Question

Calculate the height of a satellite from the surface of the earth if its period is 12 h. Compare this height with the radius of the earth.

Answer 1

$$m\frac{v^2}{r}=m\frac{GM}{r^2}$$

$$v=\frac{2\pi r}{T}$$

So,

$$r^3=(R+h)^3=\frac{GM}{4\pi^2}T^2$$

$$(6.38\cdot 10^{6}+h)^3=\frac{(6.67\cdot 10^{-11})(5.97\cdot 10^{24})}{4\pi^2}(12\cdot 3600)^2$$

$$h=20.22\cdot 10^{6}\ m$$

$$\frac{h}{R}=\frac{20.22\cdot 10^{6}}{6.38\cdot 10^{6}}=3.17$$