Question

A system consists of a thin ring of radius $r$ and of mass $M$ and a straight wire of linear mass density $\lambda$ of infinite length placed along the axis of the ring with one of its ends at the centre of the ring. Find the force of interaction between the wire and the ring.

Answer 1

The force of interaction due to ring on the element $dx$ is
$dF=\frac{GM\left(dm\right)}{r^2+x^2)}\cos \alpha$ where, $\cos \alpha =\frac{x}{\sqrt{x^2+r^2}}$
$\Rightarrow dF=\frac{GMx\lambda dx}{(r^2+x^2{)}^{3/2}}$
$\Rightarrow F=GM\lambda {\int }_0^{\infty }\frac{xdx}{(r^2+x^2{)}^{3/2}}=GM\lambda {\int }_0^{\pi /2}\frac{r^2\tan \theta {\sec }^2\theta d\theta }{r^3{\sec }^3\theta }$
$=\frac{GM\lambda }{r}{\int }_0^{\pi /2}\sin \theta d\theta =\frac{GM\lambda }{r}|-\cos \theta {|}_0^{\pi /2}=\frac{GM\lambda }{r}$.