Say I have two gears, with a 2:1 gear ratio. The larger gear is spinning at almost the speed of light,so what would happen to the smaller gear since it cannot move faster than the speed of light? This is just my own curiosity.
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1Or this, this, this, this. – Chemomechanics Sep 26 '22 at 02:16
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Related: https://en.wikipedia.org/wiki/Ehrenfest_paradox – PM 2Ring Sep 26 '22 at 03:28
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Gears in contact rotate with the same tangent velocity, not a variable one dependent on gear ratio.
g s
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what would happen to the smaller hear since it cannot move faster than the speed of light?
It would break. Or the larger gear would break, which ever was weaker.
This is not a result that can be avoided by building the gears out of unobtanium. It is a fundamental limit.
Dale
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That is correct speaking realistically apart of any ideal thought experiment. Another physical conclusion could be that at c speed there cannot be any mass object but only light (or other massless particles). – Markoul11 Sep 27 '22 at 06:23
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Angular velocity $ω$ (or also known as angular frequency) of the smaller gear in contact, will be greater. However, both gears would share the same tangential velocity $v$ (see answer by g s):
$$ \omega=\frac{d \phi}{d t}=\frac{v_{\perp}}{r} $$
Where $v\perp$, the perpendicular velocity vector component to the radius $r$ of any point on the gear thus the tangential velocity.
Assuming as a thought experiment that the gears could handle the mechanical stress and the larger gear of the two is spinning with tangential velocity almost at speed $c$ (i.e. c is the speed of light limit) and a 2:1 gear ratio, then there is no violation of special relativity speed limit $c$ by the smaller gear, since SR in this case refers to the tangential $v$ speed value which is the same for both gears.
Markoul11
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