Question 9.2: Figure 9-35 shows a three-particle system, with masses m1 = ......
Figure 9-35 shows a three-particle system, with masses m_{1} = 3.0 kg, {{m}}_{2} = 4.0 kg, and m_{3} = 8.0 kg. The scales on the axes are set by {x_{s}} = 2.0 m and y_{{s}} = 2.0 m. What are (a) the x coordinate and (b) the y coordinate of the system’s center of mass? (c) If m_{3} is gradually increased, does the center of mass of the system shift toward or away from that particle, or does it remain stationary?
Our notation is as follows: x_{1} = 0 and y_{1} = 0 are the coordinates of the m_{1} = 3.0 kg particle; {{x}}_{2} = 2.0 m and y_{2} = 1.0 m are the coordinates of the m_{2} = 4.0 kg particle; and x_{3} = 1.0 m and y_{3} = 2.0 m are the coordinates of the m_{3} = 8.0 kg particle.
(a) The x coordinate of the center of mass is
x_{\mathrm{com}}={\frac{m_{{1}}x_{{1}}+m_{2}x_{{2}}+m_{3}x_{{3}}}{m_{{1}}+m_{2}+m_{3}}}={\frac{0+\left(4.0\mathrm{\ kg}\right)\left(2.0\mathrm{\ m}\right)+\left(8.0\mathrm{\ kg}\right)\left(1.0\mathrm{\ m}\right)}{3.0\mathrm{\ kg}+4.0\mathrm{\ kg}+8.0\mathrm{\ kg}}}=1.1~\mathrm{m.}
(b) The y coordinate of the center of mass is
y_{\mathrm{com}}={\frac{m_{1}y_{1}+m_{2}y_{2}+m_{3}y_{3}}{m_{1}+m_{2}+m_{3}}}={\frac{0+\left(4.0\mathrm{\ kg}\right)\left(1.0\mathrm{\ m}\right)+\left(8.0\mathrm{\ kg}\right)\left(2.0\mathrm{\ m}\right)}{3.0\mathrm{\ kg}+4.0\mathrm{\ kg}+8.0\mathrm{\ kg}}}=1.3\,\mathrm{m.}
(c) As the mass of m_{3}, the topmost particle, is increased, the center of mass shifts toward that particle. As we approach the limit where m_{3} is infinitely more massive than the others, the center of mass becomes infinitesimally close to the position of m_{3}.