Deriving the expression orformula #1

TheGravitational Forcebetween the earth and the satellite =Fg= (G.M.m)/r2 ……………… (1)

Thecentripetal forceacting on the satellite =Fc= mV2/r ……………………….. (2)

Here, M is the mass of earth and m is the mass of the satellite which is having a uniform circular motion in a circular track of radius r around the earth. V is the linear velocity of the satellite at a point on its circular track.

Now this r is the sum of the radius of the earth(R) and the height(h) of the satellite from the surface of the earth.

r = R + h

Now equating, equation 1 and 2 we get,

Fg= Fc

=> (G.M.m)/r2= mV2/r

V = [(GM)/r]1/2……………………………….. (3)

This is the first equation or expression of Orbital Velocity of a satellite. Herer = R +h

Deriving the expression orformula #2

For a mass of m on earth’s surface, the following is true:

mg = (GMm)/R2………………………. (4)

Note, on earth surface h=0 and r = R. Andgravitational forceon amassis equal to its weight on the surface.

From equation 4 we get this equation,GM = g. R2…………………….. (5)

Substituting this expression of GM in equation 3 (Orbital velocity), we get,

V = [(gR2)/r]1/2

V = R . (g/r)1/2……………………. (6)

This is the second expression of Orbital Velocityof a satellite.Here, as said earlier, r = R +h