If one could (Theoretically) dig a hole straight thru the Earth with a shovel, would you come out feet first on the other side?
What about digging thru a smaller body, like an asteroid, where heat and pressure, are not a factor?
The answer depends on how far you want to take your theoretical argument.
The gravitation field in a solid sphere is computed using the sphere below you. So, if you on a 6000 kilometer radiused sphere of rock and iron, and you’re 2000 kilometers into it, the field is computed with the 4000 kilometer radiused sphere below. So, assuming a uniform density for the earth, roughly the graviational acceleration (g’) would be (quick calculations omitted):
g’ = g (r’/r) = 9.81(4/6) = 6.54 M/s/s
r is radius of earth (I’m calling it 6000 kilometers)
g is sea level graviational acceleration 9.81 M/s/s
r’ radius when you’re 2000 K in a hole (4000 K)
So, as you dig into the Earth, your weight drops off linearly with depth. At the center, you weight nothing. As you continue digging, gravity will start pulling you back toward the center. It would be as if you were digging straight up.
Now, here’s where the limits of theory sets in. If you assume you can dig straight up with a shovel, with your feet planted on the ceiling of the shaft, and some provision to support you, yes you will come out feet first. Otherwise, you’ll come out head first.
On asteroids, there’s so little gravity that your orientation would be based more on how you arrange for an “equal and opposite” force to counteract the force require to push your spade in the rock.
Yup.
Just because you’ve gone al the way through a planet doesn’t mean the laws of geometry change. If you dig a hole all the way through the earth to China (assuming that you start on the side diametrically opposite China) and go through it you will, in fact, pop out on the other side feet first, surprising the Chinese family eating dinner on the other side.
Of course, you can’t do it because of just the things you mention – heat, pressure, the tendency of the sides of the hole to all back in, a molten core. Little things like that.
Nonetheless, the idea of a hole through the earth is a fascinating one. A common problem given to undergraduates is to ask what the motion of something dropped through such a hole in the earth is. Ignoring heat, pressure, etc. you find that you oscillate back and forth through the center, slowing down through friction with the air. I imagine you would have a SERIOUS case of ears popping.
If you drill a hole through the earth that doesn’t go through the center and put down frictionless rails you have a motorless transporation system – gravity pulls you down, then momentum carries you up the other side until you end up at the other end of the hole. The time for such a journey turns out to be independent of the length of the hole (assuming a straight path, and that all journeys are on the same size sphere).
There’s a bit on this in one of Martin Gardner’s books – I think it’s the chapter in “Fads and Falacies in the Name of Science” that’s devoted to hollow earth theories. Gardner says that around the turn of the last century there was quite a bit of interest in this question, with lurid pictures of holes through the earth appearing in popular magazines.
I imagine it would be pretty similar, if less dramatic, if you dropped through a hole in an asteroid. You’d fall a lot slower, and if the asteroid is not round, your motion would be more eccentric.
If you dug a hole straight through the earth, you would obviously be captured in Hell, and ruled over by Satan for eternity. Now, who wouldn’t want that?
From the US: If you went straight through, got a pass to go through Hell, you’d come out in the Indian Ocean west of Australia.
Except for a place in Colorado and another nearby state, which are located opposite a couple tiny South Pacific islands. So choose your starting place carefully, and bring lots of levels and plumb bobs.
CalMeacham said:
What do you mean by that? The laws of geometry don’t change, but the direction gravity pulls will change. Thus, as pointed out above, when you reach the center you reverse and are digging up, not down.
From looking at my cheapo atlas at the office:
If you were digging in northern Alaska, you would end up in Antarctica.
If you were digging in Hawaii, you would end up in Africa.
Unca Cece covered this general topic in What if you fell into a tube through the earth?, but never got around to figuring out how long it would take to fall from one end of the hole to the other. Can anyone help him out?
About 42 minutes in each direction, if it’s safe to make the simplifying assumptions that the density of the earth is uniform, that the earth is a perfect sphere, and that the earth isn’t spinning. (None of those assumptions are true, of course, but I don’t think it will change the answer very much). All physics is done in the tradition of “Assume the cow is a sphere . . .”
T=(3pi/4GD)[sup]½[/sup]
My quibble is with your first sentence, there. You did your calculation with a sphere of constant density, I guess, but for the real Earth, your weight does not drop off linearly with depth. In fact, it’s nearly constant down to the core-mantle boundary, where there is an increase.
If i am digging up, what am i standing on?
If I am digging up, what am I standing on?
BTW, thank you for the great responses. This place truly is the home of some very smart people.
madd1, what are you standing on? Well, I would assume you brought some sort of equipment to support you, probably a platform that mounts to the sides of the hole, or maybe climbing gear.
The point is that “down” would be toward the center of the earth. Once you pass the center, down reverses. If you don’t have any support, I don’t know how you plan to dig straight up, but realize that you will need something or you’re stuck.
Of course I’m drastically simplifying lots of details like the density distribution of the earth, the presence of the liquid core, etc.