Rotating cylinder on x'-axis in S' frame. Find twist per unit length in S frame

  1. 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)
    Lorentz equations
    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
    ω= dθ'/dt',
    and then transposing we get
    dθ'=ωdt'
    because theta is in the z-y plane we can say that dθ'=dθ ???

    So subbing in dt'=y(dt-vdx/c2) we get

    dθ= ωy(dt-vdx/c2)

    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 :)
     
  2. jcsd
  3. 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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