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)#
