Question

Find the centre of mass of a uniform semi-circular ring of radius $R$ and mass $m$.

Answer 1

Consider the centre of the ring at origin. Consider a differential element of length $dl$ subtended by the length $dl$ is $d\theta$ at the centre, then $dl=Rd\theta$. Let $\lambda$ be the mass per unit length. Then mass of this element is $dm=\lambda Rd\theta$
$x_{cm}=\frac{1}{m}\int xdm=\frac{1}{m}{\int }_0^{\pi }\left(R\cos \theta \right)\lambda Rd\theta =0$
$\Rightarrow y_{cm}=\frac{1}{m}{\int }_0^{\pi }\left(R\sin \theta \right)\lambda Rd\theta =\frac{\lambda R^2}{m}{\int }_0^{\pi }\left(\sin \theta \right)d\theta =\frac{\lambda R^2}{m}{[-\cos \theta ]}_0^{\pi }\Rightarrow y_{cm}=2R/\pi$