The intrepid fellow is taking a break from the hairsploitation circuit and wishes to investigate the nature of gravity. He drills a nice wide hole all the way through the planet, passing through the exact centre, takes his clothes off, rolls about in some mud, and jumps right in.
There’s a method to his madness. The mud dries on his all-over fur in such a manner that each hair has a little muddy dangleberry clinging to the tip, acting like a plumb-bob so that each hair points “down”. Assuming that he doesn’t touch the sides of the hole, and that conditions throughout are conducive to lycanthropic survival (i.e. no molten core, a breathable atmosphere etc.) he will oscillate about the centre until finally air resistance brings him to a halt at the exact centre of the planet. As the planet is of conveniently homogeneous construction the physical centre is also the exact centre of gravity.
What our hirsute hero wishes to know is this: Once at rest at the centre, which direction will all his hairy winnets point in? Will the tagnuts all point inwards, as the centre of gravity of the planet is now somewhere in his stomach, will the earthy clingons all point radiating outwards as the actual mass of the planet is all around him, or will the clinkers be subject to no significant resultant forces at all (like in deep space) given that he is now floating in mid-air? Or, a less appealing prospect, would the gravitational gradients be so steep that close to the centre that he would feel incredibly sick and wish he hadn’t started on the whole wretched adventure?
I normally have a good intuitive grasp of Newtonian physics, but this has got me stumped. Can we save our furry friend all this trouble? Don’t make me go to Mexico!
Fair enough, but it’s a slightly different question than what would happen to the body, or how long would it take, since, if I understand correctly, the question is, in what direction is the pull of the earth’s gravitational pull when one is actually at the centre of the earth.
I’m afraid I can’t answer it, although it’s an interesting question.
I can point the OP to a, er, documentary about the repercussions of messing with the earth’s core… (completely safe for work!)
If you are in the centre of an object, with the mass of the object symmetrically distributed around you, then there is a net gravitational attraction of zero between you and the other object. You’d just float there, and would not be attracted to the side of the tube.
Once at the exact center of the planet, WolfBoy will feel no overall gravitational pull at all from the planet (assuming the planet is perfectly spherical, and the hole is small enough to be ignored).
The mudballs will be (very slightly) attracted more-or-less towards WolfBoy’s center of mass, because of the gravitational attraction from WolfBoy’s body.
One item that has not yet been addressed yet is the gravitational effect of other bodies in the solar system.
Assuming that the wolf boy weighs in at 150 lbs, and the mudballs are on average 8" from his center of gravity, the gravitational effect of the sun will be around 54000 times stronger on the balls than wolfboy’s own mass. The moon will also exert a stronger pull, although only 300 times stronger.
Wolfboy’s location in the center of the spherical earth will cancel the effect of any terrestial gravity, and so the mudballs will attracted towards the sun.
See posts 32 and 32 in Exapno Mapcase’s third link.
The Sun is also pulling on the Earth. The net effect of the Sun is its tidal effect, which for an object the size of the boy is very tiny, smaller than the direct force of the Sun (which I think is what you’re calculating, but I’m too lazy to verify) by a factor that’s on the order of the boys size divided by the distance to the Sun.
Would it make a difference that the hair is not at the center of gravity of either wolfboy or the Earth? Let’s say, for example that wolfboy’s body is centered at the center of the earth and his head is pointing towards the north. That means a hair on his head is about three feet from the center of the Earth so there is more mass to its south than there is to its north and that hair should be attracted to the south. A hair on wolfboy’s foot would have the opposite effect and be attracted to the north. Overall, every hair should be attracted in the direction of wolfboy’s center.
Yes, I calculated the direct gravitational force of the sun. I don’t follow, however, why the sun’s gravitational pull on the earth is a consideration. Please help me figure out where I’m wrong… due to geometrical symmetry, wolfboy is weightless wrt the earth; he is neither touching it, nor gravitationally attracted to it, and we can remove it from our free body system. Wolfboy is orbiting the sun (at the same speed as the earth) due to its gravity. This gravity may induce a tiny tidal effect on wolfboy’s skin, but it also produces a direct first order effect on the nearby mud particles, which are tied to wolfboy by ultrastrong, massless, wolfhair tethers.
And writing to this point, I think I see the problem in my understanding - the sun’s gravitational pull on the mud particles results in the particles orbiting the sun (at the same speed as wolfboy and the earth). From this orbiting frame of reference there is no observable increased pull towards the sun.
Correct. There’d still be a slight difference, since the mudballs aren’t exactly the same distance from the Sun as is Wolfboy’s center of mass, but that slight difference is the tidal effects.
Cool, thanks. Is the answer then that the mud particles will point at the moon? It isn’t as intuitively apparent to me that the moon’s effects disappear in the frame of reference.
Thank-you to all for the answers, and Mr Mapcase’s links were most entertaining, but our Mexican wolf boy - let us call him Dave - remains unenlightened.
As this is a thought experiment and we can do all kinds of crazy things like playing with torches on trains travelling at the speed of light and suchlike, let us assume the following:[ul]
[li]There are no other gravitational forces at play apart from Dave and the planet. Dave’s universe is mostly empty now that he’s left Mexico.[/li][li]The planet is completely motionless.[/li][li]We can conveniently ignore the dirty great hole Dave made (though this will make a small difference, as has been rightly pointed out). If you like, the planet can exhibit perfect symmetry in all planes, and Dave is magically transported somehow into a spherical void in the centre, no tunnels involved.[/li][li]Dave has an unfortunate glandular problem, and is perfectly spherical. But still hairy. Bummer.[/li][/ul]
As gravitational forces of any mass can be modelled as a single point source of zero dimension, infinite density and finite mass, would this still apply if this single point source were inside Dave’s unfortunate innards? Or does this model break down if the centre of gravity is inside the observer? Especially as the actual mass encompasses Dave, as opposed to giving him kicking indigestion.
Is being pulled in all directions simultaneously the same as not being subject to any external forces at all? While being in the centre of a planet and being in deep space may both leave a body floating, it doesn’t quite seem the same to me. And gravity can be counter-intuitive, like the example of someone holding aloft a helium balloon inside a car while it’s cornering hard. Which direction does the balloon move in? That example is easier to prove, wrapping oneself around the centre of gravity of a planetary mass is a trickier practical experiment.
The model breaks down within the volume of the mass. As you guessed, being pulled every-which-way simultaneously is the same as not being pulled at all: http://en.wikipedia.org/wiki/Shell_theorem
I’ve always been curious, and since the question has already been asked and answered (for the nth time), let me rephrase things a bit, if no one minds:
What happens if the tube is off center? My understanding is that even if the tube is off center, the effect will still be the same…you’ll come to rest basically in the center of the tube, half way between where the straight line bisects the sphere. Correct? Also, if the tube is frictionless, you will still end up oscillating between the two ends indefinitely…yes?
(completely safe for work!)
