A system consists of a thin ring of radius R and a very long uniform wire oriented along axis of the ring with one of its ends coinciding with the centre of the ring. If mass of ring be M and mass of wire be `lambda` per unit length, calculate interaction force between the ring and the wire.

Explanation

The force of interaction due to ring on the element `dx` is
`dF=(GM(dm))/((r^(2)+x^(2)))cosalpha` where, `cosalpha=(x)/(sqrt(x^(2)+r^(2)))`
`impliesdF=(Gmxlambdadx)/((r^(2)+x^(2))^(3//2))`
`impliesF=Gmlambda int_(0)^(oo)(x dx)/((r^(2)+x^(2))^(3//2))=GMlambda int_(0)^(pi//2)(r^(2)tan theta sec^(2) theta d theta)/(r^(3) sec^(3) theta)`
`=(GMlambda)/(r )int_(0)^(pi//2)sin thetad theta=(GMlambda)/(r )|-costheta|_(0)^(pi//2)=(GMlambda)/(r )`