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Main
Date: 29 Apr 2007 05:06:22
From: moorthy
Subject: gravity below the surface of earth
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i want to know, why acceleration due to gravity (g) decreases as we
move below the surface of the earth?
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Date: 01 May 2007 20:12:23
From: Dr J R Stockton
Subject: Re: gravity below the surface of earth
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In sci.astro.amateur message <bvqdndldZMx98qrbnZ2dnUVZ_j6dnZ2d@comcast.c
om >, Tue, 1 May 2007 10:50:39, Shawn <s.currynospam@bresnan.net> posted:
>
>That occurred to me too. It brings up the question, which is way
>beyond the "Physics for Biology Majors" (one step up from "Physics for
>Poets") that I took in college: If you could drill a tunnel such that
>you wouldn't touch the sides as you fell, how would the tunnel curve,
>relative to the earth? Also: Would it look substantially different if
>it were drilled at the equator vs. 40 deg N in Boulder, CO, where I
>took "Physics for Biology Majors? ;-)
Your path would be an ellipse centred on the Earth's centre, AISB. In
non-rotating co-ordinates, the tunnel would be an ellipse too.
Since the period of the orbit is presumably incommensurate with a
sidereal day, a new tunnel section would be requited for each orbit.
Therefore, starting from the Equator, the tunnels become a very deep
thin cutting, reaching to within probably about one eighteenth of the
Earth's radius of the centre.
Starting from 40 degrees of latitude, ISTM that all the material within
the tangents from that latitude to the R/18 sphere would need to be
removed; but my visualisation is unclear.
--
(c) John Stockton, Surrey, UK. ?@merlyn.demon.co.uk Turnpike v6.05 MIME.
Web <URL:http://www.merlyn.demon.co.uk/ > - FAQqish topics, acronyms & links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News.
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Date: 01 May 2007 14:09:17
From: Dr J R Stockton
Subject: Re: gravity below the surface of earth
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In sci.astro.amateur message <f16kph$qi0$1@praesepe.isi.edu >, Mon, 30
Apr 2007 22:56:01, Brian Tung <brian@isi.edu > posted:
>
>If there is friction, then yes. If not, you would go all the way
>through the Earth to the antipodes of your starting point, then make it
>all the way back to the starting point, then over again, etc. The
>period of oscillation is often quoted to be about 84 minutes, but I
>*think* that assumes a uniformly dense Earth.
It does - although there must be an infinite number of spherically
symmetrical but otherwise implausible mass distributions that give the
same result.
Consider a homogeneous sphere, and another of the same mass but with the
mass concentrated at the centre. In the second case the acceleration
towards the centre is always greater than the first, therefore the
period is less. Now consider a spherical shell of the same mass; the
period is, I think, infinite. A combination of the two, in the right
proportion, will give any desired period.
To show that it is 84 minutes - we know that the period of low Earth
orbit is 84 minutes, and can easily prove that without calculus.
Compare a satellite in low polar orbit, and a body falling in a tunnel
between the poles of a homogeneous Earth.
When at either pole, they have the same acceleration towards the
equatorial plane - being in the same location. It's easy to see that
the acceleration towards the equatorial plane is in each case linear
with distance to that plane; therefore it is the same in each case for
the same distance from the plane, and the periods must match.
In effect, a LEO satellite is simply mimicking a pair of objects falling
independently in internal tunnels, but so phased as not to need the
tunnels.
Particles sliding in frictionless chordal tunnels in such a body have
the same period, too.
Particles orbiting in, but not touching, tunnels in a homogeneous body
follow ellipses with their centres at the centre of the body, with (I
think) that period.
The chordal tunnels do not need to reach the surface.
My <URL:http://www.merlyn.demon.co.uk/gravity1.htm#MiS > refers.
--
(c) John Stockton, Surrey, UK. ?@merlyn.demon.co.uk Turnpike v6.05 MIME.
Web <URL:http://www.merlyn.demon.co.uk/ > - FAQqish topics, acronyms & links;
Astro stuff via astron-1.htm, gravity0.htm ; quotings.htm, pascal.htm, etc.
No Encoding. Quotes before replies. Snip well. Write clearly. Don't Mail News.
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Date: 30 Apr 2007 23:39:48
From: canopus56
Subject: Re: gravity below the surface of earth
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On Apr 30, 10:21 am, "Peter Webb"
<webbfam...@DIESPAMDIEoptusnet.com.au > wrote:
> "Greg Neill" <gneill...@OVEsympatico.ca> wrote in message
> Just for the record, you would be "weightless" from the moment you jumped,
> and you would feel no different at the centre to anywhere else in the
> tunnel.- Hide quoted text -
Just for the record, since the poster said he only drilled the hole to
the center of the Earth - he'd be carried to the bottom by his
momentum and go "splat", since he forget to drill the other half of
the hole through to China. -:) - Canopus56
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Date: 01 May 2007 05:56:27
From: Pieter
Subject: Re: gravity below the surface of earth
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No - actually I did say I drilled it "through" the center of the earth -
not "to" the center. I was assuming a hole from here in a perfect vertical
through the center and out the other side.
I would guess you would, as you suggest, continue past center, then slow,
stop, reverse and fall back towards the point at which you started. The
forces of air friction would rapidly slow you until you can to rest at the
exact center of the earth. Here gravity is exactly the same in all
directions so you would be stuck "floating" at that point.
"canopus56" <canopus56@yahoo.com > wrote in message
news:1178001588.412876.10980@p77g2000hsh.googlegroups.com...
> On Apr 30, 10:21 am, "Peter Webb"
> <webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
>> "Greg Neill" <gneill...@OVEsympatico.ca> wrote in message
>> Just for the record, you would be "weightless" from the moment you
>> jumped,
>> and you would feel no different at the centre to anywhere else in the
>> tunnel.- Hide quoted text -
>
> Just for the record, since the poster said he only drilled the hole to
> the center of the Earth - he'd be carried to the bottom by his
> momentum and go "splat", since he forget to drill the other half of
> the hole through to China. -:) - Canopus56
>
>
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Date: 01 May 2007 19:55:28
From: William Hamblen
Subject: Re: gravity below the surface of earth
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On Tue, 1 May 2007 05:56:27 -0400, "Pieter" <pvcl@*nospam*plitch.com >
wrote:
>No - actually I did say I drilled it "through" the center of the earth -
>not "to" the center. I was assuming a hole from here in a perfect vertical
>through the center and out the other side.
>
>I would guess you would, as you suggest, continue past center, then slow,
>stop, reverse and fall back towards the point at which you started. The
>forces of air friction would rapidly slow you until you can to rest at the
>exact center of the earth. Here gravity is exactly the same in all
>directions so you would be stuck "floating" at that point.
You'd need one strong pipe to line the bore hole to keep it open. Not
to mention asbestos underwear.
Because the density of the core is greater than the density of the
crust, for the first part of your drop g in the bore hole will
increase as you get closer to the center of the Earth.
Bud
--
The night is just the shadow of the Earth.
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Date: 01 May 2007 07:16:19
From: SkySea
Subject: Re: gravity below the surface of earth
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Just to continue the sillines... I imagine the tunnel would have to be
curved to accommodate the rotation of the earth. Otherwise, you'd
scrape somewhere onthe way down.
And of course, we're ignoring terminal velocity altogether.
>"Pieter" <pvcl@*nospam*plitch.com> wrote:
>No - actually I did say I drilled it "through" the center of the earth -
>not "to" the center. I was assuming a hole from here in a perfect vertical
>through the center and out the other side.
>
>I would guess you would, as you suggest, continue past center, then slow,
>stop, reverse and fall back towards the point at which you started. The
>forces of air friction would rapidly slow you until you can to rest at the
>exact center of the earth. Here gravity is exactly the same in all
>directions so you would be stuck "floating" at that point.
>
>"canopus56" <canopus56@yahoo.com> wrote in message
>news:1178001588.412876.10980@p77g2000hsh.googlegroups.com...
>> On Apr 30, 10:21 am, "Peter Webb"
>> <webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
>>> "Greg Neill" <gneill...@OVEsympatico.ca> wrote in message
>>> Just for the record, you would be "weightless" from the moment you
>>> jumped,
>>> and you would feel no different at the centre to anywhere else in the
>>> tunnel.- Hide quoted text -
>>
>> Just for the record, since the poster said he only drilled the hole to
>> the center of the Earth - he'd be carried to the bottom by his
>> momentum and go "splat", since he forget to drill the other half of
>> the hole through to China. -:) - Canopus56
=============
- Dale Gombert (SkySea at aol.com)
122.38W, 47.58N, W. Seattle, WA
http://flavorj.com/~skysea
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Date: 01 May 2007 11:53:24
From: Chris McMahan
Subject: Re: gravity below the surface of earth
|
Also, you would slow down as you descend as a large bulk of the
Earth's mass beings pulling at you from above.
- Chris
SkySea <SkySea@aolREMOVE.com > writes:
> Just to continue the sillines... I imagine the tunnel would have to be
> curved to accommodate the rotation of the earth. Otherwise, you'd
> scrape somewhere onthe way down.
>
> And of course, we're ignoring terminal velocity altogether.
>
>
>>"Pieter" <pvcl@*nospam*plitch.com> wrote:
>>No - actually I did say I drilled it "through" the center of the earth -
>>not "to" the center. I was assuming a hole from here in a perfect vertical
>>through the center and out the other side.
>>
>>I would guess you would, as you suggest, continue past center, then slow,
>>stop, reverse and fall back towards the point at which you started. The
>>forces of air friction would rapidly slow you until you can to rest at the
>>exact center of the earth. Here gravity is exactly the same in all
>>directions so you would be stuck "floating" at that point.
>>
>>"canopus56" <canopus56@yahoo.com> wrote in message
>>news:1178001588.412876.10980@p77g2000hsh.googlegroups.com...
>>> On Apr 30, 10:21 am, "Peter Webb"
>>> <webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
>>>> "Greg Neill" <gneill...@OVEsympatico.ca> wrote in message
>>>> Just for the record, you would be "weightless" from the moment you
>>>> jumped,
>>>> and you would feel no different at the centre to anywhere else in the
>>>> tunnel.- Hide quoted text -
>>>
>>> Just for the record, since the poster said he only drilled the hole to
>>> the center of the Earth - he'd be carried to the bottom by his
>>> momentum and go "splat", since he forget to drill the other half of
>>> the hole through to China. -:) - Canopus56
>
> =============
> - Dale Gombert (SkySea at aol.com)
> 122.38W, 47.58N, W. Seattle, WA
> http://flavorj.com/~skysea
--
(. .)
=ooO=(_)=Ooo=====================================
Chris McMahan
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Date: 01 May 2007 16:06:32
From: Chris L Peterson
Subject: Re: gravity below the surface of earth
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On Tue, 01 May 2007 11:53:24 -0400, Chris McMahan
<first_initiallastname@one.dot.net > wrote:
>Also, you would slow down as you descend as a large bulk of the
>Earth's mass beings pulling at you from above.
If there were no air, your velocity would continue to increase during
the entire descent. Until you reach the center, there is always more
mass below you than above you.
If you allow for air, you will slow down as you approach the center
because the terminal velocity decreases. This happens because the
gravitational force decreases and the air density increases. I wouldn't
be surprised if this was a case of over-damped behavior, and you
couldn't even get to the center by falling, just approach it
asymptotically. Keep in mind that the phase of the air would transition
from gas to liquid long before you reached the center.
If there was just a tiny amount of air at the surface, say what you have
on the Moon, the density would increase enough towards the center that
you'd burn up like a meteor!
_________________________________________________
Chris L Peterson
Cloudbait Observatory
http://www.cloudbait.com
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Date: 01 May 2007 13:34:01
From: lal_truckee
Subject: Re: gravity below the surface of earth
|
Chris L Peterson wrote:
>
>
> If there were no air, your velocity would continue to increase during
> the entire descent. Until you reach the center, there is always more
> mass below you than above you.
Well, the acceleration would continue; but not because there's "more
mass" below you. It's because the mass in the spherical shell of radius
greater than your position exactly cancels itself, so you only feel the
mass of the sphere of radius less than your position.
IIRC it's about the first calculation a raw calculus student does that
applies to the real world.
And it's why there's no gravity inside a hollow earth - I don't
understand how Professor Challenger got it wrong. Maybe he didn't really
penetrate to the center of the earth? That Doyle - he always was a liar.
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Date: 01 May 2007 22:50:33
From: Chris L Peterson
Subject: Re: gravity below the surface of earth
|
On Tue, 01 May 2007 13:34:01 -0700, lal_truckee <lal_truckee@yahoo.com >
wrote:
>Well, the acceleration would continue; but not because there's "more
>mass" below you. It's because the mass in the spherical shell of radius
>greater than your position exactly cancels itself, so you only feel the
>mass of the sphere of radius less than your position.
Different ways of saying the same thing. The way you state it is the
simplest way to handle it mathematically, but I was just putting it into
the same sort of words as the person I was replying to.
_________________________________________________
Chris L Peterson
Cloudbait Observatory
http://www.cloudbait.com
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Date: 01 May 2007 10:56:57
From: Shawn
Subject: Re: gravity below the surface of earth
|
Chris L Peterson wrote:
> On Tue, 01 May 2007 11:53:24 -0400, Chris McMahan
> <first_initiallastname@one.dot.net> wrote:
>
>> Also, you would slow down as you descend as a large bulk of the
>> Earth's mass beings pulling at you from above.
>
> If there were no air, your velocity would continue to increase during
> the entire descent. Until you reach the center, there is always more
> mass below you than above you.
>
> If you allow for air, you will slow down as you approach the center
> because the terminal velocity decreases. This happens because the
> gravitational force decreases and the air density increases. I wouldn't
> be surprised if this was a case of over-damped behavior, and you
> couldn't even get to the center by falling, just approach it
> asymptotically. Keep in mind that the phase of the air would transition
> from gas to liquid long before you reached the center.
>
> If there was just a tiny amount of air at the surface, say what you have
> on the Moon, the density would increase enough towards the center that
> you'd burn up like a meteor!
While looking for the answer to the question I asked elsewhere in this
thread (no luck), I found a site that claimed one would end up floating
in the liquefied air only a few tens of miles down, never falling much
past the earth's crust!
Shawn
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Date: 30 Apr 2007 00:31:24
From: Knap
Subject: Re: gravity below the surface of earth
|
above mass vs inside mass ..... unless you are MIN!
moorthy wrote:
> i want to know, why acceleration due to gravity (g) decreases as we
> move below the surface of the earth?
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Date: 30 Apr 2007 07:09:20
From: Pieter
Subject: Re: gravity below the surface of earth
|
So if one were to drill a perfectly straight hole exactly through the center
of a perfectly round earth ( work wih me here, I know all the minor :"but
its not"s), and if one were to jump into this hole feet first, not touching
any side at any time, you would eventually come to rest at the center of
the earth in a state of weightlessness (gravity equal in all directions)??
Would this feel different than weightlessness in spcae where there is little
or no force being exerted dur to the distance of the masses?
"Knap" <gro@mchsi.com > wrote in message news:46357F2C.8E5CC5B7@mchsi.com...
> above mass vs inside mass ..... unless you are MIN!
>
>
>
> moorthy wrote:
>
>> i want to know, why acceleration due to gravity (g) decreases as we
>> move below the surface of the earth?
>
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Date: 30 Apr 2007 22:56:01
From: Brian Tung
Subject: Re: gravity below the surface of earth
|
Pieter wrote:
> So if one were to drill a perfectly straight hole exactly through the center
> of a perfectly round earth ( work wih me here, I know all the minor :"but
> its not"s), and if one were to jump into this hole feet first, not touching
> any side at any time, you would eventually come to rest at the center of
> the earth in a state of weightlessness (gravity equal in all directions)??
If there is friction, then yes. If not, you would go all the way
through the Earth to the antipodes of your starting point, then make it
all the way back to the starting point, then over again, etc. The
period of oscillation is often quoted to be about 84 minutes, but I
*think* that assumes a uniformly dense Earth. If so, it would in fact
be a little less than 84 minutes.
> Would this feel different than weightlessness in spcae where there is little
> or no force being exerted dur to the distance of the masses?
No, it would not. As long as you are in free fall, you are weightless,
even if you are close to a massive object like the Earth. Astronauts
orbiting the Earth are weightless. A child jumping off the monkey bars
is (momentarily) weightless.
Incidentally, unless the hole is drilled from the north or south pole,
a straight hole won't work because of the Earth's rotation. I assume
you're not concerned about that wrinkle. :)
--
Brian Tung <brian@isi.edu >
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.html
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Date: 01 May 2007 10:50:39
From: Shawn
Subject: Re: gravity below the surface of earth
|
Brian Tung wrote:
> Pieter wrote:
>> So if one were to drill a perfectly straight hole exactly through the center
>> of a perfectly round earth ( work wih me here, I know all the minor :"but
>> its not"s), and if one were to jump into this hole feet first, not touching
>> any side at any time, you would eventually come to rest at the center of
>> the earth in a state of weightlessness (gravity equal in all directions)??
>
> If there is friction, then yes. If not, you would go all the way
> through the Earth to the antipodes of your starting point, then make it
> all the way back to the starting point, then over again, etc. The
> period of oscillation is often quoted to be about 84 minutes, but I
> *think* that assumes a uniformly dense Earth. If so, it would in fact
> be a little less than 84 minutes.
>
>> Would this feel different than weightlessness in spcae where there is little
>> or no force being exerted dur to the distance of the masses?
>
> No, it would not. As long as you are in free fall, you are weightless,
> even if you are close to a massive object like the Earth. Astronauts
> orbiting the Earth are weightless. A child jumping off the monkey bars
> is (momentarily) weightless.
>
> Incidentally, unless the hole is drilled from the north or south pole,
> a straight hole won't work because of the Earth's rotation. I assume
> you're not concerned about that wrinkle. :)
That occurred to me too. It brings up the question, which is way beyond
the "Physics for Biology Majors" (one step up from "Physics for Poets")
that I took in college: If you could drill a tunnel such that you
wouldn't touch the sides as you fell, how would the tunnel curve,
relative to the earth? Also: Would it look substantially different if
it were drilled at the equator vs. 40 deg N in Boulder, CO, where I took
"Physics for Biology Majors? ;-)
Assuming no air resistance and negligible relativistic effects.
Shawn
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Date: 30 Apr 2007 07:25:08
From: Greg Neill
Subject: Re: gravity below the surface of earth
|
"Pieter" <pvcl@*nospam*plitch.com > wrote in message
news:4635ce51$0$15169$4c368faf@roadrunner.com...
> So if one were to drill a perfectly straight hole exactly through the
center
> of a perfectly round earth ( work wih me here, I know all the minor :"but
> its not"s), and if one were to jump into this hole feet first, not
touching
> any side at any time, you would eventually come to rest at the center of
> the earth in a state of weightlessness (gravity equal in all directions)??
> Would this feel different than weightlessness in spcae where there is
little
> or no force being exerted dur to the distance of the masses?
Nope. Momentum would carry you past the center and
"up" the other side of the hole. If there were no
air friction you'd have a simple harmonic oscillation,
repeatedly falling down one side and back up the other.
If you allow friction then the amplitude of the
oscillations would decay until you came to rest at the
center. There you would float in a weightless condition
just like in remote space, only you'd have air to breath
(but probably at far too high a pressure for your
comfort).
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Date: 01 May 2007 02:21:06
From: Peter Webb
Subject: Re: gravity below the surface of earth
|
"Greg Neill" <gneillREM@OVEsympatico.ca > wrote in message
news:4635d1b2$0$30411$9a6e19ea@news.newshosting.com...
> "Pieter" <pvcl@*nospam*plitch.com> wrote in message
> news:4635ce51$0$15169$4c368faf@roadrunner.com...
>> So if one were to drill a perfectly straight hole exactly through the
> center
>> of a perfectly round earth ( work wih me here, I know all the minor :"but
>> its not"s), and if one were to jump into this hole feet first, not
> touching
>> any side at any time, you would eventually come to rest at the center of
>> the earth in a state of weightlessness (gravity equal in all
>> directions)??
>> Would this feel different than weightlessness in spcae where there is
> little
>> or no force being exerted dur to the distance of the masses?
>
> Nope. Momentum would carry you past the center and
> "up" the other side of the hole. If there were no
> air friction you'd have a simple harmonic oscillation,
> repeatedly falling down one side and back up the other.
>
> If you allow friction then the amplitude of the
> oscillations would decay until you came to rest at the
> center. There you would float in a weightless condition
> just like in remote space, only you'd have air to breath
> (but probably at far too high a pressure for your
> comfort).
>
>
Just for the record, you would be "weightless" from the moment you jumped,
and you would feel no different at the centre to anywhere else in the
tunnel.
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Date: 29 Apr 2007 08:51:03
From: Brian Tung
Subject: Re: gravity below the surface of earth
|
moorthy wrote:
> i want to know, why acceleration due to gravity (g) decreases as we
> move below the surface of the earth?
Effectively, only the part of the Earth closer to the center than you
are is pulling on you; the pulls from the various parts further than you
are cancel out.
The Earth has a radius of about 6,400 km. When you are at the surface,
all of the Earth's mass is pulling down on you. Although some of the
pull is "sideways," the sideways pulls cancel out and only the downward
pull remains.
If you were to descend 100 km, say, then the part of the Earth above you
is pulling up. There isn't very much Earth pulling up at you, but since
it's relatively close to you (compared to the rest of the Earth), its
effect is somewhat magnified, due to the inverse-square form of the law
of gravity.
It can be shown that if you take a shell made up of the upper 100 km of
the Earth, those pulls cancel out exactly, and only the pull from the
inner part of the Earth--the part with a radius of just 6,300 km--still
remains. The math to show this is a *little* involved, but only a
little.
(The simplest way I know matches every patch on the shell with a
corresponding patch diametrically opposite you. The sizes of both of
the patches is inversely proportional to the square of its distance from
you, so that the pulls of the two patches on you cancel each other out.)
In principle, as you descend further and further, the part of the Earth
closer to the center than you are decreases as the inverse cube of your
distance from the center, whereas gravity is only an inverse-square law.
Since the volume goes down faster than the square of the distance, the
force of gravity upon you decreases as you descend.
In reality, the central parts of the Earth are denser than the outer
parts, so that even though the volume goes down as the inverse cube of
your distance from the center, the mass goes down rather slower than
that. The ultimate effect is that the force of gravity would actually
go *up* slightly as you descended first--only when you reached the
densest parts of the Earth would it then go down.
At the center of the Earth, in reality as in principle, all parts of the
Earth pull out at you, balancing themselves out, so the force of gravity
upon you there is zero.
--
Brian Tung <brian@isi.edu >
The Astronomy Corner at http://astro.isi.edu/
Unofficial C5+ Home Page at http://astro.isi.edu/c5plus/
The PleiadAtlas Home Page at http://astro.isi.edu/pleiadatlas/
My Own Personal FAQ (SAA) at http://astro.isi.edu/reference/faq.html
|
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Date: 29 Apr 2007 05:24:48
From:
Subject: Re: gravity below the surface of earth
|
On Apr 29, 8:06 am, moorthy <cmkmoor...@gmail.com > wrote:
> i want to know, why acceleration due to gravity (g) decreases as we
> move below the surface of the earth?
Because it is difficult for an object to continue falling through
solid rock? :-)
Any object at the very center of the Earth is being acted upon equally
in all directions, therefore it experiences no acceleration. An object
about halfway to the center is being "pulled" upward by the matter
located above it, and there is also less matter below it than there
would be if the object were still at the surface.
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Date: 29 Apr 2007 12:16:51
From: Sam Wormley
Subject: Re: gravity below the surface of earth
|
moorthy wrote:
> I want to know, why acceleration due to gravity (g) decreases as we
> move below the surface of the earth?
>
o there is decreasing mass (and therefor gravitation) below you
o there is increasing mass (and therefor gravitation) above you
o at the center... the would be equal gravitational pull in all
directions.
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