Question

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?

Answer 1

 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