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I want to see (actually: show to my kids) evidence of the $1/r^2$-decay of gravitational acceleration by measurement, in a way as simple as possible. So I was thinking of measuring gravitational acceleration during a plane flight. With the radius of the earth $R_E = 6371\,km$ and a flight height of $H_p = 11\,km$ one would expect a difference of gravitational acceleration by $1-R_E^2/(R_E+H_p)^2 \approx 0.34\%$. Conventional smart phone accelerometers are currently not precise enough to measure that difference, right? But an object showing $1\,kg$ on ground should show roughly $997\,g$ at the height of $11\,km$. Wow, that should be measureable with simple means. Has anyone tried something like that? Any practical hints on how to actually perform such a measurement?

Note: I am aware that also the centrifugal acceleration has to be taken into account to actually interpret the measurement result. Depending on the direction of the flight the velocity of the plane (e.g. $v_p = 700km/h \approx 194m/s$) has to be combined with the velocity of the ground around the earth's axis of rotation (e.g. $v_{ground} \approx 297m/s$ at Frankfurt), which amounts to an effect in the measurement in the same order of magnitude. But the necessary information (height, speed and direction of flight) should be available from the on-board flight information system with sufficient accuracy.

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  • $\begingroup$ You may know of this website, apologies if you do, but it gives pretty accurate data on most commercial aircraft flights across the world : flightradar24.com the height, direction, speed of each aircraft plotted seem to be pretty accurate based on its predicted versus actual arrival times. $\endgroup$
    – user108787
    Commented Oct 22, 2016 at 12:09
  • $\begingroup$ I wonder if the issue with smart-phone accelerometers is their accuracy, or just their precision? If the latter, you might consider asking the flight crew to participate in the experiment, and see if many passengers would like to be involved. Then they could average together all their results and see what they get, with a few dozen participants you might be able to beat down the noise. Even if the problem is accuracy of calibration, you might see something in the differential effect. Or try many measurements with your own phone. $\endgroup$
    – Ken G
    Commented Oct 22, 2016 at 12:22
  • $\begingroup$ @CountTo10 I have heard of flightradar24.com, but never taken a closer look. So during the measurement on the plane one only needs to record the time. thanks for the hint $\endgroup$
    – coproc
    Commented Oct 23, 2016 at 7:46
  • $\begingroup$ @KenG you are right: the problem is the precision (noise), especially since I am more interested in the difference of measurements. And also the resolution is a problem: my smart-phone's accelerometers seem to have a resolution of ~0.03 m/s^2. Averaging is also a good idea: I have found an app with a smoothing average that reduces the noise to +/- 0.003 m/s^2. So I will try a first attempt with smart-phone accelerometers. $\endgroup$
    – coproc
    Commented Oct 23, 2016 at 7:56
  • $\begingroup$ @coproc I think they get their data, directly or indirectly, from the transponders on the aircraft. Test it first without buying anything, use their data to get an ETA on any aircraft, then look to your local airport arrival times to see is it correct. I think the adfree version is about 10 dollars, otherwise it's a bit too much loaded up with ads. Although it's still cheaper than what you really want, the Vomit Comet thespacegal.com/blog/2014/4/25/…. Have a nice flight. $\endgroup$
    – user108787
    Commented Oct 23, 2016 at 11:52

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You won't be able to demonstrate this. The value of r is measured from the centre of the earth. Even if you are several km high, this will have virtually no effect on r.

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  • $\begingroup$ It's perfectly possible in principle. It just requires a high-precision measurement of g. $\endgroup$
    – user4552
    Commented Mar 26, 2018 at 1:33

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