Find the coordination of the centre of mass of a uniform semicircular wire of radius R and mass M.

Explanation

Figure shows a uniform semicircular wire of radius r and mass M . Its centre O is taken as the origin , the line joining its ends as x - axis in the plane of the wire is taken as y - axis
Let us take radius making an angle `theta ` with the x - axis . Now consider a small element of length `edtheta ` of the wire . Its coordinates are ` ( r cos theta , r sin theta )`
Its mass `dm M/(pir) = M/(pir) . rd theta = M/pi d theta `
The centre of mass lies on the axis of symmetry .
The coordinate of the centre of mass are
`X _(CM) = 0 `
`Y_(CM) = 1/M int y dm = 1/(M_(0)) underset (0) overset (1) int ( r sin theta ) M/pi d theta `
` = r/pi [ - cos theta ] _(0)^(pi) = (2r)/pi`
Hence the centre of mass is `(0,(2r)/pi)`