Find the center of mass of the system of objects having masses $4$, $6$, and $9$ kg, located at the points $(-2,1), (3,-2)$, and $(4,3),$ respectively.

Solution See Figure 68. Where is a good estimate for the center of mass? Certainly, it will lie within the rectangle $-2\leq x\leq 4$; $-2\leq y\leq 3$.

460

To find the exact position of the center of mass, we first find the moment of the system about the $y$-axis, $M_{y},$ and the moment of the system about the $x$-axis, $M_{x}$. $$\begin{align*} M_{y}=\sum\limits_{i=1}^{3} m_{i}x_{i} =4(-2)+6(3)+9(4)=46\\ M_{x}=\sum\limits_{i=1}^{3} m_{i} y_{i} =4(1)+6(-2)+9(3)=19 \end{align*}$$

Since the mass $M$ of the system is $M=4+6+9=19,$ we have $$\begin{equation*} \bar{x}=\frac{M_{y}}{M}= \frac{46}{19},\qquad \bar{y}=\frac{M_{x}}{M}=\frac{19}{19}=1 \end{equation*}$$

The center of mass of these objects is at the point $\left( \dfrac{46}{19},1\right)$. Notice that the center of mass lies in the rectangle $-2\leq x\leq 4$; $-2\leq y\leq 3$.