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Find the proper length of a rod if in the laboratory frame of reference its velocity is v=c//2, the length l=1.00m, and the angle between the rod and its direction of motion is theta=45^@. |
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Answer» Solution :In the rest frame, the COORDINATES of the ENDS of the rod in terms of proper length `l_0` `A:(0,0,0) B: (l_0costheta_0, l_0sintheta_0, 0)` at time t. In the LABORATORY frame the coordinates at time `t'` are `A: (vt^', 0, 0), B: (l_0costheta_0srqt(1-beta^2)+vt', l_0sintheta_0,0)` THEREFORE we can write, `locstheta_0=l_0costheta_0sqrt(1-beta^2)` and `lsintheta=l_0sintheta_0` HENCE `l_0^2=(l^2)((cos^2theta+(1-beta^2)sin^2theta)/(1-beta^2))` or, `=sqrt((1-beta^2sin^2theta)/(1-beta^2))` 
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