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,
= πa^{4}/4
∴x¯ =3a/8