Find out the centre of mass of a composite object shown in figure. Object consists of a cone with its base joint with the base of a hemisphere. The dimensions of the object are shown in figure. Assume uniform density of the system.

Explanation

The shown object is made up of joining a solid cone and a hemisphere. We already know the location of the centre of mass of a cone and that of a hemisphere. The masses of the two are in proportion of their volume. The masses of cone and hemisphere are
Mass of cone is `"m"_(1)=rho(1)/(3)piR^(2)h`
and that of hemisphere is `"m"_(2)=rho(2)/(3)piR^(3)`
Now we apply the result of two body system to find the centre of mass of the composite body. Let `l` be the distance between the independent centre of mass of the bodies cone and hemisphere, then
`l=(3R)/(8)=(h)/(4)`
The position of centre of mass `m_(2)` is
`x=(m_(1)l)/(m_(1)+m_(2))=(rho(1)/(3)piR^(2)h((3R)/(8)=(h)/(4)))/(rho(1)/(3)piR^(2)h+rho(2)/(3)piR^(3))`
`x=(h(3R+2h))/(8(h+2R))`