6, -3)
C) (-12, -8)
D) (3, 6)

Question

Quizplus · Accepted Answer

The center of mass of the system of three masses is given by the formula:$$(x_c, y_c) = \frac{m_1(x_1, y_1) + m_2(x_2, y_2) + m_3(x_3, y_3)}{m_1 + m_2 + m_3}$$where $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ are the coordinates of the three masses, and $m_1$, $m_2$, and $m_3$ are their respective masses.In this case, we have:$$(x_c, y_c) = \frac{3 kg(0, 8) + 1 kg(12, 0) + 4 kg(x_3, y_3)}{3 kg + 1 kg + 4 kg}$$$$(x_c, y_c) = \frac{(0, 24) + (12, 0) + (4x_3, 4y_3)}{8 kg}$$$$(x_c, y_c) = \frac{(12 + 4x_3, 24 + 4y_3)}{8}$$In order for the center of mass to be located at the origin, we must have:$$12 + 4x_3 = 0$$$$24 + 4y_3 = 0$$Solving these equations, we get:$$x_3 = -3$$$$y_3 = -6$$Therefore, the coordinates of the 4 kg mass are $(-3, -6)$.

A 3 Kg Mass Is Positioned at (0, 8), and a 1