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In his measurement of the speed of light, why did Fizeau use a sawtooth gear with $n$ teeth and $n$ gaps? Why not use only one gap, so when light passes through the gap we will know that it is that unique gap and we wouldn't have to calculate $\theta=2n/2\pi$?

The experiment looks, schematically, as follows:

Image source

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  • $\begingroup$ How would this help? You would still have to determine how many revolutions the gear had undertaken in the round trip. $\endgroup$ Mar 2, 2017 at 11:46
  • $\begingroup$ Also, just to fix a common ground, can you confirm that you're referring to this schematic apparatus? $\endgroup$ Mar 2, 2017 at 11:48
  • $\begingroup$ @EmilioPisanty yes i'm refering to that shematic my question is why didnt he had only one gap so he can be sure that light passes at that same gap and not other $\endgroup$
    – Bardeen
    Mar 2, 2017 at 11:52
  • $\begingroup$ Is it possible to argue that this should be in the engineering SE? I'm not sure: I think the reason is engineering, but it's a physics experiment. $\endgroup$
    – user107153
    Mar 2, 2017 at 12:40

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The more teeth you have the slower the rotation speed of the wheel can be. He had 720, with a single tooth you would need to rotate the thing 720 times faster. The description of the experiment says that the wheel could rotate at hundreds of Hz, so a single-tooth wheel might need to spin at, perhaps, 100,000Hz or more. This would be challenging to put it extremely mildly.

In fact, based on a 16km path, the lowest rotational speed is $3\times 10^8/(16\times 10^3\times 720) \approx 26\:\mathrm{rps}$ for a 720-tooth wheel. This is perfectly mechanically viable, although it would have been fairly demanding in 1851. 720 (or in fact 360, see below) times this would be extremely challenging now. Thanks to Emilio Pisantry for the numbers.

(A single-tooth wheel is also asymmetric -- it's a cam, really -- which would make things catastrophically worse: a two-toothed wheel is less mechanically insane.)

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    $\begingroup$ From the available numbers (12-mile round trip, 720 teeth), I make the first gap at 20 rps, which is plenty doable mechanically. However, using a two-notch wheel brings that up to 7800 rps, which is an entirely different proposition. $\endgroup$ Mar 2, 2017 at 12:27
  • $\begingroup$ What's your heuristic for physically realistic rps ? I would be interested to run Fizeau's wheel at 100 rps ($4\times$ faster than his initial $26 rps$). You estimate the order of the angular kinetic energy, $L=mv^2/2=m (r\omega)^2/2$ ? Otherwise I find your answer very useful here. $\endgroup$
    – JHM
    Apr 30, 2022 at 14:48
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Basically it is because the speed of light is so very very high which requires the wheel to spin very fast and/or have many narrow notches.

Fizeau used the wheel to "time" light as it travelled a distance of 16km. Light passing through one notch on its outward journey was blocked by the adjacent tooth when it returned, after the tooth had moved a very small distance. Even though the wheel rotated a few hundred times per second, it was still necessary to have 720 teeth in order to get the experiment to work.

If the wheel had only one narrow notch, it would still have to rotate at the same speed as if it had 720 teeth, but only $\frac{1}{720}$ of the same amount of light would get through when the notch is aligned for the outward and returning light. This makes it much more difficult to distinguish at what point this light is completely blocked out, because there is so little contrast between "light" and "dark".

Fizeau was able to know that the light was blocked by the adjacent tooth because he increased the speed gradually, observing the light get dimmer. When the transmitted light reached a minimum the first time he could be sure it was blocked by the adjacent tooth. When the 2nd minimum came because of the next tooth, this was when the wheel spun 3x as fast. The big difference in speed made it possible to be sure which tooth was blocking the light.

Reference : wikipedia article

see also How did Fizeau make his famous speed-of-light experiment? and How did Fizeau control the cog's rotations per second?

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It is necessary for the sawtooth pattern to be regular and having some amount of rotational symmetry because the wheel needs to be spinning with uniform angular velocity.

If you have a saw-wheel with only one tooth, then it's very difficult to get the wheel to spin uniformly! It will require a variable input to counter-act the rotational torque which arises from spinning the irregular wheel.

For example, having worked with saw mills and seeing spinning saw blades in action, the blades need to be discarded after they hit a nail. Why? Because if a single saw-tooth is damaged, then the saw-blade begins to oscillate and vibrate, and the saw no longer cuts straight.

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