Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics V resit / Question 7 / b / ii / Solution

Use source polar coordinates $x^{\prime }=s\cos \theta$, $y^{\prime }=s\sin \theta$ on the semicircular disk, with $0\le s\le a$ and $0\le \theta \le \pi$. Its surface area element is $sdsd\theta$, and the separation from $(0,0,h)$ is $\sqrt{s^2+h^2}$. ThusThis Newtonian gravitational potential is half that of the full disk because the separation is independent of the polar angle. At large $h$ it approaches $-G(\pi a^2\sigma /2)/h$, with the correct total mass.