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Anderson Video - Weight on Rotating Earth

Professor Anderson
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>> And what do we already know about circular motion? We know that if an object is moving in a circle, even if it's moving at constant speed v, is that thing accelerating? Yes, it is and we call that centripetal acceleration. And which way does it point? Towards the center of the circle, ok. Centripetal acceleration is towards the center of the circle, we write it with an a sub c, and it's always towards the center of the circle, ok. Remember when we think about acceleration -- acceleration is delta v over delta t. Delta v can be change in speed or it can be change in direction. It doesn't have to be both; it can be one or the other. And if you have either of those then you have an acceleration. In this case, even if we're not changing speed we are changing direction as we go. Because if vi is right there at some later time vf would look like that. It certainly has changed direction. Alright, so centripetal acceleration becomes v squared over r in magnitude and it points toward the circle's center. So if we have an acceleration, we have to have a force. Right, what do we know about Newton's second? Newton's second says sum of the forces has to be the mass times acceleration. In this case there is only one force which is keeping us in the circle. If it was like a car on a road it would be friction, if it's a ball on a string it would be tension, if it was a person in orbit it would be gravity. That is mv squared over r. And the way we write it with a subscript is with an r right there. What we mean by this r is radial force and the sign convention is positive towards the center. This is a little bit different than you might normally think of a radial variable where positive would be out away from the center. Here we're using positive towards the center. This is of course centripetal acceleration, this is the mass, and those two things combined are what we call centripetal force. Okay, let me ask you a question, okay. You're sitting here on the earth presumably; even the people at home are probably on the earth. Right, maybe someday we'll have like students on the International Space Station following along with these Learning Glass Lectures. That'd be kind of cool right? Here you are you're sitting on the earth. What forces are acting on you, right now? Gravity and what else? Normal force. So let me ask you guys a simple question, and I'm going to ask the people at home too. Does gravity equal the normal force on you right now; does gravity equal the normal force? Think about it for a second, and discuss it, I'm going to post it to the people at home and let's give them some options. A yes, B no, C don't know. What do you guys think over here? Would you like A, B or C? >> A. >> A. Table two? A. Table three? A. Back table? A. Over here? You guys want to go with B, just to be a little different? >> Yeah. >> Yeah. Alright we'll see what the people at home thought. Okay, people at home more or less in agreement with what you guys said, right. They liked the A the most and then they like B. So let's attack this problem. How do we do this problem right here? Okay, well, let's re-draw the picture. First off are you moving in a circle right now? Yeah, absolutely right. So we're moving around because the earth is rotating on its axis, okay. What forces are acting on the person? What's the first one we always draw? Gravity - mg. What else? Normal force, okay. Somebody said centripetal force; centripetal force is not a real force. Okay, these are real forces; the sum of them is equal to the centripetal force. Okay, so centripetal force is really just a description of forces applied to circular motion. Is that it? Is that the only two forces, mg and normal force? Yes. Alright, let's identify something else here, r the radius of that circle that we are moving in. We've got our picture; we've got our free body diagram, very simple. We go to Newton's second law. Newton's second law in circular motion says the following; the sum of the forces has to be mv squared over r. The sign convention that we just talked about said that towards the circle center is positive. So mg is towards the circle center, n is away from the circle center. And that has to be mv squared over r. And so look what happens if I solve this for n, what do I get? N is mg minus mv squared over r. So is n equal to mg? No, n is not equal to mg. N is mg minus this other quantity. If you're moving at some v, which we are on the earth, and we're at some radius the radius of the earth. Your normal force is not your weight, okay. N is not equal to mg, for this problem. Okay, so that last table that wanted to be different and say B, good thought. Okay so let's see if we can solve this for you guys right now. Let's make it a little more simple though. Let's say you're standing at the equator and if you're standing at the equator you are going around, you have some mass let's give you a mass, how about I don't know, 70kg. Okay, typical mass of a human. We are at a radius of the radius of the earth, if we're at the equator. Anybody know what the radius of the earth is? You guys, sitting there with your phones in front of you, people at home. Take a Google search for radius of the earth. >> Three thousand nine hundred [inaudible] >> Okay, but we need SI units, miles aren't really going to help us here. What is it in meters? Let's see if anybody at home got it. No, not yet. 6.37 million, one million, one million meters. Sorry million's not that much, one billion. No what would he say, one trillion right? Okay, 6.37 times 10 to the sixth meters. Alright, what else do we need for this calculation? We need v. How do we get v? Anybody know the velocity that you're moving at if you're at the equator? I don't know. What we do know is that you go once around in how long? How long do you go once around if you're standing at the equator? One day, right, 24 hours. So what does this become? It becomes two pi times the radius of the earth, divided by t where t is 24 hours. And so we can punch in some numbers here. Two pi, radius of the earth we just said was 6.37 times 10 to the sixth meters, one day is what? It's 24 hours times 3600 seconds per hour. Somebody punch this into your calculator Somebody punch this into your calculator and tell me what you get. How much? >>86,400 [inaudible] >>86,400 >> Oh, that's for t, right? Yeah. That was for t, okay, I got you. What did you say, 463? 463 and its SI units so we're in meters per second. Does that sound right? Yeah, that sounds right because we said that we're moving about 1,000 miles per hour, double that you get about 1,000 miles per hour. Ok, so let's calculate the normal force now. The normal force we said is m times g, minus v squared over r, which becomes 70kg times 9.8 minus 463, squared over 6.37 times 10 to the sixth. And if you punch that in, what do you guys get? Anybody get a number for this one? Omar, did you get a number? You lost your calculator. 683, sure why not? I love how you say you lost your calculator and I love how you say you lost your calculator and yet you're sitting with a computer right in front of you that has a calculator on it. You do Apple spacebar, type in calculator and it'll pop right up. Alright 683 what are the units? Newtons, right mg is a force so this had better be in Newtons. What does that mean? What does that mean? 683 Newtons. What if you were at the North Pole? What if you were at the North Pole? What would your normal force be if you were at the North Pole? It would be mg. What is mg? What is mg? It's 70kg times 9.8, and if you punch It's 70kg times 9.8, and if you punch that into your calculator what do you get? 686 Newtons. Alright, so wait a minute, are we saying that if you're at the equator that you weigh less than if you're at the North Pole? than if you're at the North Pole? Is that what we're saying? Remember your weight; your perceived weight is how hard the ground is pushing up on you. And we just showed that the normal force at the equator is 683 Newtons, the normal force at the North Pole is 686 Newtons. And this is really what happens. If you're at the North Pole you are about 1/2% heavier than when you're at the equator. Now, you right now are neither at the North Pole or the equator so you're somewhere in between the two, okay? But you are certainly experiencing a normal force on you that is not equal to mg. N is not equal to mg; the ground is not pushing up on you with mg because you're moving in a circle. So let's go back to this picture for a second. Here's the earth, here you are, and there are forces that are acting on you. Mg is down, the normal force is up. If that normal force was exactly equal to mg, then you wouldn't be moving in a circle. Something has to have a little bit of an extra force to keep you moving in a circle and that is gravity being a little bit bigger than the normal force. Let's say we did the following, let's say we just went back to some basics for a second. If we turned off gravity, what would the normal force become? It would also be zero and what would happen to you? You would fly away, but what would your trajectory be? It would be a straight line. If those were both zero you would just fly away in a straight line. What makes them not zero is mg being a little bit bigger than n and that's what keeps you moving in this circle.