Okay, this was just a curious question that's been bobbing around in my head recently.

The formula for gravitational force between two masses is

Fg=Gm1m2r2


right?

So, gravitational force increases as you approach the center of the earth. I was wondering, what is the gravitational force at the exact center of the earth? I thought it could be approximated by

limr→0Gm1m2r2=∞


But I know that's wrong. I just don't know why. Why wouldn't there be infinite force at the center of the earth, and if there was, why wouldn't we collapse on ourself?

Hmmmm,

Anytime someone starts doing physics formulas in the forum, I take notice.

I believe most of your math and thinking are correct...the force of gravity at the center of the Earth would be sufficiently neutral and negligent given two important considerations:

1. We are only talking about the EARTH's gravitational influence on itself. However, the Earth does not exist in a gravitational vacuum by itself...at the center of the Earth, you would still feel some effects of other large gravitational masses such as the SUN, the Moon, and all the other planets of the solar system.

2. Your idea takes into consideration that Earth's gravitational center IS in the center. This would be true if the Earth was perfectly round in shape, however the Earth does not have a perfectly round and symmetrical shape...it is an ellipse.

The reason why we dont collapse on ourselves is that the force of our gravity is not sufficiently stronger than the density of the matter it is acting upon. This is also why you would weigh MUCH more on the surface of Saturn or Jupiter or any other planet that has a larger gravitational mass than Earth.

No, this is wrong.

Outside of the surface of the earth, we can somewhat usefully model the earths gravity by treating it as a point mass at the earths center of mass, but this is not the truth. It's a simplifying model that is useful in some situations, but not in others.

Yes! You are on the right track there, and you are right to question this.

Edit: by 'right track', I mean the part where you say "I know that's wrong"

The gravitational force acting on an object at some point in space is the sum of all the gravitational forces acting on the object.

To truly calculate the gravitational force of the earth on an object, we would first have to inventory every atom in the earth. We would have to account for whether your bowling ball is on the left side, or the right side, of your closet. We would even have to account for which carbon molecules were C12 and which were C14, since they have different masses.

This would be insane, and the accuracy gained would not be helpful. So instead we use simplifying assumptions.

For some calculations, we just pretend that the earth is a sphere (it isn't) with a point mass at its center equal to some estimate of the earths total mass. This is wrong, but it works perfectly well for many calculations.

For other calculations, we might start with a 3-dimensional map of estimates of the density of different regions of the earth's interior, and then sum over the all of those regions and their estimated masses.

If you go deep inside the earth, now all of the earth that is above you is no longer pulling you down towards the center. It is pulling you away from the center.

Agreed.

If there was some magical hollow cavity at the center of the earth that could protect you from the heat and not collapse you would still likely gravitate toward the walls as the mass of the earth shifted around you. Even a magical sphere that arbitrarily tried to keep you in the exact center would fail to keep you perfectly balanced without flaw. I imagine your buoyancy would be fairly extreme, and thus my vertigo would be extreme as well. The chuck factor would be off the scale.

So what do you think about the scientists speculating there is a black hole at the center of the earth? (Due to the voracious appetite of black holes I don't see how this can be).

I think they took the wrong pill.

Additional information for consideration for your information (?):


http://science.nasa.gov/science-news/science-at-nasa/2011/04may_epic/


This website may provide info and be of interest to you on a recent discovery by NASA.


RE: Black Holes http://www.youtube.com/watch?v=cW7BvabYnn8

I think that speculation is fun and harmless...as long as you remember that its just speculation.

Scientists often reject ideas that have been given a thorough and fair trial and found wanting, which leads ignorant people to accuse them of being 'closed minded'. Most of the scientists I have known were among the most open minded people I've met. I think its great that some scientists may play with this idea and see where it leads them.

You mention a voracious appetite - the theories I've heard about terrestrial black holes deal with ones so small they are only able to pull in matter that is extremely close to the hole...not your typical black hole.

And (I've heard) others have suggested that holes that size would evaporate.

I've not been keeping up with this topic - these are some ideas I read of almost 10 years ago.

Amazing! From wikipedia:

Manny,

Your question lead me on some fun thought experiments. The earth itself is too irregular for this, but suppose we built a perfectly spherical planet, with a comparable mass and size, with a hollow chamber at the center and a tunnel going straight through the planet, through the chamber.

The old stand-by is: what happens if you drop something down through that tunnel?

And how about your weight as you climb down into the tunne? You would find yourself getting lighter and lighter until you reach the center, where you would find yourself nearly weightless.

And yet, I believe you would be subject to incredible pressure from the atmosphere. Though the force of gravity acting on the air directly above you would be very weak, it would be pushed downwards by the weight of the air above it - and farther out that air is getting pulled downwards by a much stronger force of gravity.

Very cool!

Assuming that there is an equilibrium point somewere between te crust and core, and further assuming that it is possible that the speed of the ball traveling towards the center of earth is insufficent to effectively propel it through the 'equilibrium point', I would further posit that it could be the case that the ball could/would eventually reach a point where it would just float in mid air.

The "net" gravity at the center of the Earth is zero since the vector sum of all the Earth's mass would cancel each other out giving a "weight" of zero.

You all know I am in love with Sheldon from The Big Bang Theory right??

Also, to get the gravitational effects to be so extreme that the atoms would "fall inward onto themselves" takes something like ten solar masses.

In other words it takes a star ten times the size of our sun to create a black hole by running out of fuel and collapsing. Small black holes can be caused by massive pressures in explosions but they evaporate quickly.

Journey through the center of the Earth
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Suppose you could drill a hole through the Earth and then drop into it. How long would it take you to pop up on the other side of the Earth?

Your initial acceleration would be the surface acceleration of gravity

but the acceleration would be progressively smaller as you approached the center. Your weight would be zero as you flew through the center of the Earth. For our hypothetical journey we will assume the Earth to be of uniform density and neglect air friction and the high temperature of this trip.

For a spherically symmetric mass, the net gravity force on an object from that mass would be only that due to the mass inside its radius, and that would act as if it were a point mass located at the center. When this is analyzed in detail, you find that the gravity at any radius r less than REarth will be linearly proportional to the distance from the center.

This is the same form as Hooke's Law for a mass on a spring. It would cause the trans-Earth traveler to oscillate back and forth through the center of the Earth like a mass bobbing up and down on a spring.

The traveler accelerates toward the center of the Earth and is momentarily weightless when passing through the geometric center at about 7900 m/s or almost 17,700 miles/hr. The traveler would pop up on the opposite side of the Earth after a little more than 42 minutes. But unless he or she grabs something to hold on, they will fall back for a return journey and continue to oscillate with a round-trip time of 84.5 minutes.

As a further feature of this fanciful journey, suppose a satellite could be put in a circular orbit about the Earth right above the surface. Ignoring air drag and the terrific sonic boom that would accompany such an orbit, suppose it passed overhead just above the falling person as they popped up out of the hole. The period of such an orbit would be such that it would be passing overhead every time the oscillating person popped up on either side of the Earth.

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/earthole.html

I left out the math. It was too much trouble to copy and paste.

Speaking of voracious appetites of black holes here is a recent NASA article/utube entitled: "Black Hole Caught Red-handed in a Stellar Homicide" ...

Check out the video on a Black Hole's appetite with an elasped time line included.

In the end let's say of the Universe ... for it may very well be a matter of a fight to the finish episode of vorvacious appetites i.e., "Last two Black Holes Dual it to the Finish" ... with no one left to watch it.

Does new life start again, within a Black Hole or can it?

Video - http://hubblesite.org/newscenter/archive/releases/2012/18/video/b/

Article -
http://hubblesite.org/newscenter/archive/releases/2012/18/image/a/

Could be the findings of Black Holes discoveries, be what is "past" and "present" destiny, existence of life as in the beginning the universe was and becomes void, a blank slate so to speak ... the Black Hole being a Janitor, renewal system of the Universe, for are not all things/masses recyclable like for example precipitation, what goes around comes around ... just a wild thought of life and the universe... Big Bangs will always be, a probability!

No one really knows of a renewal process in cosmology other than supernovas. Larger concepts such as a collapsing universe that would then explode into another big bang does not appear to match current expansion estimates which give the universe a long slow frozen death.

A giant black hole may be manifested as a giant white hole in a parallel universe ... maybe.

Most modern theories seem to point at a granular multiverse with an ever renewing series of buds which, if seen from inside the bud, as a big bang.

The limitations of the laws of physics may keep us from ever learning the whole truth.