Problems

The gravitational force between two point particles of masses \(m\) and \(m'\) is mutually attractive and has magnitude $$ {G m m' \over r^2}\,, $$ where \(G\) is a constant and \(r\) is the distance between them. A particle of unit mass lies on the axis of a thin uniform circular ring of radius \(r\) and mass \(m\), at a distance \(x\) from its centre. Explain why the net force on the particle is directed towards the centre of the ring and show that its magnitude is $$ {G m x \over (x^2 + r^2)^{3/2}} \,. $$ The particle now lies inside a thin hollow spherical shell of uniform density, mass \(M\) and radius \(a\), at a distance \(b\) from its centre. Show that the particle experiences no gravitational force due to the shell. %Explain without calculation the effect on this result if %the shell has finite thickness \(x\).