Consider a very long stick of length 2R which extends from just above the surface of the earth to a radius 3R if initial conditions have been set up so that this stick moves in a circular orbit while always pointing radially what is the period of this orbit a represents acceleration due to gravity on surface of earth

Options

A. `6pi sqrt(R/(2g)`
B. `2pisqrt(6)sqrt(R/g)`
C. `pisqrt(6)sqrt(R/g)`
D. none of these

Answer

B

Explanation

force on this element
`df=(GM(lambdadx))/x^(2)`
total force on the stick
`underset(0)overset(f)int = Gmlambda underset(R)overset(3R)int (dx)/x^(2)`
`f=Gmlambda [(-1)/x]_(R)^(3R)= Gmlambda[1/R-1/(3R)]=GMlambda[2/(3R)]`
CM of stick is rotating in a circle of radius 2R
`GM[(2lambda)/(3R)] = lambda(2R)(2R)omega^(2)`
`omega^(2)=(GM)/(6R^(3))`
`T = 2pisqrt(6) sqrt((R^(3))/(gR^(2))) = 2pisqrt(6) sqrt(R)/g`