Answer to Question #249623 in Molecular Physics | Thermodynamics for Kelani

Question

Consider the following mass distribution where the x- and y-coordinates are given in meters: 5.0 kg at (0.0, 0.0) m, 2.9 kg at (0.0, 4.8) m, and 4.0 kg at (2.7, 0.0) m. Where should the fourth object of 9.4 kg be placed so that the center of gravity of the four-object arrangement will be at (0.0, 0.0) m?

x = ___  m

y = ___ m

Answer 1

"The center of gravity of a collection of masses is the point where all the weight of the object can be considered to be concentrated. If (x_cg,y_cg) are the coordinates of the centre of gravity of a collection of point masses m₁, m₂ , etc, located at coordinates (x₁,y₁), (x₂,y₂), respectively, then:"(https://theory.uwinnipeg.ca/physics/rot/node4.html).

$x_{cg} = \dfrac{\sum m_ix_i}{\sum m_i},\; y_{cg} = \dfrac{\sum m_iy_i}{\sum m_i}$

In our case,

$0 = \dfrac{1}{5+2.9+4.0+9.4}\cdot (5\cdot0.0+2.9\cdot0.0 + 4.0\cdot 2.7 +9.4\cdot x) \\ \; \Rightarrow x = -1.15. \\ 0 = \dfrac{1}{5+2.9+4.0+9.4}\cdot (5\cdot0.0+2.9\cdot4.8 + 4.0\cdot 0.0 +9.4\cdot y) \\ \; \Rightarrow y = -1.48. \\$