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प्रश्न
Derive an expression for critical velocity of a satellite revolving around the earth in a circular orbit.
Obtain an expression for critical velocity of a satellite orbiting around the earth
उत्तर १
Consider a satellite of mass m revolving round the Earth at a height ‘h’ above the surface of the Earth.
Let M be the mass and R be the radius of the Earth.
The satellite is moving with velocity Vc and the radius of the circular orbit is r = R + h.
Centripetal force = Gravitational force
`:.(mV_c^2)/r=(GMm)/r^2`
`:.V_c^2=GM/r`
`:.V_c=sqrt((GM)/(R+h)) ...............(Equ. 1)`
This is the expression for critical velocity of a satellite moving in a circular orbit around the Earth.
We know that,
`g_h=(GM)/((R+h)^2`
`:.GM=g_h(R+h)^2`
Substituting in equation 1, we get
`:.V_c=sqrt(g_h(R+h)^2/(R+h)`
`:.V_c=sqrt(g_h(R+h)`
where gh is the acceleration due to gravity at a height h above the surface of the Earth.
उत्तर २
Let,
M = mass of the earth
R = radius of the earth
h = height of the satellite from the earth’s surface
m = mass of the satellite
vc
= critical velocity of the satellite in the given orbit
r = (R + h) = radius of the circular orbit
For the circular motion of the satellite, the necessary centripetal force is given as
`F_CP = (mv_c^2)/r` 1
Gravitational force provides the centripetal force necessary for the circular motion of the satellite
∴ FCP = FG
Equation (4) represents the expression for critical velocity.
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