Let

#m->"mass of satellite" #

#M->"mass of Earth" #

#R->"radius of Earth" #

#G->"Gravitational constant" #

#T->"time period of satellite" #

#omega->"angular speed of satellite"=(2pi)/T #

#g->"acceleration due to gravity on "#

#" earth surface"#

Equating weight #(mg)# of satellite revolving round the earth along the orbit of radius nrearly equal to the radius #(R)# of the earth with the gravitational pull on it, we get

#mg=(GmM)/R^2#

#=>GM=gR^2........(1)#

The centripetal force (#F_c#) acting on the satellite revolving round the Earth along the orbit having radius nearly equal to radius of Earth is given by

#F_c=momega^2R#

The gravitaional force #(F_g)# acting on the satellite will provide the required centripetal force.

#F_g=G(mM)/R^2#

Now #F_c=F_g#

#=>momega^2R=(GmM)/R^2#

#=>((2pi)/T)^2R=(GM)/R^2#

#=>T^2=(4pi^2R^3)/(GM)#

#=>T=2pisqrt(R^3/(GM))#

[using relation (1)]

#=>T=2pisqrt(R^3/(gR^2))#

#=>T=2pisqrt(R/g)#