How Newton’s law of gravitation helps in understanding the motion of satellites? On what factors the orbital speed of a satellite depends?
OR
Derive the formula for the orbital speed of an artificial satellite?
Difficulty: Hard
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 vo. 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}
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