Solved Problem on Center of Mass

We have the following distribution of masses in the xy-plane: m₁ = 2 kg at position (1, -1), m₂ = 3 kg at (0, 2), m₃ = 1 kg at (-1,0), m₄ = 2 kg at (4, 3) and m₅ = 7 kg at (-11, 2). Determine the coordinates of the center of mass of this distribution and plot a graph.

Problem data:

	mass (kg)	position (x, y)
1	2	(1, -1)
2	3	(0, 2)
3	1	(-1, 0)
4	2	(4, 3)
5	7	(-11, 2)

Table 1

Problem diagram:

Figure 1

Solution

The Center of Mass is given by

\[ \begin{gather} \bbox[#99CCFF,10px] {{\vec{r}}_{cm}=\frac{\sum m_{i}{\vec{r}}_{i}}{m_{i}}} \end{gather} \]

writing the components

\[ \begin{gather} x_{cm}=\frac{\sum m_{i}x_{i}}{m_{i}} \end{gather} \]

\[ \begin{gather} y_{cm}=\frac{\sum m_{i}y_{i}}{m_{i}} \end{gather} \]

substituting the data

\[ \begin{gather} x_{cm}=\frac{2\times 1+3\times 0+1\times (-1)+2\times 4+7\times (-11)}{2+3+1+2+7}\\[5pt] x_{cm}=\frac{2+0-1+8-77}{15}\\[5pt] x_{cm}=\frac{-68}{15}\\[5pt] x_{cm}\simeq -4,5 \\[10pt] y_{cm}=\frac{2\times (-1)+3\times 2+1\times 0+2\times 3+7\times 2}{2+3+1+2+7}\\[5pt] y_{cm}=\frac{-2+6+0+6+14}{15}\\[5pt] y_{cm}=\frac{24}{15}\\[5pt] y_{cm}=1,6\ \end{gather} \]

The coordinates of the center of mass are

\[ \begin{gather} \bbox[#FFCCCC,10px] {\left(x_{cm};y_{cm}\;\right)=\left(-4.5,1.6\right)} \end{gather} \]

plotting the graph

Figure 2

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