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:
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