I know that theoretically the weight of an object will decrease as it is elevated further from the center of the Earth … what I am wondering about is whether this experiment has ever been carried out, using a hyper-sensitive weighing machine to weigh the object first at ground level and then in an aircraft at 30,000 ft altitude.
I do appreciate that the theoretical difference in weight might well be too small to confirm experimentally, even with the most sensitive equipment, but am just wondering if the experiment has ever in fact been attempted.
If my calcs are right, that’s about 0.3% difference in weight (using Newtonian gravity, and doing it directly above the North or South Pole so that there’s no effects from the Earth’s rotation). Now that’s not hard to measure on a nice stable lab bench (even if you can’t use a balance scale), but it’s going to be very tough to get that accuracy on a moving airplane.
But we don’t really need to. We’ve already proved that things ‘weigh’ less when they’re farther away from the earth, because we’ve put up satellites that don’t fall down or fly into space. Getting a satellite into a stable orbit requires knowing how big the force of gravity is.
The last sentence is correct, but it is misleading to say that an orbiting satellite is not falling. This reinforces the common misconception of zero gravity. The force of gravity is only slightly reduced for objects in low earth orbit. They are, in fact, in free fall; moving forward at a velocity that ensures they are constantly falling around the earth and not into it.
In answer to the OP, measuring the weight of an object near sea-level and then high on a mountain might be a more practical experiment. Actually, there are scales that are sensitive enough to detect weight changes at relatively* small altitude differentials.
You’d have to correct for centripetal force not just from the rotation of the Earth but from the trajectory of the jet. You’d also have to take into account reduced buoyancy from the lower air pressure.
Inertial navigation devices are incredibly sensitive. I wouldn’t be surprised if they could take the altitude into account if the difference in gravity is important.
Would the mass of the mountain affect the reading ? If you went to the top of Mt Everest to weigh the object, would the reading be the same as if you did it on a hot air balloon in the middle of the Pacific at the same altitude? Always assuming a balloon can rise that high and that you had a stable enough platform suspended under it etc etc etc …
In Newtonian mechanics, the gravitational force between two bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers of mass. So, within the precision of Newtonian physics, all that matters is the distance from the center of mass of the earth.
Um, the effect of gravity (as measured by time dilation) can be detected by a height difference of a foot or so. In essence, clocks run faster the higher they are from Earth’s gravity well.
This paper was published 5 years ago - I have no doubt that detectors are even more sensitive now.
Actually, yes, having a mountain underneath you would increase the gravitational force, compared to having a bunch of air. I’ve already done one calculation today, so I’ll let someone else figure out how much difference.
Jim – the distance from the center of the Earth is the only thing that matters if the Earth is perfectly spherical. A mountain makes it not so perfect.
Bolding added. The bolded part is incorrect in general. It only holds for spherically symmetric objects, which a mountain certainly isn’t. Even in Newtonian mechanics, the mountaineer and balloonist would measure different weights for identical masses.
Yes, it would. Since the Earth is not a perfect uniformly-dense sphere, any irregularity will have an effect (if you are close enough). Earth’s gravity is often quoted as 9.807 and the third decimal place will change depending on how close you are to a big mountain.
Would the mass of the mountain affect the reading ? If you went to the top of Mt Everest to weigh the object, would the reading be the same as if you did it on a hot air balloon in the middle of the Pacific at the same altitude?/QUOTE]
Yes, the mountain would affect the reading. If you fly around at a constant distance from the center of Earth, your scale would see the effects of all kinds of irregularities: mountains, ocean trenches, water vs. rock, hot spots in the mantle, all kinds of random density and mass changes.
As an aside, those gravity variations are one of the tools oil and mineral companies use to look for resources below the ground. Gravitometry can’t give you the full picture, but gravitometry combined with other methods like explosive sounding and sample drilling can give you more information than those methods alone.
