Two satellite `S_(1)` and `S_(2)` revolve around a planet in coplanar circular orbits in the opposite sense. The periods of revolutions are `T` and `eta T` respectively. Find the angular speed of `S_(2)` as observed by an astronouts in `S_(1)`, are observed by an astronaut in `S_(1)`, when they are closest to each other.

Options

A. `omega=(2pi(n^(-1/3)+1))/(T(n^(1/3)-1))`
B. `omega=(2pi(n^(-1/3)+1))/(T^(2)(n^(2/3)-1))`
C. `omega=(2pi(n^(-1/3)+1))/(T(n^(2/3)-1))`
D. `omega=(2pi(n^(-2/3)+1))/(T(n^(-1/3)-1))`

Answer

C

Explanation

`Toor^(3//2)` or `rooT^(2//3)`
`:. (r_(1))/(r_(2))=((T_(1))/(T_(2)))^(2//3)=G/(etaT)^(2//3)=1/(eta^(2//3))`
`omega_(21)=(v_(2)+v_(1))/(r_(2)-r_(1))=((2pir_(2))/(T_(2))+(2pir_(1))/(T_(1)))/(r_(2)-r_(1))=(2pi(1/(T_(2))+1/(T_(1)).(r_(1))/(r_(2))))/(1-(r_(1))/(r_(2)))`
`=(2pi(1/(etaT)+1/T.1/(eta^(2//3))))/(1-eta^(1/(2//3)))=((2pi)/T(eta^(-1//3)+1))/((eta^(2//3)-1))`