Question

A semicircular wire has a length $L$ and mass $M$. Find the gravitational field at the centre of the circle.

Answer 1

Consider a segment of angle $d\theta$ from any one end at an angle $\theta$. The field component in vertical direction due to this segment is

$dg=\frac{G\frac{M}{\pi }d\theta }{(\frac{L}{\pi }{)}^2}sin\theta =\frac{4\pi GM}{L^2}sin\theta d\theta$

$g=\int dg={\int }_0^{\pi }\frac{\pi GM}{L^2}sin\theta d\theta =\frac{2\pi GM}{L^2}$

Answer is $\frac{2\pi GM}{L^2}$