A sky lab of mass `2 xx 10^(3)kg` is first launched from the surface of earth in a circular orbit of radius `2R` and them it is shifted from this circular orbit to another circular orbit of radius `3R`. Calculate the energy required
(a) to place the lab in the first orbit,
(b) to shift the lab from first orbit to the second orbit. `(R = 6400 km`, `g = 10 m//s^(2))`

Options

A. `3/4mgR,(mgR)/6`
B. `3/4mgR,(mgR)/12`
C. `mgR,mgR`
D. `2mgR,mgR`

Answer

B

Explanation

Energy of the skylab in the first orbit is
`-(GMm)/(2(2R))=-(GMm)/(4R)`
Total energy required to place the skylab into the orbt of radius `2R` frm the surface of earth is
`=(GMm)/(4R)-(-(GMm)/R)=(3G/Mm)/(4R)`
`=(3gR^(2)m)/(4R)=3/4mgR`
Energy of the skylab in the second orbit `=-(GMm)//6R`
Energy needed to shift the skylab from the first orbit to the second orbit is
`-(GMm)/(4R)-(GMm)/(6R)=(GMm)/Rxx2/24=(mgR)/12`