Answer to Question #101765 in Mechanics | Relativity for AbdulRehman

Question #101765
Use the mass and radius of the earth to calculate
how much faster a clock in Denver runs than does a clock in Washington,
D.C. Denver is 1600m higher than Washington, D.C.
1
Expert's answer
2020-01-27T09:38:28-0500

Calculate the radii of trajectories that Denver and Washington will follow:


As we see from the figure:


RW=R cos4745=4283649 m,RD=R cos3944=4899476 m.R_W=R\text{ cos}47^\circ45'=4283649\text{ m},\\ R_D=R\text{ cos}39^\circ44'=4899476\text{ m}.

But Denver is 1600 m higher, that is why make it more precise:


RD=4899476+1600=4901076 m.R_D=4899476+1600=4901076\text{ m}.

How much faster does a clock in Denver run than a clock in Washington?

Use equation (7.16):


ΔτDΔτW=exp(RWRDg(r)c2dr)= =exp(GMc2RWRDdrr2)= =exp(GMc2(1RW1RD))= =exp(6.673101161024(3108)2(1428364914901076))=1.\frac{\Delta\tau_D}{\Delta\tau_W}=\text{exp}\bigg(\int_{R_W}^{R_D}\frac{g(r)}{c^2}\text{d}r\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{GM}{c^2}\int_{R_W}^{R_D}\frac{\text{d}r}{r^2}\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{GM}{c^2}\bigg(\frac{1}{R_W}-\frac{1}{R_D}\bigg)\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{6.673\cdot10^{-11}\cdot6\cdot10^{24}}{(3\cdot10^8)^2}\bigg(\frac{1}{4283649}-\frac{1}{4901076}\bigg)\bigg)=1.

As we see, that difference is insignificant. The clocks can be considered as running at the same rate.



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