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Special Relativity - time paradox question, not sure if it's right!

  1. Jul 31, 2013 #1
    1. The problem statement, all variables and given/known data

    2 identical clocks are synchronised in a lab on the equator. One clock is carried around the equator in 24 hours at a constant speed. Given that the radius of the Earth is 6.4 x 10 ^ 6 m, find the difference between the times registered by 2 clocks when the travelling clock returns to the lab.

    2. Relevant equations

    t' = t / γ

    γ = 1 / (√(1 - v^2 / c^2))

    3. The attempt at a solution

    First I found the circumference of the Earth using c = 2 pi r
    Found the circumference of the Earth to be 40212385 m.

    The clock takes 24 hours to go around - 86400 seconds

    so the speed the clock goes around the earth at is distance / time = circumference / time = 465 m/s.

    I used the gamma equation to find the Lorentz factor

    γ = 1 / (√(1 - v^2 / c^2)) = 1.
    (the v = 465 m/s was too small to impact the equation?)

    So then, subbing my Lorentz factor into the time dilation formula t' = t/y
    t' = t/1 = t, so there's no time difference?

    That doesn't sound right at all, I've done something stupid, if anyone can correct me that would be great :)
     
  2. jcsd
  3. Jul 31, 2013 #2
    Use more figures in your gamma factor, I got Y=1.000001205095829, then used t' = t/y = 86400/1.000001205095829 = 86399.90 seconds, so the difference should be about 0.1 seconds, seems correct because Y is so close to 1.
     
  4. Jul 31, 2013 #3
    As it's the difference between 1 and 1-v2/c2 that matters, even a small v does impact on the equation - γ is close to 1, but different from 1.

    Thus, either solve as baddin suggested*, or solve as Einstein did in 1905 - he did not have a pocket calculator! Derive an approximate equation for the difference (easy math, simplify 1/√(1-x2) for x<<1). See the second half of §4 of http://www.fourmilab.ch/etexts/einstein/specrel/www/

    *PS: I get a much smaller result both ways
     
    Last edited: Jul 31, 2013
  5. Jul 31, 2013 #4
    Thanks both!

    My calculator doesn't show enough figures to give me a non-1 answer, so I'm going to do it the way harrylin suggested, cheers :)
     
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