Find the centre of mass of a uniform semicircular ring of radius `R` and mass `M`.

Explanation

Consider the centre of the ring at orign. Consider a differential element of length dl of the ring whose radius vector makes an angle `theta` with the x-axis. If the angle 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`
`rArr y_(cm)=(1)/(m)int_(0)^(pi)(R sin theta)lambda Rd theta = (lambda R^(2))/(m)int_(0)^(pi)sin theta d theta = (lambda R^(2))/(lambda pi R)[-cos theta]_(0)^(pi)`
`rArr y_(cm)=2R//pi`.