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Length Contraction

We consider a rod parallel to x-axis in frame S at rest.

The length of the rod in frame 5 (Stationary frame) is

l= x2- x1

 

where x1 and x2 are the abcissae of the ends of the rod and in moving system $’ it is measured as

l’ = x2- x1

where x1 and x2 are the abcissae of the ends of the same rod. Using Lorentz transformation we have,

Hence  ,           l = x2 –x1

L = l  √ l –v2 /c2

 

 

This length is also known as proper length. Here  [ 1 –v2 /c2]. Hence the rod seems to be contracted from the measured value of length in moving frame when observed from a stationary frame. This is called as Length Contraction.

 

The body’s length is thus measured to be the maximum when it is at rest relative to the observer. As soon as it starts moving with a velocity ‘v’ relative to the observer, its measured length is contracted in the direction of its motion, whereas its dimensions perpendicular to the direction of motion remains unaffected. Fitzgerald and Lorentz proposed this idea of length contraction long before Einstein’s special Theory of relativity. So, this is also known as  Lorentz Fitzgerald contraction. In Present ·particle physics experiments in high energy accelerators the colliding particles (nucleons, proton 5, etc.) have achieved a very high velocity approaching c’, the length contraction is regularly observed.