 Is that right? Is that kind of where we were? We already were. No, we'd already part of it. And then I told you to plot the rest. So who did? Did you hear the brake slamming on right there? This is when I was ahead of the little thing that would make cricket noises. You know, you give it all the sudden this deathly silence in here. All right, we'll go over it again. The deal was we had a rock climber who had last put a little piece of protection in. That's a physical connection with the wall. So that if he falls, he doesn't fall all the way down to the bottom, which if it's a wall like El Capitan and Yosemite Valley, that's a 2,000 foot fall. And I think only one person's ever survived that. So he'll put protection in, climb somewhat above that so that if he falls, he'll only fall down to the protection that much farther and then a little bit of stretch in the rope to actually cushion the fall. It's no more than the type of things that your airbag, your bicycle helmet, or your seatbelt do for you. They bring you to a stop less abruptly to cushion any of the force required to bring you to a stop. So he had, I believe, a 10 meter distance down to the protection so he was going to fall that, plus that much more, plus a little bit of stretch in the rope. And we were looking at that. We were at the point where we were looking at the potential energy as a function of Y. Because both of those potential energy components, and what were they? There were two potential energy components. G for the gravitation of the fact that he's going from a height up here to a height all the way down here. And also then once he got to where the rope just became taut, then there was a little bit of further drop as the rope stretched, taking out the last of his speed and then bringing him to a stop. I think we even had this distance yesterday or the last Wednesday. Didn't we have 2.7 meters? We got that? Okay. So we actually had that 2.7 meters. So we were looking at this right now, and we found that it was a piecewise function, I think they call it in math, where only the gravitational potential energy plays any part in the first 20 meters of the fall because the rope is just slack and falling, trailing along behind him. And it doesn't come into it until right here where the rope becomes taut and then starts to absorb the fall. And I chose to call that y equals zero. Arbitrary choice is our origin always is. But that allowed us to separate the business above that, which was purely gravitational, from the part below that, which is now the gravitational and the elastic potential energy is coming into play. And I think we had something like that. Yeah, that looks like it. And this was for y less than or equal to zero just because that was my arbitrary dividing place. So it was actually kind of a convenient choice, which is why I did that. And then we were looking at that gravitational potential energy curve as a function of y. Numbers I have, this is kilojoules. Okay, so we started at some point, 20 meters above my origin. Did we figure out the energy at that point? Yeah, that was merely a matter of putting 20 into here. And, well, we have the numbers. I think it was 70 kilograms. 70, 80, 70, or 80. One or the other? This is the made up number anyway. So this was 15.7 kilojoules of energy he started with. As he fell from there, he was losing gravitational potential energy. And in fact, since M and G are constants, he was losing it linearly. This is a linear function, a straight line. In fact, we even know what the intercept is. It's zero, which makes sense if the gravitational potential energy is going to turn to zero at the origin. That's an arbitrary choice. Gravitational potential energy isn't something we can actually go measure. We don't have a gravitational potential energy meter. We can stick on something and see how much energy it's got. It's nothing like that at all. It's a matter of us calculating it. What we're mostly interested in is the change in that anyway. So when we look at the change, the origin has absolutely nothing to do with it. It just calculates out. But then we have this linear function down to the origin, because the intercept was zero. That's his total potential energy as he falls. And that's where we left it on Wednesday, right? That's when we got right to that point. And then we said we want to look at the rest of it. Actually, we don't need del there, do we? We can put a y in there. Here, del and y, since they both even measure from the same place and do the same thing. All right, so we wanted to look at the function from then on. In fact, I even have these values. I can't remember if we had them up there. It's just a matter of putting in the numbers in appropriate units and calculating them so that we then have the equation to look at. And so I think that one came out to be that. Okay, did anybody look at the rest of this function? Obviously, it's parabolic. And remember what it means? When this is positive, that means it curves up. So we know that it's going to keep going somehow and then curve back up. Did anybody look at that and see what the numbers were that came out with it? Stop treating your weekends like weekends. You got to get to my age, then you sleep in, and then you get some coffee, and then you putter around the yard a little bit because you're tired of wife yelling at you in the house. Then you go take a nap. All right, it is parabolic upward. It goes down something like it's rather steep. Very little of it actually nicks below there. Can anybody venture a guess of what this Y value is? That's where it returns to the original energy it had before. He returns to his original potential energy. In fact, when he was up there, that's all the energy he had because he didn't have any kinetic energy. He wasn't moving. So he's fallen, fell down to here. That's where the rope tightens up. Anybody know what this point is where the energy curve goes back up to the original 15.7? Is that this? We calculated a negative 2.7. How did we find that? Remember how we found that negative 2.7? What was it we solved for? The what? We did the work energy equation which really wasn't a whole lot different than this anyway and solved for that. That is the minus 2.7 meters right there. That's where all of his potential energy, gravitational potential energy, has turned into potential energy in the rope. It's now got the rope out at maximum stretch. The rope is storing the maximum possible potential energy it can and he's at full stretch. In fact, just to show you what the curve looks like, where's the button ahead of time and it doesn't go off. Joey, keep an eye on it now. Do you see that little green light? Joey? Joey? Don't look away. That reminds me that he keeps saying, Joey, he keeps looking at me and I'm saying, Joey, I don't remember once I put my daughter in the corner because she was bad and she turned around and looked at me and they turned around and she started crying because she was turned around in her mind. In my mind, I wanted her unturned around which was turned around in the corner. So I'm good luck being parents guys in the Alps. It's tough. It's a tough business. Alright, hopefully this will come up. I've got the 16 new buttons pushed. I've got a push. Full vision on. There's the full curve. Sorry, that's not the one on one. There's the full curve. Now I've got the axis over here rather than up through the zero while I draw it on the board. But it's essentially the same. There's almost nothing below the x-axis that it gives but then it does return from the 20, when I had the 15.7 down to the 2.7 that's just barely beyond this mark and it returns to its original energy level. What I've done then is I took this little box area and blew it up just so you can see it in a little bit more detail because what it illustrates is we've got this low point here which is right at about minus 0.32 where he's at his minimum energy at least his minimum potential energy. Remember the kinetic energy is not in this. So that's his minimum potential energy point. What is he doing right there besides screaming? But in terms of the physics what is he doing at this point right here? The rope is already starting to become taut because remember that happened at y equals 0 which is right here on the axis so he's gone a little bit past that he started to tighten up the rope what's happening to him at that point that's obviously the point of minimum total potential energy. Can you be even more specific than that? Well let's see let's see if we can go back and remember something. The total energy he's got at any time is going to be made up of three things at least three things as far as we care about in terms of mechanical energy there's chemical and thermal and caloric and nuclear except we're not dealing with that we're dealing with the type of mechanical energy we've been looking at it's made up of three parts work energy equation work energy equation what else? gravitational potential energy potential energy in fact that's the whole right hand side of the work energy equation we could have written the whole thing like that in this problem what work is being done Len said it John you said it, you said 0 anybody else? I don't want to have just I mean even when we vote for president we have two choices we don't vote for president the only way is to time do you ever see that show with Kevin Kosner where the election comes down to a dead heat tie so the one vote to be cast is Kevin Kosner he was this drunk in a trailer in Arizona somewhere which what? yeah I'll vote if my vote makes the choice then I'll vote other than that what's the point There's an open question, answer it. Oh, yeah, that was a question. How much work is being done in this problem? It's not laughing. I forgot the question. Ah, then go after yourself. How much work is being done in this problem? We had a guess of zero from our two most competent students, should I say that? So maybe they're right. Work done by what when we calculate it? Non-conservative forces, wait a second, fellow. When he falls and the rope becomes tight because of this protection, doesn't that protection exert a force on the rope? And isn't that an outside force that we should consider in this problem, in the work it does? I said, yep, I have a note. And clearly that piece of whatever it is he pounded into the wall or however he stuck that piece of the wall to stop any further fall, clearly that exerts a force on the rope. Otherwise, he could do it with a piece of gum. No, that's not, this isn't it, the rope is elastic, this isn't, this is the wall and he's found some crack here and he jammed this thing into the crack, it's called a chock. It's got a sling on it. Then he clips in a carabiner and then he runs the rope. So, when he falls and the rope comes up tight against this thing and his partner's holding onto the other end, so that this doesn't really come into it, it's only the part above that protection, clearly that thing's gonna have a huge force on it. You wouldn't want to be standing there holding that with your hand when this guy falls. Rip it right off your hand, take your fingers with it. Isn't that clearly an outside force exerted on our system, which is the rope and the climber? Yeah, it's a normal force. As the carabiner goes like that and the rope from his partner goes there and then he goes down there, that would be a normal force exerted by the carabiner on the rope. Why is it doing work? Or is it? We should put that in there. Did you just say something, John, about that? Well, why not? That's a huge force. That's gonna be several thousand pounds of force exerted on that, which is pretty pertinent to that climber. He doesn't want to use a shoe string. We expect it to hold. We'd use a paper clip from this carabiner. Does everybody know what a carabiner is? It's one of those clip snap rings, I think they're sometimes called. Climbers call them carabiners. John's got one. Is that what we're gonna see, John? You got one you wouldn't want to use. Right on. These are the little cheap ones, but that's the thing we're talking about. You clip it on the rope, then you clip it on the sling coming out of the chalk and then you clip the rope into it and it snaps shut so the rope can't come out as a way to attach the rope to the wall to cut his fall, to keep his fall from being all the way to the bottom. Why doesn't this force, this normal force that that carabiner exerts on the rope, why doesn't that do any work? It's not going anywhere. It's not going anywhere. There's no delta X. Work, remember, is F delta X in its simplest form. There's no delta X here. He's going a big delta, well, delta Y actually, but that force really isn't. Once that force is there, there's no distance that thing actually travels that would cause any outside work. There is no outside work being done on him. Delta E is zero. Therefore, E is, no? Joey, don't talk with your mouth full though. I have to call your mom in here. Yeah, you're gonna choke. E is what? What do you do in sign language, Joe? No. Oh, we're all waiting. What's he gonna say? We can't wait, it's so exciting. It's a constant. Oh, now he gets credit. If delta E is zero, then E is constant. So this quantity is constant. This part, which means if that part drops, this part must go up by the same amount. However much this blue part drops, the kinetic energy must go up. So let's see, here we're at a place where the U is at a minimum. This part here has become as minimum as it can be. Fine, I think we'll take a nap in the lab there. I can't watch this, you're gonna break your neck. I'm worried about you. Still, would you scooch over so we can follow through by the shoulder? I don't want him to get hurt. As this reaches its absolute minimum, what must happen to kinetic energy? Must reach an absolute maximum because the two together are always a constant. In fact, we know what that constant is. It's 15.7. So as this drops below 15.7, that's gonna pick up all of that extra. So this is a point of minimum potential energy and maximum kinetic energy. And how do we recognize maximum kinetic energy? Betic energy is one-half MV squared. So he's either at his maximum one-half, his maximum mass is actually losing mass and fouling himself as he falls. He must be a maximum velocity at that point. At that point, he's reached his maximum velocity. From then on, he starts slowing down. Actually, he was picking up speed all the way down here and then started picking up less quickly until finally he's now starting to lose speed as that kinetic energy turns back into mostly, well, when Tom was entirely the potential energy in the spring as he brings the spring to maximum kinetic stretch. He's got as much stored up in his rope as he can. What's this kinetic energy here? Zero. That's where he came to a stop. That's how we define that spot. Comes down to that point down there. What's he do then? He'll stop and then bounce back up like bungee jumpers do. And he'll start the curve again. And you've seen the bungee jumpers do that. And they'll come up far enough where their spring starts to go slack again. In fact, I think some of them have to actually hold it so they don't become retangled in it as they come back up their tumbling a bit. So he'll go down and if everything was ideal, which climbing ropes are not ideal elastic media, he could bounce between these two things forever. Evidently they don't because there's not a bunch of guys going on and on and on these rock walls where they've all fallen. All right, so that's what we do with the potential energy curve. There was a problem like that on the homework, I think. Very much the same kind that the points of importance are these minimums and maximums, obviously. That's where something's going on. Something big is happening there. There's also other energy levels. For example, the zero energy level might be important. However, that was pretty arbitrary that he chose it there. All right, any other questions that you wanna ask some questions for tomorrow's test? Or I could give you another work energy problem since I know you love them. Yeah, same format. A couple of multiple choice questions, but a couple, we don't wanna look at that. But a couple of problems that you have to work out or 12 or something like that, problems down there about maybe two thirds of them, three quarters of them are multiple choice. And the rest are worth the problem. Which some of the multiple choice are anyway, it's just once you've got an answer, then you look and see if it's on the list. So it's kind of one and the same for a few of them. But some of them are more concept questions. Open book, open notes. You're gonna have to melt them handicapped. Come on over. Any specific questions? Say what? I was wondering if you could address this problem and what the book really happens. Oh, yeah. Of course, it's true. Usually you fall out a little bit. The rope will actually tighten up over here and then it's a bit of an elastic pendulum into the wall. A 20 meter fall is a pretty good fall. That's quite possibly more than you want to fall. Well, I don't know if you want to fall any distance, but a conservative climber would have put more protection in that 10 meters than, rather than get 10 meters above his last protection. So thanks for making it a little more gory than it was anyway. Any questions? Sure, test tomorrow. If not, I don't want to just stand here and look pretty. Yeah, that sounds good. Pardon me? I don't want to do it. Yeah, do you want to talk over some old problems now? It's the time to do it. I can give you another work energy type of problem since those are the newest work energy problems. We had one specifically of that, I think. Well, we had both that sweep, that seat and the sling, and then we also had that cylinder where the floor drops out. Relative. I don't know. But I don't know what our, I couldn't do any of the questions on the test. That's why I put questions on the test so I hope that I can get the answers fine. I don't remember. I don't know what I put on the test that well. Plus, what do you think is hard? Do you think it's easy and do you think it's in French? Do you feel comfortable with it? No, you can go if you want. If you're sleepy, you can go. When did you go to bed last night? Or did you? Did you bet? Don't you guys want to hear the answer? I do. Your head's been going around like this a little so far today. You were so tired, you skipped drawing class. That's how I'm tired. You've got to be really tired to skip drawing class. When did you go to bed last night? Two. Two. What were you doing until two? Good answer. And how come it isn't here? It's too far up there. Wow. I'm going to bed. The fort's done. I'm getting them early. I'm finishing. The doorknob works better. You're not shot for the whole day. No questions? Comfortable? Huh? Do you want to work energy? You want one? You're tired of work energy. Do you want one of those? Len, do you like one of those? You'd stay for that? Yeah. A work energy problem. But you two would be unhappy. Anything's fine. That's the spirit. I'll do anything. Just do something. All right, apropos for the chapter here. Here's a little mounting spot on the wall to which is attached a spring. To which is attached a mast. That's 1.5 meters from one end of the spring to the other. This object is a quarter of a kilogram. It's released from rest such that it passes down here. Now this is all the length of the spring. That's what these wings are. It now falls down to that point where it's then going. Since it's a spring it's not gonna be going horizontally there like it would if that was a non-stretching cable. Since this is a spring it can stretch some. It's actually got a little bit downward velocity there. At that point it's going five meters per second. Rest length of the spring, one and a half meters. Oh, that's good. So that means we just didn't even have to stretch the spring at all to hook it up between the mounting spot and that little object there. I want you to find the spring concept. How stiff should that spring be so that it does that? If it's too soft, too stretchy, then it's going to be going down here farther. It's probably gonna be going faster. If it's too stiff it'll be going up here too high. It might not be going fast enough. I want it to go that speed, that direction. That point so I need the right spring. So I need you to tell me what K is so I can call up Earl, order the right spring from Earl. Oh, you know Earl's number? Yeah, I'll give you the answer in the last. Oh, I've got him on speed dial. That arrow that you got for five meters per second. You said that two meters, is that the bottom of the curve or is it gonna stretch further? That's where it's directly below the mounting spot. So beyond that I don't care what it does. I want it to be doing that at that point. I mean, so that arrow is actually horizontal. No, look at the picture. I'm a better drawer than that, all right? Non-stretch cable. Then you'd be doing a quarter turn. Then it would have to be horizontal. But because this spring stretches, it gets longer. It means he's following some elliptical path of some kind. Energy equation would work well here. It's a position dependent problem. As a spring in it, as gravitational stuff in it. Don't be lazy writing down the work energy equation. We've got to get all the little pieces right. So don't believe out your delta symbols as if that's too much work. Yeah, Joey, I'm off duty. I'm at the managed number three. But I'm gonna help you out here. Put your minds together. Another little bit of that'll help. And I think I've got a little there. Something completely different. Are you close? A couple million? The negative sign is problematic. The lack of units is problematic. Other than that, it goes actually within a couple million. Oh yeah, from the very first moment I introduced them to you, they had units. You were five foot six then anyway. Five foot six, 14 years old. Pretty much do what you want in preschool. Why does it stop? It can't be, it can't be right. There you go, units. Put your heads together. Oh yeah. Oh yeah, you got rid of the minus sign. Put in the wrong units and got rid of the minus sign. That's good physics. That right there is a good physics. Best of minus signs, they're gone. Spray can of the two stuff that didn't get rid of the minus sign, just spray it on the paper and the minus sign. That would be very easy. Then we could have another one. It sprays units. Well now, yeah. You don't want to tack them on the end, you want to take care of them as you go through them. There's a problem with that. That's a new meter, five by meters, even per meter. Magic with units. Just wave your hand over them and they'll become whatever you want. That's what I'm trying to be too. I think we need some groups to form here. This is a multiple head process. This really isn't any different than was our, at least in terms of the work energy equation. It's just the rock climber problem we were concerned with where he comes to a stop. This one we're concerned with some place before that, not only with blank. So I did it before you didn't like it. I know it wasn't right. So if someone's wrong before that, if your units aren't coming out right, there's something wrong in your equation. Or else you're making a mistake when you look at the units. If there's something that's got the wrong units in the problem, they can put it in. All three came up, you can do it with mirrors, especially when I felt good. All three of those things, they're all three of the delta things came up in the meter. Okay, good. So what I saw is what I should have got. The new meter's right there. Check your book. What's it saying there? If you don't believe me when I gave you the units on K, then check the book and see what it says. It's divided by eight meters. Because you can't lie when it's in print. This doesn't happen. Because you say, I got to read the newspaper, or I saw it on the web. That makes it true. It's the new meter's right there. Yeah. Yeah, that's what you do. You want to do the piece. Oh, it was per me that I got you. Right? What? Yeah, I'm looking at how. I think you're just great. I think you're all right. I think what you're good. Yes. I'm looking at how. Yes. Thanks for the question. What do you say? I'm good. I'm good. I'm good. What are you good for? I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good for what I'm good for. I'm good. I'm good for what I'm good for. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. I'm good. Let me help. Did you divide the ue all the way to the other side? Yeah, the ue stayed on the right, I subtracted delta k. And then I divide it by the constant. This is 1.125 squared divided by 1.125. Alright. So this is solved. My interest is pretty much in that game. What did you get? What's your equation for that? If not, that could be anything. You're taking your equations. Oh. Oh my God. Just make sure everything I put up on the board is the right numbers. Two, five. Oh my God. Thank you. Are there pieces, these three separate things that I can check? That's two. You don't even agree with him? Yeah, of course. Or are they negative? Negative 0.905. 0.125. I think I know what I did wrong. Alright. Yeah, there's three separate things. Calculated independently. I can check them. You got any of those three things? That's what? Delta K right there? Where's Delta K? Right there. One of the units. Oh, you're awake. No, I've been awake. That's awesome. Well, that's good. Let's see. A, it was started from rest. I did say that. What about the fact there's an angle here? Do you take just the X component? Just the Y component. Here's a kilogram. That is M for VB. What about this angle? That's what Allen was wrestling with for some time. What about it? You just ignored it? No, it's not. Oh, yeah, that's 90 degrees. This is 90 degrees there. But this angle here is not. So what did you do about it? Fudge everybody else. I don't know why I don't know about anything. You just don't know. Here's the situation. But Joey, kinetic energy has no directional component to it. This is not the vector squared. There's no such thing as the square of a vector. This is just the magnitude squared. Just like it is. What's that come out to be? We had some disagreement between people. What? 125 newton meters. I saw a couple other things. I think some people were being kind of loose about when you do this squaring and when you do this dividing by a half. You know about the order of operations. It's one of the very first things you learn. I don't know. You guys probably learned that kind of stuff back in sixth or seventh grade. You learned about order of operations. It's not even sooner. The next one, let's see. We'll do delta UG next. It's just not because it's next, but just because it doesn't involve K in it. So we'll get the ones out of the way that don't have the K in them. MG delta H. We've got all those pieces. Would you put in for G? Negative. For like the millionth time that G is a... Well, we take it to be a constant. It's not actual. It's an average of values across the Earth. What do you put in for delta H? What do you put in for delta H? Two meters. No, negative two meters in the direction of the gravitational field. Would that give us appropriate units? Newton meters right there. And so when we do that, what do you get? Negative 4.9. Any disagreements so far? Here's the K we're looking for. One squared was used in A and B. So let's be consistent with that. What's del B? Two meters is del, however, is L minus L0, and L0 is 1.5. So it's two meters minus the rest length because that's how much the spring actually stretched, which is what we want here. So we're looking for K. Del, what is that? 0.5. 1.5 meters. It is 1.5 meters long right here. So it's no stretch, no compression put in it. You're able to just set it right on the hooks there. So del A is zero. It's how much the string has been stretched since you took it out of the box. It hasn't been stretched at all at that point. So what's 1.5 times 0.5 squared? It's what? Yeah, 0.125. K's unknown. Meter squared. What must K be in for this to be Newton meters? Which it must, or I can't add them together. They must all be Newton meters. What must K be? I'm going to ask Newton's per meter times meter squared. That will give me Newton meters. Everything back together. I get zero equals delta K is 3.125 Newton meters. Delta UE. 0. Minus 4.9 Newton meters. K equal then. It's about as simple as an equation can be physics 1, 14.2 Newton's per meter. We know what's got those units because that's what it has to have. We check back here. Mike, when you've got a negative 2 meters to the length or from a change in height, how come it's negative when gravity is down and that's the way that it fell? That's where I messed up. All right, let's do two possibilities here. Starts here, ends up down here. Do one possibility where down is negative. So that means Mg H2 minus H1, well actually Hb minus Ha would be Mg. Those are both positive so we don't care about those. What's H2? If our origin's right there and down is negative. So H2 is negative 2 meters minus H1, which is zero meters. So that agrees with what we had there and that's what most people did in their minds anyway. Another possibility, let's see. Let's call that positive and it's H1. The M is positive, the G is positive so we don't care about those. H2 is zero, which is different than what we got when that's meters, so clearly it matters. What then I guess if you want to do this call down negative, I mean down positive and artificial I guess but we could call this the origin then the numbers would work out okay, wouldn't they? So it would be then zero and H1 is 2, which would give us minus 2, actually it would give us a minus minus 2. So the question, the answer is you can't just do anything you want. You have to pay attention to it. What's important to us is that it goes deeper into the gravitational field and we call that negative. G's up and down, make it off plus and minus and we're not taking into account whether it's traveling in the direction of the gravitational field or not. That's what's crucial for the gravitational potential energy. If it goes deeper into the gravitational field it's going to lower potential energy. That's not arbitrary decision one way or the other. Whereas all of this stuff is and if we flip this but don't take into account this then it just doesn't come out right. I guess we could handle it by putting a negative on there but I don't want you to do that. This is not kinematics where our choice of origin and our choice of direction didn't matter. This is kinetics which is driven by the direction of this and that has to be taken into account. The easiest thing to do is take down as negative. Just take down change as negative and up change as positive and then you're always fine and leave G as always positive. The number itself. But you've got to pay attention to your own negative signs. Does that help any? The best thing to do probably is just be consistent with what you're doing. Okay, we've almost always taken down as negative. It's always worked before so why change? You're starting to kinematics. It doesn't matter much. All that is this position. This, we're talking about something that has a directional component to it itself. What about the fact that the spring is pulling horizontally here, pulling vertically here? Can I take that into account? It doesn't matter. All we care about is the amount that spring is stretched. We don't care which direction it's stretched in which I think makes all this stuff simpler. There's no component in the direction of here, no directional component here. I have to worry about that a little bit but it's pretty easy to save those down as negative but those up as positive. Most of you getting messed up. You had this one. Most of you didn't have this one. What was growing up here? Was it algebra? Or you had two meters in here? That's why I don't like it when the book uses X because the amount of stretch or friction problem gave you one we may not actually come back to. I think we did that one. The one I don't think we ever came back to tell me if we did one where we had a one kilogram crate that was tied to the wall and a 20 Newton force on the bottom crate with a coefficient of kinetic friction at each surface, each interface, between two surfaces. Did we finish that one? I think I asked you to find the acceleration of the lower block. Find A2, we'll call it 2, the 2 kilogram block. We wanted to find the acceleration of it. Obviously the other one, the top one's not going anywhere. It's acceleration zero. But then also find the tension in that line right there. If we sum the forces on this block, we're interested in how that block accelerates. So let's look at it. Maybe we don't even have to do anything with the other block and we can be done with it straight away. Sum the forces on that has that mass to accelerate. So the best way to, when you sum the forces is to get a free body diagram. So here's our 2 kilogram block. Obviously it's got that force on it. What else? It's got a normal force where, perpendicular to the surface, which is horizontal. So it's got a normal force there. Any other forces? It's weight, which is, it's 2 kilograms times G. What's that, 19.6 newtons, something like that. That's its weight. Just so we don't get confused, I'll put W2. What else? Any other forces? So the surface is in contact, so that's easy. But is it going left or right? It's in the direction. If it goes out to the right, friction's trying to hold it back to the left. So I'll even put that right there. Why didn't you account for the weight of the box to top that though? Who said I didn't? It was by the 2 kilograms, by 9.8. Yeah. The weight of the box. But then there's also the 1 kilogram box. I didn't say we were done. Are there any other forces? There's also friction at the top dragging out from under that block. Which direction is that push? That block stays there. This block comes out from under it. So friction's trying to drag it back. Are those two the same? Are those two frictions the same? Maybe, but there's no reason to assume they are, other than it makes things a lot easier. So we'll call this, I know we'll call this FL for the lower surface. We'll call this FU, because it's always fun to do that. Any other forces? The force of gravity being a little stronger over in the corner. Yeah, the upper box is exerting a force down on it. If you didn't think so, what if that upper box was a couple of tons and your hand was underneath it? You'd think, yeah, I think there's some forces there. So there's some force from the upper box. In fact, it's the action-reaction pair. As the upper box pushes down on the lower box, the lower box pushes up on it with a normal force. It's the equal and opposite of that. How big is it? It's what? Yeah, it's just the weight of the upper box. If we had a scale right there in between those two, it would just breathe the weight of the upper box. So that's pretty easy. We'll call that W1, 9.8 Newtons. Any other forces? The 2 kilogram box is pushing up on the 1 kilogram box. So if we took the 1 kilogram box away, the 2 kilogram box would go up. Sitting on this surface right here is 3 kilograms. And I've got 2-3 kilograms pushing down. Don't outthink these. I was just saying that because you put the normal force on me before you put the weight of it on us. Oh, I put it on the order they were given to me. Not in the order of which they actually occur. Yeah, that force doesn't occur until the weights are there. But the weights are there even before we write it down. Uh-oh. No other forces. So we've got that force and those two forces in the X direction to accelerate, which we're looking for. What are those two frictions? The upper one, coefficient at that surface, the normal force at that surface. What's the normal force at that surface? If you're not sure, let's draw its weight there. What else? Let's get these drawings up real because then it's just a matter of adding things together. It's just algebra after that. What you've got the free body diagrams. Force. So is that the same as this normal force? No. No, so we better number those. We've got two different normal forces. Any other force in there? If not, this box is going to accelerate to the left. Two kilograms pulls under it. It tries to drag the one kilogram one with it. Is this a third friction force or is this really one of these two? It's equal and opposite to FU. So that then is the coefficient of friction times the normal force at that place. The normal force from this drawing is the same as W1. So that's that one. What's FL? Same coefficient of friction times what normal force? Times what? Same coefficient of force. Times N2 is N2 equal to W2. For the picture says, whatever the free body diagram says, that's the best way to get the normal force. N2 must be equal to W1 plus W2. I'll give you those numbers so you can check them. A2, 0.14 meters per second. The tension, 2.92 newtons. Wow, a lot less than the 20. The friction's picking up a lot of that. Okay.