1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    [tex]L=L_0\sqrt{1-\frac{v^2}{c^2}cos^2θ_0}[/tex]


    (b) Show that the angle that the rod makes with the x-axis is given by the expression
    [tex]tanθ=γtanθ_0[/tex]

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

    2. Relevant equations

    [tex]γ=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}[/tex]

    [tex]L=\frac{L_0}{γ}[/tex]

    [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):

    [itex]x'=L_0cosθ_0[/itex]

    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

    TSny

    User Avatar
    Homework Helper
    Gold Member

    Write out the γ2 factor in terms of v/c and simplify. It's not too bad.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted