Satellite X moves around Earth in a circular orbit radius R. Satellite Y is also in a circular orbit around earth, and it completes one orbit for every eight orbits completed by satellite X. What is the orbital radius of satellite Y?
G = gravitational constant M = mass of the earth R = radius of orbit of a satellite r = radius of orbit of a second satellite v = speed of the satellite P = period of a satellite p = period of a second satellite
Equate gravitational acceleration with centripetal acceleration g = G*M/R^2 = v^2/R
Express the orbital speed in terms of the orbit circumference and period v = 2*pi*R/P
And insert the expression for v into the first equation G*M/R = 4*PI^2*R^2/P^2 G*M/R^3 = 4*pi^2/P^2 R^3/P^2 = 4*pi^2/(G*M) = constant = C
We can do the above since G and M are constants for all earth orbits
So we can write a second equation of the same form for another satellite and equate to get: R^3/P^2 = r^3/p^2 r^3 = R^3*p^2/P^2 r = R*(p^2/P^2)^(1/3)
For the second satellite we have p = 8*P r = R*(8^2)^(1/3) = R*(64)^(1/3) = 4*R
