The motion of Artificial Satellites:

            A satellite requires a centripetal force that keeps it to move around the Earth. The gravitational force of attraction between the satellite and the earth provides the necessary centripetal force.

            Consider a satellite of mass m revolving around the Earth at an altitude h in an orbit of radius  $r\circ$ with velocity v_o. The necessary centripetal force required is given by equation $F\circ=m \: \frac{v\circ^{2}}{r\circ}$ ……..   (i)

 

This force is provided by the gravitational force of attraction between the Earth and the satellite and is equal to the weight of the satellite $w'=mg_{h}$…..   (ii)

 

From (i) and (ii) we get

$mg_{h}m\frac{v\circ}{r\circ}$

or    $v\circ^{2}= g_{h}r\cir$

or    $v\circ=\surd g_{h}r_{\circ}$ .......(iii)

 

As  $r\circ=R+h$

$v\circ= \surd g_{h\left(R+h\right)}$ ........(iv)

 

Equation (iii) gives us the velocity, which a satellite must possess when launched in an orbit of radius $r\circ=\left(R+h\right)$ around the Earth.

An approximation can be made for a satellite revolving close to the Earth such that R >> h.

 

$R+h\approx R$

And $g_{h}\approx g$

v_{\circ}=\surd gR ……..(v)

A satellite revolving around very close to the Earth has speed $v_{\circ}$ nearly 8kms^{-1}or \: 29000 \: kmh^{-1}