 I don't know about that. I think I posted on Angel. Yeah? Why can we save them on a bar as a slender rod instead of a slender rod? Well, what are you going to use for the dimensions that make up my problem list? Well, that's why you use slender rod, because it's the case where the cross-sectional dimensions are much, much less than the length, which is kind of the definition of slender. So it's integrated essentially as a line function rather than as a volume. So you can reintegrate it, I guess. Do you want me to? All right, the last little piece we have is impulse momentum for rigid bodies. Now, as we've been finding for all these things, we added rigid bodies to it. The most of the particle equations, or at least the translation equations, apply it. So we had a lot of the things that led to transitional impulse momentum then, of course, continue with the rigid body momentum. It's just generally applied to the center of gravity. So just to remind us what we had there, time rate of change of the momentum is due to an imbalance of forces, and that as well as the acceleration due to gravity. Pulling things around a little bit. Oh, yeah. Rearranged a little bit and then integrate of the sum of the forces and integrate on the other side, assuming mass is constant. And that was our impulse momentum equation for either particles or rigid bodies in pure translation. This was the impulse side and is the area under the FT curve. If it's constant, that just integrates right out, and it becomes very simple. And then this side was the change in momentum. Change, that's no different for rigid body, general planar rigid body motion. We generally apply that itself to the center of mass. We also now have to take into account that it is a rigid body, so we have to add the angular momentum component. Our book uses H, is that right? It's sort of the rotational equivalent of the linear momentum we had there. Or the time rate of change of that equals where this is i, g there, the moment of inertia with respect to the center of gravity. Well, we also know that i alpha equals the sum of whatever moments there are that are being applied. So we can take basically the same steps we took there with that other one, arrange things a little bit, put in our definition of acceleration. We can arrange again, do a little integration. We have the impulse momentum for rigid bodies in full form when we use the two of the equations to get area of the sum of the moments, the sum of the moments time curve if we happen to have that. If it's constant, then it just integrates out. And it's straight forward as these kind of integrals can be. And of course, this is the change in angular moment. Whatever particular problem we have, one or both of those can apply. We typically use planar problems. So this is generally not a vector form. But the general idea is don't change. And for our planar problems, this can actually be done in two dimensions. Jake? That's not alpha? Here? No, because we took alpha, made it into d omega dt. Then we integrate, assuming a constant moment of inertia, then the integral d omega becomes delta omega. You guys, heck, you guys could come up with that just about. We've been through the linear impulse momentum enough times that you probably could have done that. So we'll look at a problem. Imagine we've got a small motor starting up from rest of an applied torque that's going to that Christmas or at Easter, I guess, a little better. It's going to start up a larger gear to which it's attached. So this is a non-slip condition there. Some of the details, so we'll, and this is the gear on it, not the whole motor itself, just the gear on it. That's all we're rotating starting up. A radius of gyration of 80 millimeters on a radius of 100, a gear of 6 millimeters. That's trying to turn a gear that has a mass of 10 radius of gyration, radius of 250, and the whole system initially at rest. It takes the motor to come up to volumes often well solved with the impulse momentum equation. There's the motor gear. It's trying to turn the big gear because of that, there's going to be a force here from the big gear. That's the big gear pushing back on the little gear as the little gear pushes on the big gear trying to bring it up to speed. Then with this moment, the torque from the motor being applied in that direction, it's going to see a very same force opposite direction trying to bring it up to speed. Pretty much every season, time it takes the motor to come up to speed. So that'll be the integral moments being applied to it for whatever time as the, let's label that, the little 1B as per reference, that's what that business was. And then let's see, that'll involve what unknowns? What parts do we not know going into that? Some of the moments will involve what two pieces? That's the sum of the moments. Moment applied plus the force times its own radius. So the force will be unknown in that because the moments are radius are known. You're looking for the time. So we have, in just this left side, we have the two unknowns of the force. It'll be delta T if we take everything to be constant. What's unknown on the right side? Maybe not immediately known, but uncalculable from just that equation itself. We don't immediately have the moment of inertia, but we do have the radius of gyration and the mass. Omega two, we're looking for, omega one is zero. Okay, so there's two unknowns. One equation, because this is planar rotational motion. So this is only one equation. Don't we have omega two is not correct? Sorry? Yeah, I don't, omega two is not listed as an unknown. You have two unknowns. Omega two, it's not remembered in the proper form, but it's essentially known. We don't wanna mess with every single little thing that's unknown when we calculate it directly. So what's your second equation then? You have to do the very same thing on the other here. Just making sure that a piece is right. There's only one moment on that. It involves however the same force, so you're not introducing a new unknown. And this is the same delta T, or even rest. Same speed, omega two, but the reason we use different gears is so we can have it one speed on one and another speed on another with no slip contact. So you should have all the pieces there, wrap it up a little bit. Some things as you go along, if I were you, make sure that you're using the same moments of inertia, the same omega twos, everything else is pretty much set. Watch your units. End of the term for you, even though there might be days left. You don't know what units you're trying to trick you in. You gotta change it to a second first. Mine's not even that bad with the distortion on the camera. And then look at this one, that's a beauty, that one is. You have no slip on the second one. If there's 600, first one's pretty bad with the distortion on the camera. So why don't you get a read? You get a few of these. Who'll read you? You got a pencil. Well here, I'll bring you a piece of chalk. And an eraser. You want to call it a chalk? It's got that mustard yellow. No way, no way. You call it serrano's one time. What? End of serrano's note to code, oh you call it a serrano's note. You know what I mean? It's a birth date. Oh it's probably a people's date. You see white with more kind of paper, that's why it's a little cringey here. Can we help you? Just as I was drawing. I think you're being exaggerated. I think I better sell it on eBay. It's a logo. Don't you have a logo? Finger gesture that kind of represents your life. It's not quite a logo. Are the moments going to be the same? The errors of others are going to be the same. Which moments? These? You mean that's zero? If that's zero, then this is zero. Be careful. It's just a fair tale of those portions that you call an opposite. Both occur at the same point in the tangent of the two circles. It's a sketch, but sometimes those sketch can throw you off. I'm showing off the carol. We got something? Probably a lot less writing. Let's see, Alex, what have we got? I don't, because it doesn't belong there. I see that. What? Fix that. We'll be happy to do it. It's actually a no-slip condition. It's the point that this point has the same velocity as this point does. So we're setting the velocity of that, equal the velocity of that, and then we can solve for omega. What do we call it? Omega a. What? Question? For this second one, the first one has an omega zone. What's the solution? We're going to be able to do it at the same time. Yeah, we're going to be able to do it at the same time. Yeah. If all this content will come out of this integration, it'll actually be a delta T. Oh, delta T. Yeah. Oh, you got it? Yeah. We'll do the first one first, because I have some board space here. Well, let's see, what do we need? We need I, G, or B, I guess I called it. It's going to be K, KGB. How cool is that? Of the small gear. Delta need omega two, which is the 600 RPM. Three in pieces, we need it for that one. All right, so some of the moments, well, this will integrate to some of the monuments, since omega one is zero. All right, the moments are, well, we'll pick up positive direction, just to make sure. So we have M in the positive direction, minus the unknown F, and it's the delta T. That was disc B. Oh, snap. Nope. That's the whole point of having gears of different sizes. You have one turning one speed, and another turning another speed. You can think about it. If this goes around once, this one's only going to go around part way, not even half. That's the last two pieces we need then for, let's see, this is for B, for A then, the equation becomes, for the time. The time to get the motor, the small gear, to reach 600 RPM. And that influences the RPM of the big gear. The moments, positive that way, this makes a negative F, as the positive direction is minus that. Is that what most of you have up to that point? Here you have the product F delta T, and here you have the same product F delta T, so it might be easy to solve for that and just stick it in there. And then that leaves only one delta T off of that. You can solve for it. That's the answer? Well, we're not following for that trick. We're not stupid. I didn't just say the answer. She said the answer in my agreement. What was the answer? I couldn't do that to her. That worked. Jake, what do you get? For my friends, I said differently, different entirely. Try solving this for F delta T, just as a way to make the solution a little bit easier, because you also need F delta T up here, that'll take out one of the delta T's you have on both of those. Doesn't matter. We should still get the same answer. The directions are arbitrary. If you picked positive for this direction, then these two negative signs cancel, or you don't have those two negative signs whereas I have them, and they cancel delta T together, you get 40.2 on the A equation. Just stick that in here. You've got to be less than the little gear. The little gear is going to spin faster than the bigger gear. What is R, B, and R? The radius, the radius of B and the radius of A. This is saying that the contact point on A has got to have the same velocity as the contact point on B, no slope condition. If you leave F delta T together, then you can just stick them in here, solve for delta T directly. Just as an algebra step to save you some trouble, but you can still solve it either way. So did you get it? Do you know? Is that right? Something in algebra or not? You got it? 0.87. Probably algebra. Is everybody okay with the physics that went into setting up the equations? The rest is algebra. Delta T should be 0.871. You can find the force from that if it was asked. Which one? It comes out to be about 46.2 newtons. You guys are ready to forget out of class? Do me, even you. Yeah? The two-parter. We'll do that again, Bob. All right. Take that to be a 20-pound disc with a radius of a quart to it of four foot pounds. Okay, that's W. It's weight is 20 pounds. 20-pound disc, the axle for the disc. And notice from the drawing that the axle doesn't go right to the center. So the disc wobbles. Just to make it earn and get out of class. Yeah, see, that's exactly the way everybody grew it. For the most part. That's got a wobble. There's pretty much at the center, however, the gear is elliptical. The disc is elliptical. All right, so. Two things you need to find. Actually, three things. See if you take the forces out of the axle and you need X to go on out of the way. Start from rest. Find the speed after start-ups, two seconds. Remember anybody else who used it? Keep going, Bob. It's like you drew it. Are you serious about that? Everybody else took it seriously. Look at their drawings. You took it seriously. You're looking at it all wobbly. Yeah, I know where that thing's gonna go. Well, I couldn't be serious about it because you need to know how off-center it was and the major and minor axes and all kinds of other stuff. For a lie on all of your resources, ask your office mate and start moving out from there. It's a multiple momentum equation. How could that not work? It's a multiple momentum equation. A view. Even if I get to the end, I'll get you a long pencil. Okay, that was huge. That's even on a pencil. And then you see the small, look at that, that's what we're gonna take you out. Not only through this week, well, it's gonna get you through finals week. You've got all this stuff. And one pencil, that's what we're gonna have there. Column, how do you do one? You supply it. Those are vectors. Relative, velocity, realm of acceleration, equation of vector equations. How's that in the equation? You're gonna need it, huh? There you go. Accelerate anywhere. Look, I'm, get out of cause questions. You can't just say out your answers out loud. I'll make you sit here and do nothing. I'll make you sit here and do nothing. You try what answer you think it is. I hate acceleration. Don't do it then. They've shown that people who do not accelerate live longer. Yeah. It says the class that can kill you. Acceleration kills money. That's a velocity. Let's go see how Frank's doing with his freestyle units. Yes, Frank, see how those units help? Yeah. Not on the mass, wasn't it? Yeah. Yeah. You put your units in. Probably always work when you're on the unit in them. What are you doing? Acceleration, that's a velocity. That would be, that would be alpha, yeah. Sure, alpha equal to delta made over delta. That'd be alpha. Yeah. That'd be alpha. That'd be alpha, right? No, I mean, I just mean for this. Okay, well, let's figure that out. Okay, all right, here's what we're gonna do. Let's see, we're almost at 20 minutes left. So, first, if you want to get out 20 minutes, if you want me to just give you the answer with 20 minutes left, we've got each paying $20. If one minute goes by, it's $1 less. So anytime you're ready, you just buy your way out of here. I think that's a great idea. Because that's not a get out of class three question. No, I'm seeing all kinds of numbers. I haven't seen it in a long time. No? Jake, whatever you think, you don't know. No, I don't. I hope to do the acceleration at a point where to find the normal acceleration. But since it's speeding up, I don't know where you find that. No, that's a triple acceleration between, I feel like going down a dead route. Jake, maybe it's dying, but not dead yet. So you need to go farther until it dies. Something you've done many times and then you just do it again. Where the box is the book and the answer's in there. It's thinking outside of the box. That's not 30 yet. There's also a normal force in the pen. Which cancel out the weight. Oh man, it was cancel the weight. Jeez. There it is. It's not zero. It's not zero. Because otherwise it would float. You wouldn't even need that in there. You wear free body diagram help? Well, not. In your best experience of this journey. For 8,000. Free body diagram help? I love free body diagrams. I agree. Where's your limit help? You got this. Yeah. You did? Yeah. A lot of acceleration. You're good. It's real high. It's air resistance. Oh, sorry. Grappational. It's dual. Sit it out. You sure you don't want to do that dollar a minute thing? I do it. No, you got it right. You took the sneaky route. You're checking size. That's what you're doing wrong. Good. Yeah. I said, well there's a free body diagram. So start there. Start with the basics. It's got both the weight and the key that force up the rope. I want to have a force up for that, I guess. How do you do that? It's around this time, so you can use your units. Yo, Chris. You know what I'm saying? Huh? What about my car? Yeah, my car. Well, it wasn't where I left it. That's what I mean. But I didn't place it in. I don't know where I left it from in there. I'm not listening. All right. Now I'm listening. Let me help you with it. I'll do it on the free body diagram for you. Let's see what that looks like. All right, here's the test. Let's see. You got it. You got that there. And then we're looking for forces on the pin and we got the weight of the disc, right? Is that the free body diagram for it? Something like that? What's the force in the x direction? Okay, good. You're half done when you need to come up with the force in the y direction, but you don't want to. Oh my God. It's almost there. For some of the forces, if it's not accelerating, we're already at a pin. For some of the forces, it should be there on the 30. Yeah, you bet some of the forces should be there on the 30. That's not what we're trying to find. The weight didn't fit right in. You can't shout answers out yet. Show them down in the paper. You got an answer. The thing, I wrote it down like 30 minutes ago. Well, I didn't know what that was. There you go. You can't shout answers out. What? There we go. We all knew that in the first five seconds. Why didn't you give it to me? I thought we did. You just got yelled it out. We just wanted normal forces, 30 pounds and some of the forces is zero. Really? I don't know, I'm just shouting it out. I'm gonna give it to you. I'm gonna give it to you today. Why don't you shout it out? Well, you didn't shout it. You don't shout it. I have to give you that. You got me on a technicality by not shouting it. Some of the forces, 20 down. I mean, 20 plus 10 down. Gotta have 20 plus 10 up. The one I heard for sure, I mean, you might have said it to each other and then said, no, it can't be that. And went on to take a side. I know. You got trust in the equation. The force of some is zero, trust it. You get a little overdue. That's what you're gonna tell me. It's gonna be on this. We need to do some of the forces equal zero. Well, let's see. What do I like to have here? How did acceleration problem? Because I just saw that you liked those. But not a terribly complex one because I want to leave room for you to make it incredibly complex. A relative velocity or instantaneous center one. We usually can solve those either way. And then a general motion problem. Could either be impulse momentum, work energy, or just sum of the forces equals MA and the sum of the torques equals I alpha. So probably three problems. And somewhere there, you're gonna have the sum of the forces. Guarantee of that, huh? General motion. So that could be either sum of the forces equals MA, sum of the torques equals I is alpha. Or work energy or impulse momentum.