Rotating cylinder on x'-axis in S' frame. Find twist per unit length in S frame
1. The problem statement, all variables and given/known data
A cylinder rotating uniformly about the x' axis of S' will seem twisted when observed instantaneously in S, where it not only rotates but also travels forward. If the angular speed of the cylinder in S' is ω, prove that in S the twist per unit length is yωv/c(squared). Here S and S' are inertial frames of reference in the standard configuration with respect to one another. y= gamma factor
2. Relevant equations
twist per unit length = yωv/c(squared)
Inverse Lorentz equations
3. The attempt at a solution
By twist per unit length ii think it means dθ/dx where the x-axis lines up with the axis of the cylinder?.
We can write the angular speed as
and then transposing we get
because theta is in the z-y plane we can say that dθ'=dθ ???
So subbing in dt'=y(dt-vdx/c2) we get
divide thru by dx we get
dθ/dx= ωy( dt/dx - v/c2)
dθ/dx = ωy/v - ωyv/c2
The answer should be dθ/dx = ωyv/c2 . BTW I dont think it matters about the negative sign but why am i left with ωy/v ?
Would really appreciate some hints :)
Re: Rotating cylinder on x'-axis in S' frame. Find twist per unit length in S frame
It's the comparison, at fixed time t in S of elements of the cylinder separated by dx AND that correspond to elements in S' with same θ' for a fixed t'. The thing is (x,t) and (x+dx,t) correspond to two different times t1' & t2' in S'. Elements of the cylinder with θ' at t1' are at θ'+ω(t2'-t1') at t2'.
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