Answer to Question #101765 in Mechanics | Relativity for AbdulRehman

Question

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.

Answer 1

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

As we see from the figure:

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

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

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

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

Use equation (7.16):

$$\frac{\Delta\tau_D}{\Delta\tau_W}=\text{exp}\bigg(\int_{R_W}^{R_D}\frac{g(r)}{c^2}\text{d}r\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{GM}{c^2}\int_{R_W}^{R_D}\frac{\text{d}r}{r^2}\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{GM}{c^2}\bigg(\frac{1}{R_W}-\frac{1}{R_D}\bigg)\bigg)=\\ \space\\ =\text{exp}\bigg(\frac{6.673\cdot10^{-11}\cdot6\cdot10^{24}}{(3\cdot10^8)^2}\bigg(\frac{1}{4283649}-\frac{1}{4901076}\bigg)\bigg)=1.$$

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