Locate the position of the centre of mass of a hemisphere of radius R as shown in the following figure:

Explanation

Let the density of hemisphere be `rho`. The hemisphere is constructed with a large number of co-axial discs. One such elementary disc of radius and thickness dy is at a distance y from the origin as shown above. Here the radius of the elemental disc is, `x=sqrt(R^(2)-y^(2))`
Mass of the elementary disc=Volume `xx` density
`=(pix^(2)dy)=pi(R^(2)-y^(2))dyrho`
The Y-coordinate of centre of mass are given by
`Y_(cm)=(1)/(m)intdmy`
Here, `m=(2)/(3)piR^(3)rho`
`Y_(cm)=(1)/(m)intydm=(1)/(m)int_(0)^(R)ypi(R^(2)-y^(2))rhody`
`=(pirho)/(m)int_(0)^(R)(R^(2)y-y^(3))dy`
`=(pirho)/(m)[(R^(2)y^(2))/(2)-(y^(4))/(4)]_(0)^(R)`
`=(pirho)/(m)[(R^(4))/(2)-(R^(4))/(4)]=(pirho)/(m)((R^(4))/(4))`
`=(pirho)/((2)/(3)piR^(3)rho)((R^(4))/(4))=(3)/(8)R`
Also, `Z_(cm)=intzdm=0andX_(cm)=intxdm=0`
`:.` The coordinates of centre of mass are `(0,(3)/(8)R,0)`