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Homework Help: Special Relativity - Angle Transformations

  1. Sep 17, 2014 #1
    1. The problem statement, all variables and given/known data

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

    (a) Show that the length of the rod as measured by a stationary observer is given by


    (b) Show that the angle that the rod makes with the x-axis is given by the expression

    (Take the lower end of the rod to be at the origin of the primed coordinate system.)

    2. Relevant equations



    [itex]{L_0}^2=(x')^2+(y')^2[/itex] and [itex]L^2=x^2+y^2[/itex]

    3. The attempt at a solution

    Let x and y be the rod's length and height (picture the rod forming the hypotenuse of a right triangle):


    There is no movement in the y (or y') direction, so [itex]y'=y=L_0sinθ_0[/itex]

    Meanwhile, the x component will contract in the non-prime reference frame, so [itex]x=\frac{x'}{γ}=\frac{L_0cosθ_0}{γ}[/itex]

    Thus [itex]L^2=x^2+y^2=\frac{L_0^2cos^2θ_0}{γ^2}+L_0^2sin^2θ_0[/itex]

    The algebra gets messy at this point, and I'm not sure what methods I should be using to yield the required form. I looked at my trig identities but none really seemed to fit the situation. And hopefully I haven't made a silly error in the physics side of things!
  2. jcsd
  3. Sep 17, 2014 #2


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    Write out the γ2 factor in terms of v/c and simplify. It's not too bad.
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