Answer to Question #97494 in Mechanics | Relativity for Umaima

Question

In framework of Newtonian physics suppose that the earth is moving through the luminiferous ether at speed v suppose that you are Michelson & that you have an interferometer with two arms each of length L if one arm is along direction of earth's motion & one arm is in perpendicular direction .1) how long it will take a bit of light to make complete circuit along earth arm? 2) how big is the difference between these two lines ?

Answer 1

For time along earth arm

$$t_2=\frac{L}{c-v}+\frac{L}{c+v}=\frac{2L}{c} \frac{1}{1-\frac{v^2}{c^2}}$$

For time along perpendicular arm

$$t_1=\frac{2L}{\sqrt{c^2-v^2}}=\frac{2L}{c} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

Different time

$$t_2-t_1=\frac{2L}{c} \frac{1}{1-\frac{v^2}{c^2}}-\frac{2L}{c} \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

if

$$v<<c$$ $$\Delta t=L \frac{v^2}{c^3}$$