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