 All right, we, uh, we, That was a tiny bit, good, sir, if you're right. Not with your teeth. We've been going through this whole business last week or two, and by the way, there's only two weeks left. Two weeks, it's going to go very quickly. Because I said just a little while ago, there was five weeks left, and now there isn't. So, anyway, we've been doing this whole business that for everything we do, linearly, we've got an exact analog rotation. And all we have to do is substitute the variables and use the very same equations. And that, uh, held for several things, especially, uh, when we got this idea of, uh, of linear inertia, which is what mass is, it's not, it's a measure of how hard it is to get something moving when we push on it. Then we had, uh, moment of inertia did the same thing for that. Then, uh, I think we just got our rotational equivalent of f equals ma on, uh, on, uh, last week on Wednesday. I think we've just gotten to that. Maybe actually gotten to it the day before. Um, so that the right-hand side's easy to do. Wherever there's an m, put an i, wherever there's an a, put an alpha. So we had that business there. Uh, what was it, though, that makes objects accelerate in a rotational way at a certain speed? If I have some object here, and I want it to accelerate at some rotational speed, some rotational acceleration, given that it has some moment of inertia, which is, that's what I got to get accelerating is that moment of inertia. What is it that took it to do that? How could I get something rotating that wasn't before? That was torque. Remember our symbol for it? Half time. Yeah, half time. Tap. And there can be more than one of those torques. So we have to sum them up. Uh, what was torque, though? Remember what torque is? It's a force applied at some distance. That distance we call the moment arm. So I could apply a force here. The moment arm is the smallest perpendicular distance to the point of rotation. I could apply the force at some greater place. And I have a greater d, and therefore a greater torque. Well, you kind of know all this business anyway. This is, you have some experience with this. You could go out and do this with your car this afternoon. If you take your car and hold your hands very near the inside of it, it's hard to control it much because the d is so small, you need a big force and you're adjusting things all the time. You don't have as much control. If you move your hands out farther, then things are a lot easier to control. You don't need as much force. You can feel those very, very same things with torque. A lot of you have ever tried to get a very sticky nut loose with a wrench and you may realize if I've got a nut here and a wrench on it, I push on that wrench and it doesn't come loose. What's one of the things you can do? Do what? Kick it. Yeah, you kick it, you put a greater force on it. It might not work. That thing goes flying off, hits you in the nose. You can put a pipe on there. This is an old trick. Put a pipe on there. Now you can get a much greater moment arm. You can apply less force or if you apply the same amount of force, you get so much more moment arm, you get a lot more torque and then maybe it's enough to break the nut free. That's why when there's really big nuts to move, you get really big wrenches because they're going to need a lot of force to make those move. It's nothing more than torque being a force at a particular distance. Each one of these, remember, is when the force is perpendicular to the moment arm. We even had kinetic energy in here so we could find the kinetic energy, something rotating. We had that in there. There's some other things we didn't get to yet. One would be, well, do you remember the impulse momentum equation? Yeah? Or no? You've never heard it before? Some people are not and some people are shaking their head. I'm talking to the same class. We had this equation. I'll put it in our usual constant force. Oh, you said it. Well, you sort of said it. Not quite right. We've done this before. Force times the amount of time that force is applied. Didn't we do that? That's impulse. Actually, the sum of the impulses because we could have more than one force. That's going to cause the momentum to change. And that's almost what Samantha gave me. Remember, it's a vector equation because momentum itself is a vector. You might suspect that there's some equivalent of linear momentum. If we have an equivalent of angular momentum, oops, this isn't right. The m is in the p there. So you were going to catch that, weren't you? I guess you weren't going to catch it. You'd suspect that we have some rotational equivalent of momentum. If we have mass, we put in moment of inertia. We have velocity. We put in angular velocity. So what we've got here is some object with a certain moment of inertia rotating at a certain speed. That seems like something that would have angular momentum because if we want to stop that thing from spinning, we're going to have to reach in with a certain force, apply that to a certain force for a certain amount of time to get it to stop rotating or to speed it up if we need to, whatever it is we need to do with it. So this indeed is angular momentum. We just need a symbol for it. Anybody got a recommendation of a good symbol for angular momentum? Yeah, I agree. Capital L makes sense. That makes sense, I think. You guys are good soldiers. You just write it down. Alan, and we're going to work with this one a little bit today. On the rotational side, that's a good symbol. Because everything we have rotates in one plane, so it's either going clockwise or counterclockwise or plus or minus, and that's what we were doing in more of that. But in full 3D, yeah, it is a vector. In both, like mass has no vector, but all those have vectors, so yeah, it actually would be a full vector equation. Angular momentum. So you think we have impulse and momentum equation. And we do. Some of the torques applied for a certain amount of time. So if we've got a force we're applying such that it twists something, the longer we apply that twisting force, the torque, the greater the change in speed will be. I guess that makes a pretty good sense. And we assume the mass and moment of inertia problems are constant. Well, let's see. We've worked with that impulse momentum long before. So I'm just going to put it up so we can remind ourselves. Now we've got an angular equivalent to it, and I'll leave that one up without the vectors, and we'll just assume that it only rotates counterclockwise or clockwise. So it's not really a full 3D vector thing. We did a lot of work with this equation. Didn't we? The external forces were zero. Therefore, the impulse was zero. Then it went that side zero, then that side zero. Remember doing some work with that? That was all of our collisions were governed by that equation. That situation is called conservation of momentum. You might suspect then if there are either no outside torques or whatever torques there are counterbalance each other, that's the kind of thing that you're trying to twist thing one way and I'm trying to twist it the other way by the same amount so it doesn't actually move. If that situation doesn't actually accelerate, then delta L equals zero. Remember, L is this angular momentum. We're going to do a couple things. Well, we're going to do one thing and then come back to that. First, let's get this idea of a moment of inertia in our brain just so we've got it. I'm going to take a little very simple situation. I've got two objects here, both exactly the same. A meter stick with two masses at each end, both the exact same masses. Not exact, but sort of exact same masses. Looks like that. One is the exact same things in different places. Number one, call that one number two. I have three questions for it. Three possibilities you pick than I2. I1 is equal to I2. I1 is less than I2. Those three possibilities. They can't be more than any one of them so there's only one of those you can vote for. You take a second, decide which one of those you like. A, B. And then think there's some way we could prove which is which because we're only one of these is true but we also have to prove which is true. Thirty seconds of which one of those you want to vote for. Remember physics is not a democracy. Does something have to rotate to have a moment of inertia? If you remember from our moment of inertia table, the moment of inertia has to be with respect to a certain axis about which the things might rotate but they don't necessarily have to. So we'll keep it simple. Write down the center. So there's your axis of rotation. Write down the center. Which one has the greater, yeah, I'm going to rotate that one. Which one has the greater moment of inertia? If you want, if this would help, this distance is 10 centimeters and for the bigger one, it's 45. Which one has the greater moment of inertia? You've got 15 more seconds to decide then you need to commit. And this is not a presidential election that you can just sit out. This is important so you have to vote. That's how bumper stickers are pinned and we need to get some robo calls going in the middle of dinner. Moment of inertia. I'm running for Congress. Ready? Everybody who thinks A is true, raise your hand and don't look around. I'll make you put your head down for a secret ballot if I need to. One, two, I'm not going to start over. One, two, three, four, five, six, six. The second one is true. One brave soul. John, you haven't voted yet. You've got to vote. One, two, three, four, five, six, seven, eight, nine, 10 people. That means three people think this. John, Andrew, Tyler. I've got three here. No, your 15 seconds was up. You can't start anything. On election day, you can't walk in there and say I'm still thinking. Can I come back tomorrow and vote? They say no, you vote now. Well, let's see. What is moment of inertia? How do we find it for an object like this? It's an object made up of two things. First thing you do is you look in that little table and see, gosh, is this object right there in the table? Remember table 12.3 on page 382? Is it there? This object is? What do you mean something like it? There's nothing like it. It's kind of like a dumbbell. Is that in the table? Which one? Count down. I thought it smelled something burning. This object isn't in there. Remember, though, how we find the moment of inertia of an object that's made up of compound pieces? This is made up of three different pieces. If we want to know the moment of inertia of something made up of pieces, we just add all the inertia's up individually. All right? So this first one here, I1, is made up of a long slender bar rotated about its center. Is that in the table? It sure is. That is there. That's 112 ml squared. So we'll say 112 mass of the stick l squared. I don't know the mass of the stick. I certainly know its length. It's a meter stick. Do I have the right l? The l's the whole length. Not the half length. I want to figure out the moment of inertia contribution of these things. At least get an estimate. How could I do that? Well, let's just treat them as very small masses, meaning their size is much smaller than the distance they are from the axis. So it's just simply then, let's see, the mass of the block times the square of the distance they are from the axis. And that's l over 2. Nothing more than mr squared for a very small mass. By small, I mean its size is quite a bit smaller than the r we're talking about. It's certainly r squared. Oh, there's two of them. What do we do? There's two of them. We just do it twice. So we got that a little bit. Let's see, we can simplify this a bit. Mass of the stick over 12 plus 2 over 4. So mass of the block over 2 times l squared. Is that right? I took the l squared out and that was it. Because this is l over 2 squared, that's l over 4 times 2 is l over 2. And let's see, I think it turns out that just conveniently, the company did that nicely for us, the mass of the stick is about equal to the mass of each block individually. Let's see if we can take a little bit. So I can pull that out as well. So I get 112 plus 1 half times ml squared. I don't need ms, I don't need mvm, we'll do. What's 112 plus 1 half? 7 twelves? You like fractions, don't you? Yeah, 6 twelves, 112 plus 6 twelves is 7 twelves. We can estimate one inertia of the second one. Assuming the mass is about the same, I took l to be about 50 when it's really about 45, but that's not, we're just looking for an estimate. So let's just say we're one fifth of the way in from there. I can move those to 9 instead of verify that I do what I'm not cheating. You see that? So you think I'm cheating, don't you? Yeah. So estimate the moment of inertia of the second one. The stick is still there, that's no different. We don't need a subscript on the m anymore. There's still two masses. What this time though is their distance from the axis of rotation. L over 5. Yeah, about L over 5, just for an estimate. L over 5, remember that distance is always squared. What does that reduce to? 112 to 25th. Let's just make it 24th. Yeah, let's make it 224th or 112th. We're just looking for an estimate. Remember the question was just simply which one's bigger. Not exactly what are they. So this one's about 2 twelfths mL squared compared to 7 twelfths mL squared. Give or take a little bit. But clearly which one was bigger? This one's a lot bigger. About three and a half times greater moment of inertia for this one. So six people were right. They can take a lap around the classroom. Okay, we'll make the other ones take a lap around the classroom. There you go. Really, you didn't want to take a lap around the classroom. Everybody like you drive? Really. Batteries are low. Yeah, but we didn't get enough chocolate. Notice nobody got the extra credit points for bringing me chocolate. That's pretty dumb. That's like a no-brainer. Winter's over, so you can't come clear my sidewalk of snow, my driveway. Fall is over, so you can't come stack my firewood for that season. All that was left was bringing me some chocolate for extra credit points. You blew it. What was the next question I had for you about these things? Now that we've established I1 is much greater than I2 by almost four times. What can we do to test it? Yeah, spin them somehow and see what the difference in the torque required. Well, pretty easy just to hold them and spin them. You can't tell which is greater or which is less canyon. Who wants to come up and try this and see which one's harder, which one's easier? Len. Len, I heard your name. You can stop. You can stop right about here if you don't want to be on camera. Is this how I did it one time? No, no, you can do it until you feel happy. You got to hold it right at the center. I remember that's where we were looking at things. Not to our left. Feel much different than what it takes to turn them? Yeah, it's a lot different. Would you say it's 3.5 times different? Yeah, it seems about different. Try to go fast. Because remember, alpha is the angular acceleration. That's very hard. Should we make somebody else try just to confirm it? Because they don't think this is just a simple thing. Yeah, got to take a rest. John, Mike, are you worried it'll hurt your wrist? Actually, I'm worried about it. It'll probably be good for your wrist. I don't know, it might not be. It's significantly good. You don't have to come up. Just hold it about the midpoint, right about the 50. And hold it down if you like. No, level. Hold it down, hold it level. There you go. Put Andrew's eye in. I don't want to take off like a helicopter. Danger that. You can take more physics. Yeah, this is a lot. Anybody else? Samantha? Joey? Joey wants to try it. You have to do it like this. That's hard. That's even harder. Hold it to the center. Isn't that hard? All the way up. There we go. This could have a workout machine here. Kind of like that thing. If you want to give those a try, it's easy to set. Oh, here's another question. Same kind of thing. This is a bonus question. You can write a paragraph about this for extra credit. Answer this question. It's quite easy to balance that. Why is this so much harder? That is so much harder to balance. Come try it. You've got to figure it has something to do with this business. Same moment of inertia, isn't it? Same object. It is when it starts to fall. And I have to move to prevent that. This is easy. Does it seem inertia? It's not very. Oh, Joey! I told you to wear a helmet today. Then I could have gone ahead. Give it to me tomorrow for five little points of extra credit. Why? The heavy end up is so much easier. You can go home, get a broom or something. Rooms are very easy to balance when they're the heavy end up. So a couple extra credit points for that. That was an interlude to help us understand moment of inertia. It really does is heavily affected by the distribution of the mass in an object. But let's go back now to this. Our impulse momentum for rotating objects. In the absence of outside forces outside torques which are usually caused by forces if there's some moment after those forces. In the absence of outside forces then we could say that moment of inertia equals L final. L initial equals L final. Does that follow? Well, yeah, if that's 0 then that's 0 then before and after equal. So omega times omega 4 equals I after omega. Moment of inertia changes pretty easy to do. If the moment of inertia changes then what's got to happen to the angular velocity? It's what? It's got to change in the opposite direction because in some of the ratios of the two and they're in effect inverse ratios of each other where the I on the top wouldn't that be true? If I change the moment of inertia of something that's turning if I can let's say increase the moment of inertia of something that's turning then the velocity will have to go down. Is that true? If I have something spinning and a moment of inertia increases its angular velocity should decrease they're on opposite sides of the practice. So this gets bigger this whole thing's got to get bigger which means that gets smaller. Is that true? Comfortable with that? Well you've seen this kind of thing happen a thousand times. An ice rink with a figure skater on the ice rink doing a single axle. What's a single axle? I have no idea. Oh shoot I'm sorry I have to go get on a sequin leotard first so that I can really I don't have time I'll do that next year when you take this course over again. So what could I do to change my moment of inertia? If I start spinning on that like a ice skater spins I can change my moment of inertia. Well, pull my arms out bring my arms in depending on whether I want my moment of inertia to go up or down. So let's just come up with a decent model of me as a figure skater which is a great image on its own right there. We get in class right now and you'd be happy for the rest of the day. So here's me as a figure skater with my arms out. A little bit of extra mass. Oh, she can't even stay for it. See, I'm out of here. So I'm just going to hold out a little extra mass. I have pretty massive arms as everybody knows but not massive enough. I can amplify the effect with that. So there's a model of me and then I can pull my arms in hold the masses in here like that will I have a different moment of inertia? Of course I will because the mass will be distributed differently about an axis of rotation. Let's see, let's do the exciting one this will be the initial that'll be the final moment of inertia. We'll assume that compared to the masses of my body and the mass I'm holding there that my arms are pretty massless just because we don't want to belabor the calculation. So let's put a couple numbers to this. This cylinder that makes up my torso let's say has a radius of about 25 centimeters. Is that about right? Okay. A little bit, it used to be 25 centimeters now it's 30. No, no, that's the radius. I mean that was the diameter. Yeah, 25 is close to the amount. So let that represent my body as a whole and my mass will take it to be 72. 72 kilograms. My arm length quarters of a meter almost exactly. I love that term almost exactly there's a lot of power in that one phrase almost exactly. And these are two kilogram masses. So you figure out my moment of inertia with my arms out and then we'll use pretty much the same radius. We'll assume I can get those two masses in tight here and I'll pull them in close enough that'll be the same radius so now my mass goes to we'll say 74 oh sorry 76 kilograms because I pulled the two two kilogram masses into my body. Now what's my moment of inertia and then we'll see by what factor will my angular speed change. Alright, so you take a second to calculate those moments of inertia just an estimate my height I'm 6'4. 20 meters? Yeah, or 20 meters 6'4 or 20 meters but I have shoes on so I'm 6'7 or 28 meters. Did that help Joey? Access and rotation which picture is in the table there? We're doing the same problem so you should all get approximately the same thing we're making the same assumptions and estimates and it'll be plenty close enough if this goes well you'll see what I had for lunch you're sitting at the back there come down front so you can see this better it's more exciting alright everybody know what to do to get an estimate on the moment of inertia treat me as a tall cylinder of known radius and mass do you need my height? No, not the way the rotational orientation there on the third picture that's a flat cylinder rotating but the height doesn't matter this is a tall cylinder got it? Check with anybody? we should all have about the same idea here treat it as a cylinder with two masses stuck out there and then treat it as just a heavier cylinder as our good estimate on torques is zero because I'm just going to be standing up on this spinning and all I'm going to do is pull in my arms there are no external torques in life other than to get it started do you agree? with filth? with allen? with allen agree with filth? how do you get the two masses this any different than this one? remember how we did that? not up there did you write it down? oh it's just coincidence those two are the same after the integration they have the same same moment of inertia as something else anybody agree yet with their calculations? don't forget there's two masses I'm holding out there will be the moments of inertia for each one so that we can find out the ratio then determine what the speed is or at least what the factor of speed increase will be all the numbers you need are there my body is this one my body is this one my body is this one my body is this one my body is this one two masses held out at some some big distance finally on both we have to have agreement on both now then figure out the ratio of the two of i over f because that will be the ratio of my speed to f over i okay we have some agreement that was one everybody agree everybody agree with somebody Andrew no you're not talking to anybody don't lie to me you guys agree? I just said you have agreement up here Joey's trying to bridge to bridge no for the initial for this with my arms out 4.6 units kilogram meters squared the units will cancel top and bottom so we're just getting a factor of increase and the unit the ones that pull my arms in about 2.4 we'll call it 2.3 just to make it easy a factor of 2 so by simply pulling my arms in should double my angular speed balancing is that what you see the figure skaters do that they pull their arms in and they really start spinning not even the figure skating you will be after today everybody ready Joey you'll catch me if I start to throw up I will alright was that about twice maybe even look like even a little bit more maybe because we didn't remember we didn't take into account the mass of my arms uh oh Joey where are you no don't there's plenty fast enough thank you wow we'll do it cruising ok that was plenty that was within the ballpark it was markedly faster pulling my arms in when I pulled my arms in were there any outside torques was this true I needed some outside torque to get me going I had to push off the table or have somebody come up and give me a spin but I don't trust students to do that anymore I'm only stupid once but when I pull my arms in there's no outside torques all I'm doing is changing decreasing my moment of inertia which increases my angular speed alright here's another test you predict what you're going to have I've got here a spinning or a bicycle wheel I can spin so I'm going to spin it that way I'm going to pull down on the side towards me just because that's the easiest thing to do and then I'm going to turn it sideways like that what will happen if anything I'm going to pull it down turn it sideways what will happen if anything simple question just pull it down I'll turn it from 90 to 0 but the wheel will be going toward me will be going across from my left to my right because I'm pulling it down and I'll turn it this way so it's going left to right if anything here's my friendly cylindrical object is the me you so have grown to love self-portrait alright we've got a couple seconds to come up with what's happening and then as I look at you I'll have this wheel that is rotating that way and then we'll rotate this way where that's this side closer to me as it is that side closer to me what will happen if anything you ready we've got an idea three possibilities well let's just do it see what happens I spin it I turn it sideways nothing happened who bet nothing would happen I'm sorry did I say I'd be standing on that I guess I did so I'm going to stand on this now this is going to be plenty thank you what will happen if anything let's see when I do that is there any outside torque being applied to the system the system being me and the wheel when I do that is there any outside torque applied to the system is there to the wheel yeah of course there is there's me pushing up on one end pulling down on the other so it turns over sideways but for the system as a whole for me and the wheel there's no outside torque therefore what should be true if the sum of the torques is zero then what's true then the angular momentum should be conserved delta L equals zero the before momentum angular momentum should equal the after angular momentum all those three things just fall right from each other those two things fall from the first so let's see here's me to start with that's the wheel in a vertical plane spinning my angular momentum what's the angular momentum of the system well I have to be a little more specific because this platform can only spin in one way so what's my angular momentum about the axis through that platform axis what's my angular momentum right now about an axis that goes down all the way through me through the platform here zero there's nothing spinning around that axis at that time I have zero angular momentum about that axis next it's been that way where farthest away from me is going across from my left to my right which gives us an omega in that direction of the wheel once I turn it sideways what must my angular momentum be there's no angular momentum here I turn it sideways now what must my angular momentum be must be zero there's no outside torque all this turning of the thing I'm doing is inside the system there's no external torque being applied so if all of a sudden there's some angular momentum in one direction what will I have to do as a system so that the total angular momentum is still zero so if the wheel has some angular momentum that way I'm as a system I'm going to have to get some angular momentum that way to counteract it if the wheel is going to spin this way I'm going to have to spin this way turn it the way I did it the wheel the wheel is now spinning this way I'm going to have to spin the other way so the total angular momentum is zero the angular momentum of the wheel one way the angular momentum of me the other is that what we think is going to happen that's what the physics says so if I turn it this way that's just what happens precise control this is much more pleasant to do than the other one was and it works either way this is much more fun to do now the wheel is starting to slow down did just what we thought that angular momentum was conserved whatever angular momentum I added in one direction had to be taken away some other some other angular momentum sets that the total was still zero even though I was turning one way the wheel was spinning the other so that the total angular momentum was zero which is what it was when it started now there's angular momentum sideways is that conserved because the wheel starts with angular momentum we use our right hand rule it's turning that way we have angular momentum in that direction we can serve sideways how come when I turn it I don't also start flipping head over heels which would be awesome well remember angular momentum can serve in the absence of outside torques there's friction between my feet and the platform to prevent me from twisting up head over heels I can feel like I just felt that that's why I don't actually do this with ice skates but if we were in outer space and I did this there would be three dimensional conservation of angular momentum first thing when I do this in outer space I'd start spinning the other way automatically because I just added angular momentum in that direction then when I turn it I didn't have angular momentum in that direction I'd start turning the other way I'd start turning the same that's why I won't go to space I love that conservation of angular momentum I'll wait for you when you turn it sideways that's because I used to catch my beard in it if that wheel was even further out before they actually worked with the way torque does if you had more friction on that just put that wheel further away but against the pressure of the wheel it's generally better it's hard to do it with any appreciable difference because I have to hold it a certain way away to keep my chin hairs and chest hairs from getting caught in it and I just can't I don't have very much latitude in that way but yes you're right it would that would be a good question for dynamics next year it's beyond this course alright tomorrow's lab let me introduce it now because several people have to leave early can't stay there three hours have to be gone in two so I need you to be finished in two tomorrow what we're going to do is a a torque problem there is an applied torque and we have to use that to determine what the moment of inertia is we're going to apply some known forces figure out what the angular acceleration is use that to figure out what the moment of inertia is in the lab you'll see across the front wall and across the back wall near the top some gray discs it's actually a gray disc a black disc on it and I need you to figure out the moment of inertia of those your object will be to find i of those two discs together now I'll tell you what the mass are you can measure the radius you can calculate that as a good estimate but I also want you to measure here's how you're going to do it I'm going to hang a known mass from the inside from the edge of the smaller disc we'll call that R1 we'll let the bigger disc be R2 if we need it I'm going to let that drop from rest so you're going to have it set up just like this and then let it drop from rest when it does so the mass will start to accelerate down and the disc will start to accelerate have angular acceleration as the mass falls so if we look at just the disc and by the disc I mean the two discs together as a system is there any torque being applied to that disc of course there is and it's pretty easy to figure out what that is if you know what the tension in the line was if you knew what T was you could very easily know what the torque is if we know what the angular acceleration was then we can calculate what I is anybody have any idea what the tension in the line is there's the mass it's got some weight to it as all mass on earth does and it's attached to the very same string this T equal to W so the mass is falling the mass is it has to be more than just falling it's accelerating so you had the right track just one word short because of that the weight doesn't equal the tension in the line we need the tension in the line to figure out what the torque is R1 times T how could we figure out what the acceleration is if we can figure out what the acceleration is and we know what the weight is we can figure out what the tension is if we know what the tension is we can figure out what the torque is so how could we figure out what the acceleration is what's proportional to what what's proportional to what what's your conference of this but I was saying we're getting to that hang on what if we did this by the way how awesome are these colors they're great aren't they all of known distance we time that fall we figure out the acceleration from that if we assume it's constant acceleration we sure can three things you know the distance traveled the time it takes and constant acceleration problem initial velocity is zero so we can figure out the acceleration that will lead to the tension the tension will lead to the torque once we have torque let's see what the deal is the sum of the torques equals I alpha how do we find the angular acceleration the fact that the wheel or the mass is slipping so that's R sorry A alpha if we've got this acceleration of the falling mass we've got the angular acceleration of the disc torques are being applied to that disc because it's the sum of the torques that causes that acceleration what torques are being applied this one right here any others Alan you said there were a second ago friction where in the bearings those discs don't spin forever sooner or later they slow down because there's friction in the bearing so that's a torque in the opposite direction I'll call that tau sub B for the bearing torque we'll call this tau sub A for the applied torque so the sum of the torques is tau A minus tau B the bearing torque equals I alpha acceleration once we figure out the acceleration we know the angular acceleration so that will come directly from our measurements in a few calculation steps once we know the acceleration we can figure out the tension the tension line that gives us the torque so we can change this around a little bit that's just rewriting it algebraically because here's what we're going to do we'll run this that will give you an A that will give you an alpha so this is the independent variable because we can control that we can change the masses we could there are a couple little things we can do once we determine what A and alpha A is we'll know what the tension is the tension is that will set the torque being applied that makes this the dependent variable because the other two things are constant the moment of inertia doesn't change torque doesn't change you've seen an equation that has the form dependent variable equals constant times independent variable plus constant that's y equals mx plus B dependent variable by simply picking different masses and rerunning the test once you've picked a different mass and get a particular acceleration that will give you a particular applied torque and when you graph those you should get a straight line with a intercept that's the bearing torque it'll look something like this the bearing torque is going to be very small so you have to be careful tomorrow good straight simple measurements you'll be able to come up with them very good numbers and the slope of that line is going to be the moment of inertia of the disk system which you can compare to your estimates because we'll know the mass of those disks and their radius and then just figure out what their moment of inertia is that way so you can compare those two I'll review that real quick tomorrow but that's the bulk of it so if you think about it a little bit tonight you won't be screwed up tomorrow I can't wait to grade your homework you already left