Question

Suppose we want a satellite to revolve around the earth 5 times a day. What should be the radius of its orbit? (The mass of the earth is 5.97 × 10^24 kg, G = 6.67 × 10^-11 N · m^2/kg^2, and you can neglect the presence of the moon.)

Answer 1

To solve this exercise we must apply the concepts related to the balance of forces.

For this particular case the force caused on the satellite from the planet must be equivalent to the Centrifugal Force, that is

$F_g = F_c$

$\frac{GMm}{r^2} = m\omega^2 r$

Where,

G = Gravitational Universal Constant

M = Mass of earth

m = mass of satellite

r = Distance/Orbit

Using the concept of Period through the angular velocity we have,

$\frac{GM}{\omega^2} = r^3$

$\frac{GMT^2}{4\pi^2} = r^3$

Finally re-arrange the equation to find the orbit and replacing, we have,

$r^3 = \frac{6.67*10^{-11}*5.97*10^{24}}{4\pi^2}(\frac{24*3000}{5})^2$

$r^3 = 3.011*10^{21}$

$r = 1.44*10^7m$

Therefor the orbit would be 14400 kilometers