A comet of mass m moves in a highly elliptical orbit around the sun of mass M the maximum and minium distacne of the comet from the centre of the sun are `r_(1)` and `r_(2)` respectively the magnitude of angular momentum of the comet with respect to the centre of sun is

Options

A. `[(GM_(1))/(r_(1)+r_(2))]^(1//2)`
B. `[(GMmr_(1))/(r_(1)+r_(2))]^(1//2)`
C. `[(rGM^(2)r_(1)r_(2))/(r_(1)+r_(2))]^(1//2)`
D. `[(2GMm^(2)r_(1)r_(2))/(r_(1)+r_(2))]^(1//2)`

Answer

D

Explanation

Using law of conservatin of angular momentum at location A and B we get

`L=mv_(1)r_(1)=mv_(2)r_(2) rarr v_(2)=(v_(1)r_(1))/(r_(2))`
using the principla of conservation of total energy at A and B
`1/2 mv_(1)^(2)-(GMm)/(r_(1))=1/2mv_(2)^(2)-(GMm)/(r_(2))`
or `v_(2)^(2)-v_(1)^(2)=2 GM (1)/(r_(2))-(1)/(r_(1))`
Putting the values from Eq i Eq ii and solving we get
`rarr L=mv_(1)r_(1)=m[(2GMr_(1)r_(2))/(r(1)+r_(2))]^(1//2)`