Question

define critical velocity of satellite and obtain an expression for it​

Answer 1

Critical Velocity of a Satellite

Critical Velocity of a satellite is simply the velocity with which a satellite revolves around the earth along its orbit. It is also known by the name 'orbital velocity'.

Expression for Critical Velocity of a Satellite

Consider a satellite of mass m placed at a height h above the surface of the earth of mass M and radius R, where the orbital radius of the satellite is r. Then we can say that,

$r=R+h\quad\longrightarrow\quad(1)$

Since the satellite is revolving around the earth with a critical velocity v, its gravitational force of attraction due to the earth provides the necessary centripetal force acting on the satellite. Thus,

$\dfrac {mv^2}{r}=\dfrac {GMm}{r^2}\\\\\\\boxed {v=\sqrt {\dfrac {GM}{r}}}$

From (1),

$\boxed {v=\sqrt {\dfrac {GM}{R+h}}}$

Here we get that the velocity depends on the height of the satellite from the surface of the earth, since G, M and R are constants.

Suppose the satellite is revolving around the earth nearly to its surface. Then the height of the satellite from the surface can be neglected. Then,

$v=\sqrt {\dfrac {GM}{R}}\quad\longrightarrow\quad (2)$

But we know that,

$GM=gR^2\quad\iff\quad g=\dfrac{GM}{R^2}$

Thus (2) becomes,

$v=\sqrt{\dfrac {gR^2}{R}}\\\\\\\boxed {v=\sqrt{gR}}$

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Answer 2

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