Question

In Fig. 9-37, three uniform thin rods, each of length $L=22cm,$ form an inverted U. The vertical rods each have amass of $14g$ ; the horizontal rod has amass of $42g.$ What are (a) the x coordinate and (b) the y coordinate of the system’s center of mass?

Answer 1

We will refer to the arrangement as a “table.” We locate the coordinate origin at the left end of the tabletop (as shown in Fig. 9-37). With +x rightward and +y upward, then the center of mass of the right leg is at (x,y) = $(+L,–L/2),$ the center of mass of the left leg is at (x,y) = $(0,–L/2),$ and the center of mass of the tabletop is at (x,y) = (L/2, 0).
(a) The x coordinate of the (whole table) center of mass is
$x_{com}=\frac{M(+L)+M\left(0\right)+3M(+L/2)}{M+M+3M}=\frac{L}{2}$
With $L=22cm$ , we have $x_{com}=\left(22cm\right)/2=11cm$ .
(b) The y coordinate of the (whole table) center of mass is
$y_{com}=\frac{M(-L/2)+M(-L/2)+3M\left(0\right)}{M+M+3M}=-\frac{L}{5}$
or $y_{com}=-\left(22cm\right)/5=-4.4cm$.
From the coordinates , we see that the whole table center of mass is a small distance $4.4cm$ directly below the middle of the tabletop.