Anderson Video - Gravity at the Center of the Earth

Professor Anderson
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>> Hello class. Professor Anderson here. Let's talk about the gravity that you experience at the center of the Earth. Okay? Let's pretend somehow you dug a hole and you got yourself to the center of the Earth and you stood there and you wanted to measure the gravitational force that is acting on you. What do you guys think? What do you think the gravity is at the center of the Earth? Let's say you did this experiment, you dug out a hole, you stood in the middle of that thing and you said all right, what do I feel? What do you think? Yeah, Ryan, what do you think? What's the gravity at the center of the Earth? >> I don't know an exact number but it's going to be a lot stronger than the 9.8. >> Okay. A lot stronger than 9.8. why do you think it's going to be a lot stronger than 9.8? >> Because your radius isn't going to be like on the crust, it's going to be zero since you're right in the middle. >> Okay. Since we know that gravity goes like one over r squared, if I go down to r equals zero, it seems like it should go up a lot, right? And in fact shouldn't it go to like infinity if we're at the center of the Earth? Okay that's not quite right because why? >> Well I think, like based on the equation you showed us earlier like r squared was on the bottom so if that technically is zero, you're going to divide everything by zero. >> Right. >> So the force should be zero. >> Well no if it's dividing by zero then it's going high right? It's [inaudible] right? And that is in fact what happens if I take all that matter and condense it into a very small point. If I took the entire Earth and condensed it into one point, right, I would need something bigger than the Earth to make this of course. But let's say I had something that was ten times the mass of the sun and I compressed it all into one point. Those do exist in the universe. What are those things called? >> Black holes? >> Black holes, right? That's a black hole. And the force of gravity increases a whole lot when you get near a black hole. But this is a little different right? Because I in fact carved out a little space in there so I'm not really up against the mass of the Earth. I'm standing in the center of this hole. Any other thoughts on what the gravitational force is if I do that? Any other guesses out there? Come on, let's come up with some wild guesses. Let's go back to this picture for a second. If I'm standing here at the surface of the Earth, what is the force of gravity on me? Thomas, what's the force of gravity on me up there? >> Negative G. >> Okay. Negative we draw with an arrow down. It's got a value of G after you multiply it by the mass of course. And so it's MG. What about when I'm over here? Austin, what do you think? What's the force there? >> MG then upwards towards the center. >> MG upwards. And look if I'm over here, then it's MG that way and if I'm over there, then it's MG that way. So Austin, what do you think? What's the force at the center of the Earth? >> I would think it would encompass you and cancel each other out. >> Exactly right. The force at the center of the Earth is equal to zero. There is no gravity there. Specifically there's no gravitational force. Why is that? Because every chunk of the Earth is pulling on you, but I have a chunk of the Earth that's pulling me that way. And I have a chunk of the Earth that's pulling me that way. And I have a chunk that's pulling me that way. And I have a chunk that's pulling me that way. And they all cancel out. And so if you are in this little cavity at the center of the Earth, there is in fact no gravity at all which is kind of cool right? We should do that. That'd be a great laboratory. I heard it's a little hot down there though, so, it might be hard to get there. Okay, let's ask a follow up question now. We know that the force due to gravity as a function of radius eventually is going to do this. If this is the radius of the Earth, it's going to fall off like one over r squared. Right? If I start at the surface of the Earth and I go up, it's going to fall off like one over R squared. But what about in between the center of the Earth and the surface of the Earth? We know the surface of the Earth has a force equal to that. We know the center of the Earth has a force equal to zero. How does it get from there to there? It turns out-- this is a complicated proof but not impossible-- it turns out it is linear. It increases linearly as you go up this tunnel towards the surface of the Earth. It reaches a maximum when you are at the surface of the Earth and then it falls off again like one over R squared as you go away from the surface of the Earth. And you can prove this using something like Gauss's Law. So if you keep going in physics, look out for something called Gauss's Law. It's kind of a nice little proof. All right, questions? Any abstract questions about doing experiments like this? Yeah, Thomas? >> Since the gravity's decreasing as you get to the center of the Earth, could you theoretically like, if you had a humongous hole and heat wasn't an issue, could you jump through the center of the-- jump through one side of the Earth and would you stop in the middle or would you just keep going to the other side like [inaudible]? >> Okay, excellent question. Let's take a look at that picture and see if we can make some sense of it.
>> Hello class. Professor Anderson here. Let's talk about the gravity that you experience at the center of the Earth. Okay? Let's pretend somehow you dug a hole and you got yourself to the center of the Earth and you stood there and you wanted to measure the gravitational force that is acting on you. What do you guys think? What do you think the gravity is at the center of the Earth? Let's say you did this experiment, you dug out a hole, you stood in the middle of that thing and you said all right, what do I feel? What do you think? Yeah, Ryan, what do you think? What's the gravity at the center of the Earth? >> I don't know an exact number but it's going to be a lot stronger than the 9.8. >> Okay. A lot stronger than 9.8. why do you think it's going to be a lot stronger than 9.8? >> Because your radius isn't going to be like on the crust, it's going to be zero since you're right in the middle. >> Okay. Since we know that gravity goes like one over r squared, if I go down to r equals zero, it seems like it should go up a lot, right? And in fact shouldn't it go to like infinity if we're at the center of the Earth? Okay that's not quite right because why? >> Well I think, like based on the equation you showed us earlier like r squared was on the bottom so if that technically is zero, you're going to divide everything by zero. >> Right. >> So the force should be zero. >> Well no if it's dividing by zero then it's going high right? It's [inaudible] right? And that is in fact what happens if I take all that matter and condense it into a very small point. If I took the entire Earth and condensed it into one point, right, I would need something bigger than the Earth to make this of course. But let's say I had something that was ten times the mass of the sun and I compressed it all into one point. Those do exist in the universe. What are those things called? >> Black holes? >> Black holes, right? That's a black hole. And the force of gravity increases a whole lot when you get near a black hole. But this is a little different right? Because I in fact carved out a little space in there so I'm not really up against the mass of the Earth. I'm standing in the center of this hole. Any other thoughts on what the gravitational force is if I do that? Any other guesses out there? Come on, let's come up with some wild guesses. Let's go back to this picture for a second. If I'm standing here at the surface of the Earth, what is the force of gravity on me? Thomas, what's the force of gravity on me up there? >> Negative G. >> Okay. Negative we draw with an arrow down. It's got a value of G after you multiply it by the mass of course. And so it's MG. What about when I'm over here? Austin, what do you think? What's the force there? >> MG then upwards towards the center. >> MG upwards. And look if I'm over here, then it's MG that way and if I'm over there, then it's MG that way. So Austin, what do you think? What's the force at the center of the Earth? >> I would think it would encompass you and cancel each other out. >> Exactly right. The force at the center of the Earth is equal to zero. There is no gravity there. Specifically there's no gravitational force. Why is that? Because every chunk of the Earth is pulling on you, but I have a chunk of the Earth that's pulling me that way. And I have a chunk of the Earth that's pulling me that way. And I have a chunk that's pulling me that way. And I have a chunk that's pulling me that way. And they all cancel out. And so if you are in this little cavity at the center of the Earth, there is in fact no gravity at all which is kind of cool right? We should do that. That'd be a great laboratory. I heard it's a little hot down there though, so, it might be hard to get there. Okay, let's ask a follow up question now. We know that the force due to gravity as a function of radius eventually is going to do this. If this is the radius of the Earth, it's going to fall off like one over r squared. Right? If I start at the surface of the Earth and I go up, it's going to fall off like one over R squared. But what about in between the center of the Earth and the surface of the Earth? We know the surface of the Earth has a force equal to that. We know the center of the Earth has a force equal to zero. How does it get from there to there? It turns out-- this is a complicated proof but not impossible-- it turns out it is linear. It increases linearly as you go up this tunnel towards the surface of the Earth. It reaches a maximum when you are at the surface of the Earth and then it falls off again like one over R squared as you go away from the surface of the Earth. And you can prove this using something like Gauss's Law. So if you keep going in physics, look out for something called Gauss's Law. It's kind of a nice little proof. All right, questions? Any abstract questions about doing experiments like this? Yeah, Thomas? >> Since the gravity's decreasing as you get to the center of the Earth, could you theoretically like, if you had a humongous hole and heat wasn't an issue, could you jump through the center of the-- jump through one side of the Earth and would you stop in the middle or would you just keep going to the other side like [inaudible]? >> Okay, excellent question. Let's take a look at that picture and see if we can make some sense of it.