Okay, so we have a very long problem here. Um, I took the liberty of drawing it, so I'm going to try to 0.2 things in blue as necessary as I'm reading. Um, also, I did use Microsoft Excel to solve this problem, and I can show this to you as well. It says Figure 10.32 which I tried to draw on the screen. She was an apparatus used to measure rotational inertia is of various objects in this case, spheres of varying masses, M and radi. I are The spheres are made of different materials and some are hollow, while others are solid. Um, now, one thing that I know is that inertia for ah, hollow sphere is Let me look for that hollow sphere is 2/3 ammar squared and inertia for a solid sphere is to fifth m r squared. So as it spoke, the question spoke about hollow spheres and solid spheres. It made me think of the different coefficients that come in front of m R squared. It's either 2/5 or 2/3. Okay, let me continue reading to perform the experiment. A sphere is mounted to a vertical. Axl held in a frame with essentially frictionless bearings. So up here is the sphere. Ah, spool of radius. Be so I wrote radius beyond there equals 2.5 centimeters. So I should write that down. But I'm going to write point zero to five meters. Okay, is also mounted to the axle and a string is wrapped around this school. The string runs horizontally over and essentially frictionless pulley and is tied to a mass m so we can see that over here. Tying to em and um is 77.8 grams. But I would like to write it as 0.778 kilograms. I generally like to write things in S I base units as the mass falls the string in parts of torque to the spool slash axle slash disc combination resulting in angular acceleration so that mass is going to accelerate downward. And at the same time the spool is going to spin. Ah is going to accelerate its spinning. Ah, and the mass is tied to the pool by that Axl. Okay, The mass of the string is negligible, but the combination of excellent spool has non negligible rotational inertia. I subzero. We're going to read that. We need to figure out what that I subzero is. Um that value is not known in advance. In each experimental run, the mass is suspended at a height off 1.0 meters. So I wrote that down here. Looks like I should have had 20 is Okay, so that site above the floor and the rotating system is initially at rest. So our initial angular velocity and our initial speed are both zero. The mass is released and experimenters measure the time to reach the floor. Results are given in the tables below. Determine an appropriate function of Time T which, when plotted against other quantities, including M and R, should yield two straight lines, one for the hollow spheres and one for the solid ones. Plot your data established best fit lines and use the resulting slopes to verify the numerical factors. 2/5 and 2/3. Um, and I've already written those down there in the expressions for the rotational and nurses of spears given in table 10.2. You should also find the value of the rotational inertia of the axle and the drum. Together they don't mean drum there. I think that's a typo. They mean the spool, the axle and the spool. So, um, the axle and the spool are rotating together, so they're going toe, Have, um, one total inertia. Okay, that seems like a lot. Nevertheless, the first thing that I'm going to dio is I'm going to write the some of the forces equals mass times acceleration. And I'm going to draw that. This are these air the forces one the mass element. So one that mass element there is wait, which is m times G. And there's also tension. Now we know that it's going to accelerate downward. So I'm going to set my axis so that a is downward. So what that's going to give me is when I write this down, the some of the forces are tee minus MGI or rather, MGI minus T. Because I'm taking downward as the positive direction. So let me redo that. Ah, going there. You stress again m g minus t. And that is going to equal the mass time to the acceleration. We can solve this for acceleration and we get a equals m g minus t over em, which again shows us that a equal simplifying it a little bit more g minus t over em, and we are going to use this later. Okay? Now I'm going to draw some of the torques. So if I draw the spool, I'm gonna draw it from above, and I'm gonna change my colors here, maybe read. Draw the spool from above the torque. Um, which is, which is the tension times the distance is pulling from above, and then our acceleration is defined to be in the same direction. So some of the torque equals I health. Ah, Twerk some of the torque his eye Alfa The torque is detention times B, which is the radius of the spool. And that equals I off she be equals. We know that I is some constant. I'm gonna call that constant K que is either going to be 2/3 or two fists times m r squared Now, um, good. However, our inertia is not just que times m r square. There's also the small inertia that we have to add on. Um so the k times m r squared is the inertia of the sphere. That small inertia I subzero is the inertia of the spool and maybe of the rod. Okay, so now we need to take that times Alfa the gay. Next thing that we need to know and I'll change colors here is that why equals this is for movement of the mass. Why equals y subzero? Um, plus V y subzero t plus 1/2 a t squared. However, I do find a to be downward. And so I need to change that to minus 80. Squared nips one of that to be green. Okay, Visa zero is zero. Um, why at the end is also zero zero equals? Why subzero? Why subzero was age, which is one, um, minus 1/2 A t squared. And we can solve that for a a equals to h over t squared. Really? Cutting the Children room here to h over t squared. So we're gonna need that later, and we're going to need this later. There's three things we're gonna need later. Um, we're also gonna need something else, So let me go to another page. A equals off, uh, are we don't have in our in this problem as you look back at how we draw on this problem. Um, there's no lower case r r is raised this fear, but this this are that I'm talking about now is the radius of this school which is actually be so a equals alfa be. And so if I solve that for Alfa, we see that olive ah is a over B. Okay, so now I'm going to go back to the equation that said a equals T over AM minus G r a g minus t over on my bed right here ache with G minus t over em. So let me write it over here. And that should be a lower case. M is the mess of the object. Okay? But we know what T is. So a equals. Let's go back and look what he is. I think I'm going back here. Okay, So from this equation, right here T is K m r squared plus ice. Subzero times. Alfa Overbey, K M r squared plus ice of zero already for gotten K M R squared. Okay, so we got that k m r squared ice subzero times, Alfa over. I m okay, but ah, we know what Alfa is from over. Oh, no, I have it on this page. Alfa is a over B. So that's going to give us G minus k m r squared. Plus I subzero over em a over B. And let's look at what else we got here. We know down here that a is to age over t squared. Okay? I'm not gonna use that quite yet. What I'm gonna do is I'm gonna add this to both sides, and I'm going to get that g equals a plus k m r squared. Plus I subzero over MB a Okay, m r squared. Plus I subzero. Okay, good. Can factor out the A So that's going to be a times one plus k m r squared. Plus I subzero over and be okay, but now we know a trying to go back to it. A is two h over t squared. She is to h over t squared times one lus Okay, m r squared. Plus I subzero over and be Now what I want to do is I want toe Ah, factor out the MBI the one over m b from the inside. So that's gonna give me two h over m b t squared times, MB. I'm just realizing way back, way back when I wrote this equation right here, I forgot to divide by be on both sides. So I should have written k m r squared. Ah, plus I subzero Alfa Overbey. So when you realize you have a mistake, just go back and fix it. That's going to put it. Was writing right in here K m r squared. So that's gonna put a B down here. It's just going to give me another be here. It's going to give me a B squared here or B squared here, B squared here, B squared here, b squared here. Okay, now we're now we're doing good. Um, plus, let me think about what we're doing here. Okay? Um que m r squared? Plus I subzero. Okay, um r squared. Plus I subzero. Okay, um, now I'm going to invert this and multiply it onto both sides. I'm also going to rearrange so that ah, these two are going to reverse their order. And let me write it again. Next page G I m b squared over to h t squared equals I subzero plus m b squared plus k m r squared. This is what I wanted to come up with. So this 1st 1 I am going to plot this. So I'm gonna take the t value that's given in the, um, tables. Take the T value, and I'm going to calculate this, Okay? I'm going to plot it. Verses. Um, m r squared. So m r squared is going to be I'm gonna write verses this tous okay, on the x access. So I'm going to calculate Elmar squared for every single one of them. Um, all right, that's going to give me that. The ah slope is gonna be K and the Y intercept is going to be I subzero plus and B squared. So again, I have to calculate this for every single row, every single column, actually in the table. I also have to calculate this for every single column and then plot them this one on the X axis, this one on the Y axis. And then I can read from the graph the slope on the Y intercept. So I did this using Microsoft Excel. So here we have it. I put in the first table up there on the top. I calculated in column D I calculated m r squared, um, that second tables on the bottom and I calculated I'm r squared there also. Then I also calculated t squared times g m B over to H. That is in column E. I'm all applied by 1000 because in excel, it's a little bit hard to see if you didn't ah, have reasonable, reasonably large numbers. And so I plotted the points and you can see the points there. There are 10 points. I then sorted out the points so you would see that the top table is not the top table that you saw in the book. And the bottom table is not the bottom table that you saw in the book. I sorted them out because, um, some of them are hollow and some of them are solid. Um, now we can look at the Orange Line and the blue line. You put a line through the orange ones. You can see how it says seven e negative. Five X with a slope is seven. Really. The slope is 70.6666667 So let's go back to the white board. The slope of one of them was 66667 which is the same as 2/3. And if we look back here, 2/3 goes with hollow, hollow orange. Okay, so the orange line on the spreadsheet represents all of the hollow spheres. And again, they were from different columns. They weren't all in the first table. They weren't all in the second table. Ah, some of the hollow ones air in the first table, and some of the hollow ones are in the second team. All right, you will notice that, um, the blue line 0.4. So the blue line slope is is 0.4 or a multiple of 0.4, an order of magnitude of point for so slope number two four. If we look back, 2/5 is the same as point for. So those are the solid spheres. And so that was the blue line. All right, so we got our slope. Now we've got to do our Why intercept Now, notice that our Y intercept is ice subzero plus M b squared. Next page I subzero plus and B squared equals. Let's go back to the spreadsheet. The Y intercept. Now the Y intercept is shown to be point 978.978 However, um, masses in grams. And, um actually, that doesn't matter. Hey, it's ah, point 09 70. So let me write this down. But I have to divide that by 1000. And the reason why I have to divide that by 1000 is on the spreadsheet In column E. I multiplied by 1000 so that I could see the number for the Y intercept. If I didn't do that, that Excel would not be showing me the number very well. So I need to divide by 1000. All right, if we solve this using the values of M and B. So again, M was 0.778 be was 0.0 to 50 Put those into an equation and solve for I subzero. You will get I subzero equals 4.9 times 10 to the negative fifth power kilogram meter squared. That's just off by a little bit. But from what it says on the book, there's there's got to be a rounding error. Um, perhaps in my use of Microsoft Excel, Um, perhaps in the makers of the book and what they did. Um, but nevertheless, that is how you would solve the problem. So what I want to do now is I want to make some changes here. Teoh, Um, my spreadsheet. Notice that Put mess Ingram's. I'm gonna change that to mass in kilograms just to show you what's gonna happen to the and she. But we are done, by the way. That was the answer. Um, 0.73 0.432 0.677 0.947 0.8 to 1. And so now, if we can see, we have here for the, um, slope again, it's a little hard to see. Um, exactly this this first number that that it's, uh, 6667 Whatever. So I just want to try. Let's try changing this two meters. You're a 6 to 5 when you're 386 Hey, now, what just happened? Looks didn't see what that said. 942 Still can't see it. Um, man, let's try. Tried to small applying this by 10 trombone playing this times. No, uh, back toe. Would we have 1000 and multiplying that by 10,000? I'm not even sure that I'm messing with the right one right now. Um, trying to make it so we could see that easier. Um, that's pretty funky. Try multiplying this times. 1000. All right. Um, nevertheless, I clearly just made a mistake in something that I was doing there. Um, it was right up until I started just messing with the spreadsheet. Let's go back, Teoh. Here. Um, we said that the orange one is the one that was 10.6667 and I think it was messing with the wrong one anyway. So let's see what I can do with this orange one. I'm going Teoh go here. And instead of multiplying by 1000 I'm gonna multiply by 10,000 then drag this down. You know, I'm gonna get rid of the blue because I don't know what I did to the blue one, but whatever I did was not good. Um, 0.7 All right, so rise overrun. So I need to make this a much bigger number. Let's just give it three more zeros. I just want to see that slope. There we go. 6674 Finally. So if you do see a trick of multiplying. Um, then you can see, um, the slope and the Y intercept. And that's what I wanted to show you at 667 All right, So, um, thank you for watching that. Um, took some time and we went through three pages, but we came up with the answers, so I thank you for watching.