Silly Thought Experiment

I suppose the would fall to the center and beyond. Then continue oscellating back and forth till they stabilized at the very center.

Of course, they would not survive the heat, and such a hole can't be bored.

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Novas Neil deGrasse Tyson has an explanation video Just click on the blue text.

Rap

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Larry Niven wrote a short story about a tiny black hole that someone dropped. It fell through the floor and kept going, accelerating toward the centre of the earth then decelerating as it neared the surface on the other side of the planet then repeating it's movement in reverse like a bizarre pendulum dancing with the gravitic attraction of the earth's mass.

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hingehead wrote:

Larry Niven wrote a short story about a tiny black hole that someone dropped. It fell through the floor and kept going, accelerating toward the centre of the earth then decelerating as it neared the surface on the other side of the planet then repeating it's movement in reverse like a bizarre pendulum dancing with the gravitic attraction of the earth's mass.

Of course, this was before they determined that tiny black holes would be blazing infernos of radiation.

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raprap wrote:

Novas Neil deGrasse Tyson has an explanation video Just click on the blue text.

Rap

deGrasse: "Unless somebody catches me I'll fall back down the earth again and yo-yo back and forth forever."

I don't think so. Eventually the yo-yoing would diminish until the body stabilized at the center of gravity.

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Sheesh!!

Nothing is stable. How bourgeoise twitterington can you get?

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Dr Huff--

Only if there was some method of losing energy. Degrasse covered that in his assumptions---no air resistance and no rotational losses (banging off of the sides of the hole)

Consider the force acting on the body

F=-GMeM/r^2
G is Newtons gravitational constant
Me is mass of Earth
M is mass of falling body

but F decreases as the body falls toward the center of the earth, because the mass of the earth (Me) decreases with volumn
M=Ve*density (rho)

Now I'll assume that the density (rho) of the earth is constant and assume the earth is round.

So its volumn is Ve=4/3*pi*r^3 where r is distance from center

Now the effective attractive mass (Meff) of the earth as you approach (and pass the center) is

Meff =4/3*pi*r^3*rho

This can be expressed as a function of the Me to get rid of the 4/3, pi and rho terms. So

Meff=Me*(r/R)^3

Where R is the radius of the earth sphere

So the function of the changing force on the falling body can be expressed with r (distance from center) as the independant variable

F=-G*Meff*M/r^2=-G*Me*(r/R)^3*M/r^2=-G*Me/R^3*M*r

Look at the constants here G is gravitational constant, Me is mass of earth--constant, R is radius of earth--constant, M is mass of falling body (assumed constant--starvation is neglected) so
GA*Me/hr^3*AE is a constant--for convenience sake I'll call it k

and

F=-(G*Me/R^3*M)*r=-k*r

Which is Hooke's law

So the hole in the Earth problem can be expressed as an oscillating spring and, if the spring is perfect (no internal friction/fatigue) it will oscillate until stopped by an external force.

Rap

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That happened to me rap.

It's no good.

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raprap wrote:

Dr Huff--

Only if there was some method of losing energy. Degrasse covered that in his assumptions---no air resistance and no rotational losses (banging off of the sides of the hole)

Consider the force acting on the body

F=-GMeM/r^2
G is Newtons gravitational constant
Me is mass of Earth
M is mass of falling body

but F decreases as the body falls toward the center of the earth, because the mass of the earth (Me) decreases with volumn
M=Ve*density (rho)

Now I'll assume that the density (rho) of the earth is constant and assume the earth is round.

So its volumn is Ve=4/3*pi*r^3 where r is distance from center

Now the effective attractive mass (Meff) of the earth as you approach (and pass the center) is

Meff =4/3*pi*r^3*rho

This can be expressed as a function of the Me to get rid of the 4/3, pi and rho terms. So

Meff=Me*(r/R)^3

Where R is the radius of the earth sphere

So the function of the changing force on the falling body can be expressed with r (distance from center) as the independant variable

F=-G*Meff*M/r^2=-G*Me*(r/R)^3*M/r^2=-G*Me/R^3*M*r

Look at the constants here G is gravitational constant, Me is mass of earth--constant, R is radius of earth--constant, M is mass of falling body (assumed constant--starvation is neglected) so
GA*Me/hr^3*AE is a constant--for convenience sake I'll call it k

and

F=-(G*Me/R^3*M)*r=-k*r

Which is Hooke's law

So the hole in the Earth problem can be expressed as an oscillating spring and, if the spring is perfect (no internal friction/fatigue) it will oscillate until stopped by an external force.

Rap

Thank you. You're absolutely correct.

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rap wrote-

Quote:

So the hole in the Earth problem can be expressed as an oscillating spring and, if the spring is perfect (no internal friction/fatigue) it will oscillate until stopped by an external force.

Everybody knows that don't they. Dematerialised material acts like that.

It seems to me to be an application of cosmological conditions to conditions on earth. As such, pointless.

Those deep space probes will eventually "come to rest" somewhere won't they?

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spendius wrote:

As such, pointless.

Galileo first proposed this solution to the "hole in the earth" problem. I guess by spendi's account Galileo was pointless.

Spendi you are, in effect, a modern 15th century luddite.

Rap

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I would have thought rap that with no internal friction or fatigue the oscillations would continue for ever with the body returning to its original start position after going and coming back assuming no imperfections in the sphere.

And the Luddites have not yet been proved to be in error from any sort of evolutionary position. A luxurious lifestyle, if that is what we have, is not necessarily a guarantee of the survival of our kind and many say it is a positive danger.

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spendius wrote:

I would have thought rap that with no internal friction or fatigue the oscillations would continue for ever with the body returning to its original start position after going and coming back assuming no imperfections in the sphere.

That is what Degrasse, Galileo and, in my own bumbling way, I have been saying.

And for the sake of polar bears, if luddites had their way it would have been to their benefit, as the age of coal and, for that matter, oil would never occur.

Rap

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It's that phrase "no internal friction or fatigue" that got me thinking. As those forces are operative, outside of dematerialised matter, it would come to rest, vis-a-vis the earth I mean, hurtling around the sun and spinning slowly.

Wouldn't it? I only meant it wouldn't come to rest if there were no internal forces or fatigue involved.

Which there would be and if you dropped down the hole as I would ring up a mate in Australia to tell him to catch you with a hook and take you for a pint or two with some frisky Sheilas. I'd tell him that you would be as thirsty as Tantalus.

I would have the time wouldn't I?

You could check it out with a dog first.

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Re: Silly Thought Experiment

Shapeless wrote:

Suppose there were a large hole in the ground, one that went down, down, down--straight through the earth's core, right to the other side of the planet. Suppose someone were to jump into the hole. What would happen?

They'd get stuck in the middle . . . and burned to a crisp . . . you go first . . .

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Quote:

And for the sake of polar bears, if luddites had their way it would have been to their benefit, as the age of coal and, for that matter, oil would never occur.

Some people think that more than polar bears would be better off without coal and oil.

If all your banks crash and the DOW hits the dong button you might well come around to that view yourself.

You had better not let that happen then had you?

Some might think Galileo was a heretic after all. Such would be the mess. The financial system is well known to function without spiritual guidance. It sees that as a hindrance. Just as Galileo did.

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raprap wrote:

Now I'll assume that the density (rho) of the earth is constant and assume the earth is round.

If we took into account the fact that the earth is not perfectly spherical, would this significantly alter the results or would it throw a few variables askew to a basically negligible degree?

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Re: Silly Thought Experiment

Setanta wrote:

They'd get stuck in the middle . . . and burned to a crisp . . . you go first . . .

Sure, I'll go. I'm fairly certain I saw a white rabbit jump in just a moment ago, and I confess that curiosity is getting the better of me.

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Shapeless wrote:

raprap wrote:
Now I'll assume that the density (rho) of the earth is constant and assume the earth is round.

If we took into account the fact that the earth is not perfectly spherical, would this significantly alter the results or would it throw a few variables askew to a basically negligible degree?

Not significant to the first five or so orders---at least for the spherical assumption. This is a third or fourth order assumption. The part about not turning to a crispy critter as the falling body neared a radiant heat source at several thousand degrees is the first order assumption. The second order assumption, is probably 'no rotation.'

Interestingly, since the ossicilating body is, in effect, a really long pendulum the body careening off of the sides of the walls if an axial hole could be used to estimate rotational velocity, a Focault's pendulum in a bizarre way.

Rap

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Silly Thought Experiment

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