Question

Obtain an expression for the period of a satellite in a circular orbit round the earth.

Answer 1

Let the mass of satellite is m , satellite moves in circumference of radius r around the earth of mass $M$ and radius of earth $R$

at equilibrium,
centripetal force = gravitational force
$\frac{mv^2}{r}=\frac{GMm}{r^2}$

$v=\sqrt{\frac{GM}{r}}$....(1)

here v is orbital velocity.
now, the satellite travels around the entire circumference of the circle — which is 2πr.
if r is the radius of the orbit — in the period, T. This means the orbital speed must be $\frac{2\pi r}{T}$.....(2)


from equations (1) and (2),
$\sqrt{\frac{GM}{r}}=\frac{2\pi r}{T}$

$T=2\pi\sqrt{\frac{r^3}{GM}}$

hence, time period is given by $T=2\pi\sqrt{\frac{r^3}{GM}}$

Answer 2

Answer:

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