Anderson Video - Spinning Wheel Angular Momentum

Professor Anderson
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>> Hello class Professor Anderson here, let's see how we can calculate the angular momentum for something like a spinning wheel. So what does a wheel look like? A wheel looks like this. You've got an inner radius, a very close outer radius and then you have some spokes. Right, that are going out to the wheel. And the approximation that we typically make for a wheel is that all the mass. Is out at the edge. Okay, all the mass in that wheel is basically at the radius, capital R we're going to ignore any mass or moment of inertia from the spokes. So if we're interested in calculating the angular momentum, we need to know a couple things. One we need to know the moment of inertia of the wheel and two we need to know how fast it is going in angular velocity. So let's take a regular bicycle wheel and let's say you're racing down the road. And omega is not given, but what is given is that you are going at 360 rpm. Okay, this is how fast that wheel is rotating. And let's see if we can calculate this, L equals i omega. So first off how do we calculate moment of inertia? Moment of inertia remember is given by an integral, i is equal to the integral of r squared dm. The way you do that for a wheel is you pick out a little mass element, dm, you figure out how far away from the axis it is, which is a distance R given by the radius of the wheel. And now we can just stick this right into the integral, because that mass element is capital R away from the axis of rotation, but as we integrate around the wheel, that R doesn't change. We're always at the same axis of rotation and so this comes right out of the integral and now the integral of dm is just the total mass of the wheel. And so this whole thing becomes R squared times M. What's the mass of a spinning wheel? It is R squared times M, that is if you're ignoring any mass in the center, no mass in the spokes, all the mass at the rim of the wheel. Okay what about omega? We know that this thing is going at 360 rpm. What is that? 360 revolutions per minute. If I want to convert that to omega, all I have to do is multiply it by 1 each time. So 1 revolution is 2 pi radians. Okay so now we're going to get rid of revolutions and we'll convert to radians. We got to get minutes into seconds, but we know how to do that. 1 minute is 60 seconds. Okay and so this whole thing becomes what? Omega is now 360 times 2 pi divided by 60, the revolutions canceled out, the minutes canceled out and we ended up with radians per second. 360 over 60 is just 6, 6 times 2 is 12 and so we get 12 pi radians per second. Okay we just arbitrarily chose 360. All right so now let's calculate the angular momentum. Angular momentum is i which is MR squared times omega, MR squared and omega we said was this, 12 pi and actually let's back up a step, let's put in some values for a wheel. Okay. So we're going to say that the mass, what's the mass of a bicycle wheel? It's not that heavy, it's maybe, I don't know maybe 1 or 2 kilograms, let's say it's 2 kilograms. Okay. And let's say that the radius of the wheel, how big is the wheel? It's about like that in diameter, so the radius is probably maybe about 20 centimeters. Okay. 20 centimeters is of course 0.2 meters, so let's punch all these numbers and we'll see what we get. Okay so mass of 2 kilograms, radius of 0.2 and we got to square that and then we have an omega of 12 pi and let's see what the units are on this angular momentum. We had kilogram, then we had meters squared, radians is of course unitless, but we had seconds in the bottom. So angular momentum has kind of these funky units kilogram, meters squared, per second. Let me approximate the answer here and you guys can punch it into your calculator. Erik you want to punch into your calculator and tell me what you get exactly and I'll approximate it here? So .2 squared is going to be .04, if I multiply that by 2 I get 0.08 and then I'm going to multiply that by 12 pi, .08 is really close to .1 and so if I do .1 on this I'm going to get 1.2 pi and it's going to be a little bit more than that so we'll say 1., I don't know how about 2 3 we'll just take a guess. 1.23 pi and 1.23 pi, that pi is about 3, so I'll multiply this by 3, I get 3.69 and then we want to go up a little bit more, so I want to approximate it as 3.8. Okay. 3.8 kilogram meter squared per second, Erik did you get a real answer for that one? >> (student speaking) 3.86. >> 3.86 all right good so our guess was pretty close. Kilogram meter squared per second, okay. This would be the angular momentum of a bicycle wheel. All right hopefully that's clear, if not come see me in office hours. Cheers.
>> Hello class Professor Anderson here, let's see how we can calculate the angular momentum for something like a spinning wheel. So what does a wheel look like? A wheel looks like this. You've got an inner radius, a very close outer radius and then you have some spokes. Right, that are going out to the wheel. And the approximation that we typically make for a wheel is that all the mass. Is out at the edge. Okay, all the mass in that wheel is basically at the radius, capital R we're going to ignore any mass or moment of inertia from the spokes. So if we're interested in calculating the angular momentum, we need to know a couple things. One we need to know the moment of inertia of the wheel and two we need to know how fast it is going in angular velocity. So let's take a regular bicycle wheel and let's say you're racing down the road. And omega is not given, but what is given is that you are going at 360 rpm. Okay, this is how fast that wheel is rotating. And let's see if we can calculate this, L equals i omega. So first off how do we calculate moment of inertia? Moment of inertia remember is given by an integral, i is equal to the integral of r squared dm. The way you do that for a wheel is you pick out a little mass element, dm, you figure out how far away from the axis it is, which is a distance R given by the radius of the wheel. And now we can just stick this right into the integral, because that mass element is capital R away from the axis of rotation, but as we integrate around the wheel, that R doesn't change. We're always at the same axis of rotation and so this comes right out of the integral and now the integral of dm is just the total mass of the wheel. And so this whole thing becomes R squared times M. What's the mass of a spinning wheel? It is R squared times M, that is if you're ignoring any mass in the center, no mass in the spokes, all the mass at the rim of the wheel. Okay what about omega? We know that this thing is going at 360 rpm. What is that? 360 revolutions per minute. If I want to convert that to omega, all I have to do is multiply it by 1 each time. So 1 revolution is 2 pi radians. Okay so now we're going to get rid of revolutions and we'll convert to radians. We got to get minutes into seconds, but we know how to do that. 1 minute is 60 seconds. Okay and so this whole thing becomes what? Omega is now 360 times 2 pi divided by 60, the revolutions canceled out, the minutes canceled out and we ended up with radians per second. 360 over 60 is just 6, 6 times 2 is 12 and so we get 12 pi radians per second. Okay we just arbitrarily chose 360. All right so now let's calculate the angular momentum. Angular momentum is i which is MR squared times omega, MR squared and omega we said was this, 12 pi and actually let's back up a step, let's put in some values for a wheel. Okay. So we're going to say that the mass, what's the mass of a bicycle wheel? It's not that heavy, it's maybe, I don't know maybe 1 or 2 kilograms, let's say it's 2 kilograms. Okay. And let's say that the radius of the wheel, how big is the wheel? It's about like that in diameter, so the radius is probably maybe about 20 centimeters. Okay. 20 centimeters is of course 0.2 meters, so let's punch all these numbers and we'll see what we get. Okay so mass of 2 kilograms, radius of 0.2 and we got to square that and then we have an omega of 12 pi and let's see what the units are on this angular momentum. We had kilogram, then we had meters squared, radians is of course unitless, but we had seconds in the bottom. So angular momentum has kind of these funky units kilogram, meters squared, per second. Let me approximate the answer here and you guys can punch it into your calculator. Erik you want to punch into your calculator and tell me what you get exactly and I'll approximate it here? So .2 squared is going to be .04, if I multiply that by 2 I get 0.08 and then I'm going to multiply that by 12 pi, .08 is really close to .1 and so if I do .1 on this I'm going to get 1.2 pi and it's going to be a little bit more than that so we'll say 1., I don't know how about 2 3 we'll just take a guess. 1.23 pi and 1.23 pi, that pi is about 3, so I'll multiply this by 3, I get 3.69 and then we want to go up a little bit more, so I want to approximate it as 3.8. Okay. 3.8 kilogram meter squared per second, Erik did you get a real answer for that one? >> (student speaking) 3.86. >> 3.86 all right good so our guess was pretty close. Kilogram meter squared per second, okay. This would be the angular momentum of a bicycle wheel. All right hopefully that's clear, if not come see me in office hours. Cheers.