I'm trying to measure the time difference $(\Delta t)$ between two cities (London, Moscow) (they aren't at the same latitude) but I'm facing problem because the speed of earth rotation ($\nu$) depends on latitude.
I have:
1- London latitude and longitude:
(51.50 N, -0.12 W)
2- Moscow latitude and longitude:
(55.75 N, 37.62 E)
3- The distance between the longitude for the two cities:
$d=4201.10$ $km$
4- The speed of the earth rotation at London latitude:
$\nu_l=1039.44$ $km/h$
5- The speed of the earth rotation at Moscow latitude:
$\nu_m=939.74$ $km/h$
So I have $\nu_l$ , $\nu_m$ and $d$ . How can I get the real ($\nu$) for earth rotation to use it in ($\Delta t=\frac{d}{\nu}$ ) ?
Notice: Please don't give me another method to measure the time difference between two cities. I want to use this sample law ($\Delta t=\frac{d}{\nu}$ ) but just give me the method to get the real speed of earth rotation if it possible.
If we assume that the earth spins about an axis normal to the lines of latitude, then what you want is the distance along the latitude line. The reason is that the time is the same along a longitude line.
Note also that the speed of rotation depends on latitude because the angular speed is constant. The relation is
$$v(\lambda) = R_e \omega \cos{\lambda}$$
where $R_e$ is the radius of the earth, $\omega$ is the angular speed, and $\lambda$ is the latitude. You can verify from your numbers above that the value of $R_e \omega$ is the same for both London and Moscow ($1670 \,\text{km/hr}$).
The time difference between two cities is then
$$\Delta \tau = \frac{\Delta \mu}{\omega}$$
where $\Delta \mu$ is the longitude difference (in radians). So for the Moscow-London example, $\Delta \mu = 37.74 \pi/180 \approx 0.6569 \,\text{rad}$. $\omega$ we find from the radius of the earth being $6371$ km:
$$\Delta \tau \approx 0.6569 \frac{6371 \, \text{km}}{1670 \, \text{km/hr}} \approx 2.513 \,\text{hr}$$
is the approximate time difference between London and Moscow (at least mathematically and not as specified by politics, etc.).