Determine the centre of mass of a uniform hemisphere of radius R.

Explanation

`X_(cm)=0` (by symmetry)
`Y_(cm)` can be calculated by considering a small element (taken by drawing two circles on its surface at angles `theta` and `theta + d theta` as shown).
Surface mass density `= sigma = (M)/(2pi R^(2))=(dm)/(dA)`
`dm=(M)/(2pi R^(2))2pi R cos theta R d theta= M cos theta d theta`
`Y_(cm)=(1)/(M)int dm = (1)/(M)int y M cos theta d theta = int_(0)^(pi//2)R sin theta cos theta d theta " " (because y = R sin theta)`
On solving, we get
`Y_(CM)=R//2`
`therefore` C.M. of thin uniform hemispherical shell is `(0, (R )/(2))`