SR, Doppler effect on rotating disk

[bigerst]

Homework Statement

taken directly from Rindler
A large disc rotates at uniform angular velocity ω in inertial frame S. Two observers O1 and O2 ride on the disc at radial distances r1 and r2. They carry clocks C1 and C2 they adjust to keep with clocks time with S, i.e., they have been adjusted so the readings on the agrees with the clock in S. Prove that when O1 sends a light signal to O2 the light is Doppler shifted to $v2/v1 =\gamma_2/\gamma_1$.

Homework Equations

Well, since it says the clocks have been adjusted, I'm assuming only Newtonian transformations are applicable here, so t=t'.

The Attempt at a Solution

I tried doing this problem in terms of increasing wavelength, but from geometry, I found that each photon travels the exact same distance. Since it says the clocks are calibrated, time dilation isn't relevant either, so I'm guessing I did something wrong.

 

[TSny]

The question might be a little confusing in the way it is worded. The "adjusted" clocks C1 and C2 are just auxiliary devices for deriving the Doppler shift as observed by unadjusted clocks carried by O1 and O2.

The idea is to first deduce the ratio of $\nu_2/\nu_1$ as measured by the adjusted clocks and then use that to get the frequency ratio that would actually be observed by normal clocks carried by O1 and O2.

From your comments in "the attempt at a solution" I think you might already see what the frequency ratio is for clocks C1 and C2 (but I'm not sure).

 

[bigerst]

Ok that makes much more sense, so it's an effect purely based on time dilation ergo the gamma factors. Thanks.