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Relativity with stationary rod

  1. May 2, 2009 #1
    An observer in S' sees a stationary rod lying in the x'-y' plane and making an angle [itex]\phi'[/itex] with the x' axis. Show that
    [itex]\tan{\phi'}=\frac{1}{\gamma} \tan{\phi}[/itex]

    where [itex]\phi[/itex] is the angle the rod makes with the x axis according to an observer in S.

    so far I have [itex]\tan{\phi'} = \frac{y'}{x'}=\frac{y}{\gamma(x-vt)}=\frac{1}{\gamma} \frac{y}{x-vt}[/itex]

    if i could just get rid of that vt, i'd be there but i can't seem to get it to go away...
     
  2. jcsd
  3. May 2, 2009 #2

    dx

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    Re: Relativity

    I assume the rod is moving in the x direction. In the S frame, let the length in the x direction be Lx and it's length in the y direction be Ly. In the S' frame, the length in the y direction will be the same i.e. L'y = Ly, and the length in the x direction will be shorter by a factor of √(1 - v²), i.e. L'x = √(1 - v²) Lx. So we get

    tan θ = Ly / Lx

    tan θ' = L'y / L'x = Ly/Lx√(1 - v²)
     
  4. May 2, 2009 #3

    dx

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    Re: Relativity

    Your mistake was that you took tan θ to be y/x. If the coordinates of the tips of the rod are (x1, y1) and (x2, y2), then tan θ will be ∆y/∆x where ∆x = x2 - x1 and ∆y = y2 - y1. You should apply the Lorentz transform to both (x1, y1) and (x2, y2) to obtain ∆x' and ∆y'. tan θ' will then be ∆y'/∆x'.
     
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