Question

Find the center of mass of uniform semi-circular ring of radius R

Answer 1

As the wire is uniform, the mass per unit length of the wire is $\frac{M}{\pi R}$. The mass of the element is, therefore,

$dm=\left(\frac{M}{\pi R}\right)\left(Rd\theta \right)=\frac{M}{\pi }d\theta$

The coordinates of the center of mass are

$X=\frac{1}{M}\int xdm=\frac{1}{M}{\int }_0^{\pi }\left(R\cos \theta \right)\left(\frac{M}{\pi }\right)d\theta =0$

$Y=\frac{1}{M}\int ydm=\frac{1}{M}{\int }_0^{\pi }\left(R\sin \theta \right)\left(\frac{M}{\pi }\right)d\theta =\frac{2R}{\pi }$

Hence, position of center of mass, $(0,\frac{2R}{\pi })$