Question

A Wireframe is made of a wire or uniform cross-section. which is shown in figure $ABC.HGF$ and $DIE$ are semicircular arcs of radius $r.CD=DO=OE=EF=r$ and ${}^{\prime }{O}^{\prime }$ is the centre of circle. Then centre of mass of frame is :

Answer 1

assume origin at $O$

Cordinate of COM of ring $CBA$

$(r_{1}2r+\frac{2r}{\pi })$

Cordinate of COM of ring $DIE$

$(-\frac{2r}{\pi },o)$

Cordinate of COM of ring $HGF$

$(-r,-(2r+\frac{2r}{\pi }))$, assume mass of ring of each is '$m$'

$X_{cm}=\frac{m{X}_1+m{X}_2+m{X}_3}{3m}=\frac{mr+m(-\frac{2r}{\pi })-mr}{3m}$

$=-\frac{2}{3}\frac{r}{\pi }$

$Y_{cm}=\frac{m{Y}_1+m{Y}_2+m{Y}_3}{}=\frac{0}{3m}=0$

COM of frame is at $(-\frac{2r}{3\pi },0)$