Anderson Video - Hole Through the Center of the Earth

Professor Anderson
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>> Hello, class, Professor Anderson here. Let's take a look at a hypothetical. Let's say we dig a hole all the way through the center of the earth and then we jump in it, okay? We're standing here and we're going to jump into that hole and see what happens. What we said a minute ago, was the force of gravity at the center of the earth is equal to zero, okay? And so, the question is, when I jump in, I of course, have a force mg like that. When I get to the center, the force is equal to zero. What about when I get to the other side? And will I get to the other side? Well, one way to think about these problems, is just reverse the picture. If I rotate this whole picture around 180 degrees and now I jump in from here, I'm certainly going to fall with the exact same force, mg. So, what happens to this first guy? What do you think, Thomas? When he jumps in, does he stop at the center? >> No because he has -- the gravity never actually becomes positive. >> Exactly right. Gravity was mg pointing down. It goes to zero, but it's pointing down the whole way. It of course, gets smaller and smaller, eventually coming to zero. But it never points the other way, until he gets on the other side of the center. And once he gets on the other side of the center, by symmetry, that force has to get bigger and bigger in magnitude, pointing back towards the center. So, the guy that jumps in the hole here, what is he going to do? He is going to accelerate and still accelerate and still accelerate and still accelerate, all the way down to here. And then his acceleration goes away. And then he's moving at a really high speed, okay? When he gets to that position, he is flying at some high-speed v. And then the force changes direction. And so, he starts to decelerate and decelerate and decelerate. And he just gets to the other side and his feet stick out the hole. And all of sudden he starts to fall again. And so, what does that person do? He goes through and then he comes back. And then he goes through and then he comes back. And he just keeps going up and down forever and ever and ever. Unless, we include air resistance. If there's air in the tunnel, that air is dissipating that energy. It's going to slow him down. We're also assuming he doesn't, like, hit the sides of the tunnel as he flies through, you know. That would tend to slow you down quite a bit, as well. If there's no air resistance, if there's nothing to dissipate that energy, he would just go up and down like that forever and ever and ever. That's kind of weird to think about, right? And this is a good argument for -- in physics, what we like to do often is employ symmetry. There has to be symmetry of the problem. When one person jumps in, it has to look exactly the same as somebody jumping in from the other side. Because everything else about the problem is in fact symmetric. Good question.
>> Hello, class, Professor Anderson here. Let's take a look at a hypothetical. Let's say we dig a hole all the way through the center of the earth and then we jump in it, okay? We're standing here and we're going to jump into that hole and see what happens. What we said a minute ago, was the force of gravity at the center of the earth is equal to zero, okay? And so, the question is, when I jump in, I of course, have a force mg like that. When I get to the center, the force is equal to zero. What about when I get to the other side? And will I get to the other side? Well, one way to think about these problems, is just reverse the picture. If I rotate this whole picture around 180 degrees and now I jump in from here, I'm certainly going to fall with the exact same force, mg. So, what happens to this first guy? What do you think, Thomas? When he jumps in, does he stop at the center? >> No because he has -- the gravity never actually becomes positive. >> Exactly right. Gravity was mg pointing down. It goes to zero, but it's pointing down the whole way. It of course, gets smaller and smaller, eventually coming to zero. But it never points the other way, until he gets on the other side of the center. And once he gets on the other side of the center, by symmetry, that force has to get bigger and bigger in magnitude, pointing back towards the center. So, the guy that jumps in the hole here, what is he going to do? He is going to accelerate and still accelerate and still accelerate and still accelerate, all the way down to here. And then his acceleration goes away. And then he's moving at a really high speed, okay? When he gets to that position, he is flying at some high-speed v. And then the force changes direction. And so, he starts to decelerate and decelerate and decelerate. And he just gets to the other side and his feet stick out the hole. And all of sudden he starts to fall again. And so, what does that person do? He goes through and then he comes back. And then he goes through and then he comes back. And he just keeps going up and down forever and ever and ever. Unless, we include air resistance. If there's air in the tunnel, that air is dissipating that energy. It's going to slow him down. We're also assuming he doesn't, like, hit the sides of the tunnel as he flies through, you know. That would tend to slow you down quite a bit, as well. If there's no air resistance, if there's nothing to dissipate that energy, he would just go up and down like that forever and ever and ever. That's kind of weird to think about, right? And this is a good argument for -- in physics, what we like to do often is employ symmetry. There has to be symmetry of the problem. When one person jumps in, it has to look exactly the same as somebody jumping in from the other side. Because everything else about the problem is in fact symmetric. Good question.