Three particles are placed in the xy plane.A 30-g particle is located at (3,4)m,and a 40-g particle is located at (-2,-2)m.Where must a 20-g particle be placed so that the center of mass of the three-particle system is at the origin?
A)(-3,-1)m
B)(+1,+3)m
C)(+3,-1)m
D)(-1,-3)m
E)(-0.5,-2)m

Question

Quizplus · Accepted Answer

To find the position of the third particle, we need to take the weighted average of the positions of the two particles already given, using their masses as the weights:
$$x_{cm}=\frac{m_1x_1+m_2x_2+m_3x_3}{m_1+m_2+m_3}=0$$
$$y_{cm}=\frac{m_1y_1+m_2y_2+m_3y_3}{m_1+m_2+m_3}=0$$
Substituting the given values, we get:
$$\frac{(30)(3)+(40)(-2)+20x_3}{30+40+20}=0\Rightarrow x_3=\frac{11}{2}$$
$$\frac{(30)(4)+(40)(-2)+20y_3}{30+40+20}=0\Rightarrow y_3=8$$
Therefore, the position of the third particle is (5.5,8), which corresponds to choice (E).

Three Particles Are Placed in the Xy Plane