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Homework Help: Length contraction

  1. Jan 29, 2012 #1
    1. The problem statement, all variables and given/known data

    A rod of length L_0 moves with a speed v along the horizontal direction. The rod makes an angle of θ_0 with respect to the x'-axis.

    a. Show that the length of the rod as measured by a stationary observer is given by L = L_o [1 - (v^2 / c^2) cos^2 (θ_o) ]^(.5)

    2. Relevant equations

    L = L_p/ gamma

    3. The attempt at a solution

    I have a few questions:

    Is L proper L_o (the S frame?)
    When I am trying to find the length of the rod as measured by a stationary observer do I refer to the graph to the right?

    Horizontal length is all I need to worry about, correct?

    L = L_o / gamma

    x' = L cos θ_o

    x' = x ( 1 - v^2/c^2 )^(-1/2)

    L cos θ_o = x ( 1 - v^2/c^2 )^(-1/2)

    L = x / [ (1 - v^2/c^2 )^(1/2) cos θ_o ] where x = L_o

    L = L_o / [ (1 - v^2/c^2 )^(1/2) cos θ_o ]

    Am I on the right track? I can't seem to make it look like:
    L = L_o [1 - (v^2 / c^2) cos^2 (θ_o) ]^(.5)

    Attached Files:

  2. jcsd
  3. Jan 29, 2012 #2
    Wait, huh, is the picture you attached one given by the problem or something you made? The picture is really different from how I read the problem.

    The way you're approaching the problem seems fine. But if you're going to find the x coordinates, you need to find two x coordinates because length is x2-x1. Looks like you're doing a strange mixing of length contraction and x coordinates. Either approach is okay, but you'll probably confuse yourself going between them (confused me at least).
  4. Jan 29, 2012 #3
    Yes I think that is my problem. I don't know how to draw the diagram. Can you explain me what you understood? I have read it several times and I just don't get it
  5. Jan 29, 2012 #4
    So you have an S' that moving relative to the S frame; their x axis is collinear. In S' we have a proper length L_o, and the length makes an angle with respect to the x' axis. So the part that will be contracted is L_o*cosØ'. L_o*sinØ' will stay the same in both frames. S even sees a different angle than S' does.
  6. Jan 29, 2012 #5
    Thank you, I figured out the problem. I was making it way too complicated.
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