Trying some physics i don't even know about.

  • #1

Main Question or Discussion Point

Sometimes I get so bored that I just try to calculate certain experiments I've heard without know the solution. Like using the basic formula to calculate the speed of the moon around the Earth. This time I tried to calculate the differences between 2 clocks, on put on the highest building in the world (Burj Khalifa) with 828m of the ground and one in its basement so 0m.

First of all, I went searching for some variables I need to use :
  • Rotation of the Earth = 23h 59min 04s = 86164 s
  • Radius of Earth = 6378 x 103 m
  • Speed of light = 299 792 458 m/s

I also found the formula I needed to use :





To find the Velocity, I used the simple formula
and


So, I'm trying to find the velocity of those 2 clocks relative to the center of the earth. And because one clock is further (as we see in the formula, if the angular speed is constant) it would have a faster velocity so it should go faster in time.

Velocity clock basement :

gif.latex?v_%7B1%7D%3D%5Cfrac%7B2.%5Cpi%7D%7B86164s%7D%20.6378.10%5E3m%20%3D%20465.gif


Velocity clock on top :

f.latex?v_%7B2%7D%3D%5Cfrac%7B2.%5Cpi%7D%7B86164s%7D%20.%286378.10%5E3m+828m%29%20%3D%20465.gif


Then I chose for a timeframe of an hour = 3600s

That means I need to calculate the space they have done with their velocity :

gif.latex?x_%7B1%7D%3D%20465.0916379%5Cfrac%7Bm%7D%7Bs%7D.3600s%3D1674329.gif


gif.latex?x_%7B2%7D%3D%20465.1520167%5Cfrac%7Bm%7D%7Bs%7D.3600s%3D1674547.gif



Now we should have all the variables for the formula I first mentioned, so let's fill it in :D

For the clock in the basement :

5Cfrac%7Bm%7D%7Bs%7D%7D%7B%5Cleft%20%28%20299792458%5Cfrac%7Bm%7D%7Bs%7D%20%5Cright%20%29%5E2%7D.gif



gif.latex?3599%2C9999999956680s%20%3D%203600s%20-%204%2C33192.gif


So 4.33192 nanoseconds faster than the center of the earth.



Now for the clock on top :

5Cfrac%7Bm%7D%7Bs%7D%7D%7B%5Cleft%20%28%20299792458%5Cfrac%7Bm%7D%7Bs%7D%20%5Cright%20%29%5E2%7D.gif



gif.latex?3599%2C9999999956667s%20%3D%203600s%20-%204%2C33329.gif


So 4.33329 nanoseconds faster than the center of the earth.


There seems to be a theoretical a difference of 1.37 picoseconds after 1h on the clocks.

First of all, this is my first time trying to solve such a question, I just graduated high school. But I'm really interested in this field of physics.

If I made a mistake somewhere I would love to hear it, also English grammar mistakes will be visible, my apologize for those also.

Have a great day further.
 

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  • #2
PeroK
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I also found the formula I needed to use :




The first formula relates the time of an event in two reference frames related by the Lorentz Transformation for a relative velocity ##v## in the x-direction, and isn't relevant here.

The second formula, for the gamma factor, is relevant here.

Note you you may also want to take gravitational time dilation into account. See:

https://en.wikipedia.org/wiki/Gravitational_time_dilation

Note also that "moving clocks run slow" in your reference frame.

You may also be interested in the famous Hafele-Keating experiment:

https://en.wikipedia.org/wiki/Hafele–Keating_experiment
 
  • #3
Oh well, I've still got a lot to learn, but thx for the referrals, I'll try to understand them :)
 
  • #4
PeroK
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Oh well, I've still got a lot to learn, but thx for the referrals, I'll try to understand them :)
I think gravitational time dilation is by far the greater factor in this case.
 

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