If you were to remove all the water from the Mariana trench would you experience more g's at the bottom of the Mariana trench or at the top of mount Everest?
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$\begingroup$ To first order: en.wikipedia.org/wiki/Shell_theorem . There is, of course, the fact that Earth is not completely homogeneous. One of the effects of the shell theorem is that you wouldn't even have to remove the water in the trench. It has no gravitational effect on you to begin with (again with the caveat that the Earth is not homogeneously covered in water). $\endgroup$– FlatterMannCommented Jun 12, 2023 at 0:11
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$\begingroup$ What is meant by remove all water from the Mariana trench? You mean remove and seal it so it won't overflow from the ocean? Otherwise if you drain the entire ocean that's a big difference... The mass of the water within the Mariana trench alone is only of order $10^{18} \text{ kg}$, see here compared with the entire earth which is of order $10^{24} \text{ kg}$ that won't do much to gravity. $\endgroup$– AmitCommented Jun 12, 2023 at 18:14
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NASA mapped the gravity across the surface of Earth. You can see the map here.
As you can see on the map, the gravity at Mt Everest is notably stronger than the gravity at the Mariana trench. Draining the Mariana trench won't change that.
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$\begingroup$ Yes, I believe draining the Mariana trench would result in weaker gravity there. Between having less mass directly below you, and applying the shell theorem, I see no reason to expect that it should increase. Certainly not enough to surpass the strength of gravity at Mt Everest. $\endgroup$ Commented Jun 11, 2023 at 23:39
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$\begingroup$ I was sure that the peak of mt. Everest must have weaker gravity... since it is farther away from the earth's center. $\endgroup$– AmitCommented Jun 11, 2023 at 23:40
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$\begingroup$ That's a reasonable guess, but it seems the mass of the mountains more than makes up for the additional distance. I'm going to go with NASA's data over intuition. $\endgroup$ Commented Jun 11, 2023 at 23:44
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$\begingroup$ Ah, see I couldn't find the data you were referring to, it is just a link to Wikipedia, if you can give a more direct one that'll be great. Also, yes I do understand that in the Mariana trench also there is less mass below you so applying Gauss's law results in a weaker gravity field. It's really hard to intuitively grasp that these are small differences in comparison to the earth's radius which is why things like density, etc. really become important, and the difference won't be too big anyway :) $\endgroup$– AmitCommented Jun 11, 2023 at 23:47