Question

ques

Answer 1

Time period of a satellite at a height h from the earth, moving with horizontal velocity v
\[\frac{mv^2}{r}=\frac{GMm}{r^2} \implies v^2=\frac{GM}{r}\]\[v=r\omega \implies v^2=r^2\omega^2=\frac{4\pi^2r^2}{T^2}\]
\[\frac{4\pi^2r^2}{T^2}=\frac{GM}{r} \implies T^2=\frac{4\pi^2r^3}{GM}\]
where
\[r=R+h\]
For value of g at surface we have the equation
\[g=\frac{GM}{R^2}\]
\[\implies GM=gR^2\]\[\implies T^2=\frac{4\pi^2r^3}{gR^2}=\frac{4\pi^2}{g}.\frac{R^3(1+\frac{h}{R})^3}{R^2}=\frac{4\pi^2}{g}.R(1+\frac{h}{R})^3\]
\[(1+x)^n \approx (1+nx) \forall 0\le x <1\]
so
\[T^2\approx \frac{4\pi^2}{g}.R(1+3\frac{h}{R})=\frac{4\pi^2}{g}.(R+3h)\]\[T \approx 2\pi.\sqrt{\frac{R+3h}{g}}\]
and
exact value
\[T=\frac{2\pi}{R}.\sqrt{\frac{(R+h)^3}{g}}\]

Answer 2

try it out for the ISS

h = 400km

Answer 3

I'm getting 92.63 minutes from my exact value and 89.76 minutes from approximate formula

Answer 4

cool stuff!

Wiki makes it : Orbital period 92.69 minutes