SR, Doppler effect on rotating disk

• bigerst
In summary, the problem involves a large disc rotating at uniform angular velocity, with two observers riding on the disc at different distances and carrying calibrated clocks. The goal is to prove that when one observer sends a light signal to the other, the light will be Doppler shifted according to the ratio of gamma factors between the observers. This is a result of time dilation and does not require any other transformations.
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.

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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).

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

1. What is the Doppler effect on a rotating disk?

The Doppler effect on a rotating disk is a phenomenon in which the frequency of a sound or light wave appears to change when the source of the wave is moving in a circular motion. This is due to the relative motion between the source and the observer, causing a compression or stretching of the waves.

2. How does the Doppler effect on a rotating disk affect sound waves?

The Doppler effect on a rotating disk affects sound waves by changing their perceived frequency. As the source of the sound moves towards the observer, the frequency appears to increase, while it appears to decrease when the source moves away.

3. What is the relationship between sound frequency and disk rotation speed in the Doppler effect?

The relationship between sound frequency and disk rotation speed in the Doppler effect is directly proportional. This means that as the rotation speed of the disk increases, the frequency of the sound waves also increases, and vice versa.

4. How does the Doppler effect on a rotating disk relate to Special Relativity?

The Doppler effect on a rotating disk is a result of Special Relativity, which states that the laws of physics are the same for all observers in uniform motion. The relative motion between the source and observer causes a change in the perceived frequency of the waves, which is a fundamental principle of Special Relativity.

5. What are some real-world applications of the Doppler effect on a rotating disk?

The Doppler effect on a rotating disk has various real-world applications, including in medical imaging to measure blood flow and in radar technology to detect the speed and direction of moving objects. It is also used in astronomy to determine the rotation speed and direction of celestial bodies.

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