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?

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 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

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