Question

Calculate the gravitational field intensity at the centre of the base of a hollow hemisphere of mass M and radius R. (Assume the base of hemisphere to be open).

Answer 1

We consider the shaded elemental ring of mass
$dm=\frac{M}{\left(2\pi R^2\right)}2\pi R\sin \theta \left(Rd\theta \right)$
Field due to this ring at O,
$dE=\frac{GdmR\cos \theta }{R^3}$
(see formulae for field due to a ring)
or $dE=\frac{GM}{R^2}\sin \theta \cos \theta d\theta$
Hence, $E={\int }_0^{\pi /2}dE={\int }_{R^2}^{\pi /2}\frac{GM}{R^2}\sin \theta \cos \theta d\theta$
or $E=\frac{GM}{2R^2}$.