Location of centre of gravity of a hemisphere:

Find out the location of centre of gravity of a hemisphere of radius a.

Solution

Figure shows the cross-section of the hemisphere where DE = 2a, the diameter of the hemisphere.

Assume a thin slice AB of volume dV taken at a distance x from center O of the hemisphere.

Radius of slice = a sin θ

If dx is the small thickness of slice,

dV = π (a sin θ)²  dx

x = a cos θ

dx = - a sin θ dθ

Letting equation of moments about O,

                                    = πa⁴/4

                            ∴x¯ =3a/8