How strong is gravity in the center of the earth?

[mgb_phys]

So an object released in a tunnel through the centre of the Earth (with the obvious idealizations) would execute simple harmonic motion.?
Amusing.

It's also just a special case of a very elliptical orbit

 

[Chronos]

Interestingly enough, it takes about 42 minutes to free fall through the earth, no matter what angle the hole is pitched. Of course if the inclination angle relative to the center of the Earth is too shallow, you can expect some amount of 'rolling' time where friction will slow the journey.

 

[Bob S]

If we also adjust for the speed of the earth’s rotation at equator?
Radius 6378km – 5km = 6373km - circumference = 40.058 Km = 40058000Meter / 86400s = 463,63 m/s
Do you know the influence of this?

The Earth's surface, and especially the oceans, are at a equipotential, including both the gravitational force and the centrifugal force. So the Earth is slightly prolate.

An equatorial bulge is a bulge which a planet may have around its equator, distorting it into an oblate spheroid. The Earth has an equatorial bulge of 42.72 km (26.5 miles) due to its rotation: its diameter measured across the equatorial plane (12756.28 km, 7,927 miles) is 42.72 km more than that measured between the poles (12713.56 km, 7,900 miles).

 

[Stoonroon]

The simple harmonic motion prediction for radial free-fall through a uniformly dense spherical mass (such as the idealized Earth discussed here) is very well known. Does anyone feel a desire (or need?) to verify this prediction empirically?

Though a high orbit satellite experiment would be the ideal way to do it, a laboratory version could also be done using a modified Cavendish balance.

Wouldn't it be nice to have all the theoretical discussion about this prediction backed up by a physical demonstration?

 

[Chronos]

Gravity probe B comes to mind.

 

[Stoonroon]

Gravity probe B is in orbit over Earth's surface. It has provided nice support for predictions of General Relativity involving a couple of its minute deviations from Newton's predictions for this exterior, "weak field" circumstance.

General Relativity, of course, also predicts simple harmonic motion for the interior, through-the-center circumstance. But this particular interior field prediction remains to be tested, whether in the context of Einsteinian or Newtonian gravity.

It may be relevant to point out that in General Relativity, the prediction can be expressed in terms of clock rate. The theory says that a clock at rest at the center would have the slowest rate of all stationary clocks attached to the mass. What physical mechanism makes the central clock go slow?

I'm not saying that it doesn't. I'm just pointing out that nobody has ever proven that it does. It's easy to understand that one would extrapolate the concept of gravity (whether in terms of Newtonian "potential" or relativistic spacetime curvature) from the exterior to the interior. But theoretical extrapolation is a poor substitute for physical fact.

 

[zhangzxh]

Above calculations are meaningful only if Newton's universal law of gravitation is true in inside of the earth.
But we cannot use experimental methods to prove that Newton's universal law of gravitation is true in inside of the Earth directly at present time.

 

[Bjarne]

Above calculations are meaningful only if Newton's universal law of gravitation is true in inside of the earth. .

I agree.

 

[Stoonroon]

But we cannot use experimental methods to prove that Newton's universal law of gravitation is true in inside of the Earth directly at present time.

A tunnel through the center of Earth is obviously impractical. The next best thing would be to prove Newton's law for the insides of massive bodies on a laboratory scale. A laboratory version of the oscillation prediction is testable using a modified Cavendish balance.

Imagine the large masses having slightly arced holes through which the small masses can pass. A horizontal slice is also cut away to provide clearance for the arm connecting the small masses. A fiber would not work for the support mechanism, because the arm needs to be able to swing freely through a large angular range (no restoring force wanted). This could be done with a magnetic or fluid suspension.

In the near future a balance like this may actually be built. I've been trying to generate interest in such an apparatus for a couple years. Just recently I got some very positive feedback from the experimental physicist, George Herold at TeachSpin. His company builds experimental devices for undergraduate physics classes worldwide. When I showed him what I had in mind, he replied that he had recently been thinking about exactly that idea himself. I hope he turns the idea into a tangible product, because I think it would be really cool to see the oscillation--so often presented as a thought experiment--actually demonstrated in the real world.

 

[Bjarne]

But we cannot use experimental methods to prove that Newton's universal law of gravitation is true in inside of the Earth directly at present time.

And we can measure both acceleration due to gravity and weight inside large building, systematically floor by floor. - Above and below the gravitational centre of the buildings.

Science must be based on observation and testing if this is possible. And it is possible and even cheap and easy to test central gravity..

 

[vardhan_harsh]

in simple words it'll become quarter part halfway.

 

[Nik_2213]

A couple of gotchas:

You must take the Moon's gravity into account, as that's enough to offset the 'zero point' from the geometric centre of Earth's geoid...

Speaking of the Moon's gravity: A very low lunar orbit is unstable due to the Mascons caused by ancient impacts and lava flooding. IIRC, the early Apollo missions' re-appearance from farside was uncertain by sundry scary seconds due to the 'bumpy' orbital path...
http://en.wikipedia.org/wiki/Mass_concentration_(astronomy )

 

[Zandrin]

http://en.wikipedia.org/wiki/Gravity_of_Earth#Depth
http://imm.io/e9Yg
http://imm.io/e9YI
http://imm.io/e9Yg
http://imm.io/e9YI

 

[Elliot Smith]

I am new to the subject, but I would believe that the center of the Earth would have gravitational pull in all directions seeing that all of the Earth's mass would be around you

 

[DaveC426913]

I am new to the subject, but I would believe that the center of the Earth would have gravitational pull in all directions seeing that all of the Earth's mass would be around you

No.
Gravitational potential is the sum of forces from all massive bodies (every atom in the Earth). The sum of those forces is zero.

 

[D H]

in simple words it'll become quarter part halfway.

That's wrong, even with the simple model of constant density. Gravitational acceleration is a linear function of distance from the center in such a model. It becomes half the surface value halfway down.

Inside our real Earth with it's very dense iron core and less dense mantle, gravitational acceleration halfway down is more than the surface value. Gravitational acceleration reaches its maximum at the core/mantle boundary, which is about 300 km above the halfway point.

You must take the Moon's gravity into account

Not really. At least not at the center of the Earth, and not if the acceleration is with respect to the Earth. The acceleration of the Earth toward the Moon is equal to that of a point mass at the center of the Earth. The effects cancel in an Earth-centered frame.

I am new to the subject, but I would believe that the center of the Earth would have gravitational pull in all directions seeing that all of the Earth's mass would be around you

Gravitational acceleration inside a spherical shell of mass is zero. Google "Newton's shell theorem" for more.

 

[jacoblost]

I'm not to informed on this material, but I thought that Newtons theorems have been pretty completely superseded by Einsteins. Under general relativity gravity is defined by the curvature of space-time by a massive object. Thereby all of the Earth's mass should compound the indentation of space to be at a maximum at the planets center (like putting a bunch of bowling balls on a trampoline, the indentation is greatest at the center point)?