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If you could travel to the center of the Earth (or any planet), would you be weightless there?

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Prettified the question –  Sklivvz Jan 3 '11 at 14:40
    
Assuming we're ignoring gravity due to other bodies in the planetary system...? –  Nick Pascucci Jan 4 '11 at 8:39
    
@codeMonk it would only matter if you consider the 2nd order tidal kind of effects from those bodies. –  AlanSE Aug 1 '11 at 17:19
    
This question and my answer are closely related, though I don't think this question is an exact duplicate. –  Keith Thompson Sep 12 '12 at 0:57
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4 Answers 4

Correct. If you split the earth up into spherical shells, then the gravity from the shells "above" you cancels out, and you only feel the shells "below" you. When you are in the middle there is nothing "below" you.

Refrence from Wikipedia Gauss & Shell Theorem.

{I am using some simplistic terms, but I don't want to break out surface integrals and radial flux equations}


Edit: Although the inside of the shell will have zero gravity classically, it will also have non zero gravity relativistically. At the perfect center the forces may balance out, yielding an unstable solution, meaning that a small perturbation in position will result in forces that exaggerate this perturbation.

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@Noldorin: actually you feel the gravity of all of the Earth's mass at a radius less than your own (otherwise you'd start floating every time you step into a basement). If I remember correctly the gravitational acceleration inside a solid sphere of uniform density is proportional to $r$. –  David Z Jan 3 '11 at 19:17
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@mbq - the equilibrium will be stable not unstable. The forces drop to zero linearly from the surface to the center of mass of the Earth. There won't be a strong restoring force for small displacements but there will be a force that increases linearly towards 1g at the surface. What about this makes you say that it is "very unstable?" To me that implies that any small deviation will result in a force away from the center which is the opposite of what would happen. –  inflector Jan 3 '11 at 21:50
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@mbq: it's not an unstable equilibrium at all. Quite on the contrary it serves as almost perfect harmonic oscillator. This can be best seen by digging a tunnel through the Earth and putting a ball into it. –  Marek Jan 3 '11 at 23:12
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@JustJeff: sure it seems like that would be the case, but if you go through and do the calculation you find that it's actually not. Assuming the shell is perfectly spherical, you won't feel any gravitational force inside it even if you are close to the inner surface. Basically the reason is that the force from the small bit of mass directly above you is balanced out by the force from the large remainder of the mass below you. For details, look at the Wikipedia articles linked in jalexiou's answer. (The one on the shell theorem is probably a little easier to follow) –  David Z Jan 4 '11 at 0:27
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Everybody quit deleting comments! It makes the thread very hard to follow. –  Colin K Feb 28 '11 at 14:39
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The simplest way to think about it is that there is mass all around you in the center of the Earth so you get an equal gravitational "pull" from all directions. The pulls cancel out so you get no acceleration.

If one assumes constant density for the Earth (which isn't strictly speaking true but it is close enough for this illustration) the gravitational acceleration drops linearly from 1g at the surface to 0 at the center of the Earth. So you'd get a zero if you stepped on a scale at the center of the Earth.

The more complicated explanation is that acceleration due to gravity is the derivative of the gravitational potential. This potential is a minimum at the center of the Earth and grows quadratically up to the surface. It then continues to increase at a lower rate. Since at the exact center is flat (like the bottom of a valley), the derivative which is a measure of the rate of change is zero, and there is no acceleration.

Interestingly, even though you would be weightless there, the effects of gravity are highest at the center of the Earth. You get more gravitational time dilation, for example, than you do at the surface.

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I think this answer adds significantly to jalexiou's because it points out that a simple symmetry argument leads to weightless at the center of Earth. Therefore, the specifics of the law of gravity (its inverse-square-ness) are not really important to this question. –  Mark Eichenlaub Jan 4 '11 at 1:02
    
Also since the Earth-Moon revolves around a point 4000km from the centre of the earth you would be weightless in a donut around the centre –  Martin Beckett Sep 13 '12 at 14:51
    
I'd like you to elaborate on your claim of increased time dilation at the center. –  Fernando Oct 26 '12 at 12:56
    
@Fernando The rate of your proper time relative to a distant observer is about $1 + V/c^2$ where $V$ is the gravitational potential (which is negative). Compared to someone far from the Earth (but at the same distance from the Sun), someone at the surface of the Earth would lose about two seconds per century, and someone at the centre, about three. –  Retarded Potential Mar 14 '13 at 20:19
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I like answers that appeal to symmetry, so I answer this one with a question: If you were at the center, which way would you fall? That tells us you could stay floating there.

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You would not be weightless at the center of the Earth. In other words, the Earth does not follow a geodesic. Let me explain.

The Earth is not spherical, it is an oblate spheroid. The acceleration of a uniform non-spherical body in a spherical gravitational field does not follow an inverse square law. The acceleration of the center of mass does not equal the acceleration at the center of mass. An accelerometer fixed at the center of the Earth would read approx 1.75 pgal (1.75e-14 m/$\mathrm{s^2}$), not zero.

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What does "pgal" mean (Google was unhelpful)? And if the acceleration is non-zero at the center of mass, due to the non-uniformity of the Earth's shape, am I correct in assuming there's a point where the acceleration is zero? How far from the geometric center is that point, and in what direction? –  Keith Thompson Sep 12 '12 at 0:56
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An object released at the geocenter would accelerate in the direction opposite to the Sun.<br> –  Nick Sep 13 '12 at 10:18
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The point where the acceleration is zero is about .15 m closer to the sun. If the Sun and Earth were stationary, that point would be .22 m closer to the Sun. –  Nick Sep 13 '12 at 12:06
    
@KeithThompson "gal" is acceleration in cm/s^2, so 'g' = 9.8m/s^2 = 980 gal. pgal is pico-gal. It doesn't get used much in physics but geologists use it for gravimetric measurements –  Martin Beckett Sep 13 '12 at 14:49
    
@Nick: So an object released at the geocenter would fall away from the point of zero acceleration? That seems counterintuitive. –  Keith Thompson Sep 13 '12 at 21:27
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protected by Qmechanic Apr 24 '13 at 16:05

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