If mass density of earth varies with distance 'r' from centre of earth as `rho = kr and 'R' `isradius of earth, then find the orbital velocity of an object revolving around earth at adistance '2R' from its centre.

Options

A. `sqrt((pikR ^(3)G)/(4))`
B. `sqrt((pi k R ^(3)G)/(2))`
C. `sqrt((pi k R ^(3)G)/(8))`
D. `sqrt(pi k R ^(3)G)`

Answer

B

Explanation

Let 'M' be total mass of earth. Consider a shell of thickness 'dr' and mass 'dm' at a distance 'r' from centre inside earth,
`implies dm = rho 4 pi x ^(2) dr `
`M int dm`
`= underset(0) overset(R ) int 4 pi kr^(3) dr`
`=(4 pi k R ^(4))/(4) = pi kR ^(4)`
Let field due to earth's gravity at a distance '2R' from centre be `T.IxxA=4 pi G m _("inside).`
`impliesIxx 4pi (2R) ^(2) =4 pi G (pi kR^(4))`

`impliesI =(pikR ^(4)G)/(4R^(2))`
`implies I =(pi k R ^(4) G )/(4R^(2)) `
For a satellite of mass 'm' moving in orbit of '2R' radius.
`mI =(mv^(2))/((2R))`
`implies I =(V^(2))/(2R)`
`implies (pi kR ^(2)G)/(4) =(V^(2))/(2R)`
`V = sqrt((pi kR ^(5)G)/(2))`