Question

An optical interferometer can detect a displacement of 0.1?, or ?50nm.

At what speed would a 3.0m stick "shrink" by 50 nm? Hint: Use the binomial approximation.

Answer 1

Length contraction is governed by equation

L = L₀ * v (1 - v² / c2)

hence change in length delta L =   L₀   -   L

=   L0 * [1 - v (1 - v_2 / c_2) ]          ----- (1)

v (1 - v^2 / c_2) = (1 - x) /2   (  where  x   =   v^2 / c_2,   )   

=   (1 - x/2)        

=   1  -   v^2 / 2c_2

delta L   =   L_0 * [ 1   -   1   +   v^2 / 2 * c_ 2 ]

50 * 10-9m    = 3.0 * v^2 / 2 * c_2

v^2   =   50 * 10-9 * 2 * c² / 3.0

= sqrt (3.33 * 10-8 ) * c²

v = 1.83 * 10-4 m/s * 3.00 * 10⁸ m/s    

=  5.49 * 10⁴ m/s