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)
