A particle of mass `m` is placed on centre of curvature of a fixed, uniform semi-circular ring of radius `R` and mass `M` as shown in figure. Calculate:
(a) interaction force between the ring and the particle and
(b) work required to displace the particle from centre of curvature to infinity.

Options

A. `(a) F=(2GM)/(piR^(2)), (b) (GM)/R`
B. `(a) F=(2GMm)/(pi^(2)R) , (b) (GMm)/(R^(2))`
C. `(a) F=-(2GMm)/(piR^(2)) (b) -(GMm)/R`
D. `(a) F=(2GMm)/(piR^(2)) (b) (GMm)/R`

Answer

D

Explanation

`dM=(M/(piR))(R d theta)=M/(pi).d theta`
Required force `=int_(0)^(pi//2)2 F sin theta` (towards `C`)
`=int_(0)^(pi//2)(2Gm(dM))/(R^(2))sin theta`
`=int_(0)^(pi)(2GMm)/(piR^(2))sintheta.d theta=(2GMm)/(piR^(2))`
`(b) U=-(GMm)/R`
`:.` Binding energy `=|U|=(GMm)/R`
i.e., this much energy is required to displace the particle from centre of curvature to infinity.