How key were those, especially the one where Arkin got the ball out in the short field? There you go, you know, I mean, just trying to, you know, play complimentary football, which looked like we did in the first half was really cool, great job getting the football out.  He was able to convert and help them, you know, and we couldn't contain him all the time. But we were able to get to him sometimes as well and, you know, huge play there at the end to win it on defense when they, you know, went to the screen there on the check. Five different receivers, how much do you like to see that spread the ball around? Just like them to throw to the guy that's open, that's the most important thing. Jay, look out, how do you feel you did, I know it was a sack of burns overall? I, you know, have to be able to take a look at the tape. I don't have the best view of the left tackle, you know. But I felt like, you know, at least it was a start. And again, it probably had some good plays in there. It was, you know, great as effort and see if he was finishing and competing. And, you know, again, it's a great place to start. It was a great opportunity. You know, it's good edge rushers and good interior players. So, you know, good place to start for Jalen. You know, the play concluded wasn't perfect, but how good just to win again? Just great and it was a fan, you know, I mean, put a lot into it. I'm happy for the players. You know, happy for the coaches and our fans and the organization and everybody included. It's, you know, you put a lot into it every week. And these games are tough, they're tight, they're close. And, you know, a few plays make the difference. And, you know, would have liked to end it on offense there in a four-minute drive. But it's great to see the defense come out and end it themselves. What's the answer to that? How are you guys to be able to build on this? Stop beating ourselves. Start there, you know. Penalties on offense, M.E.'s, third down penalties or quarterback scrambles, you know. So, start there. Yeah, I mean, this is a complimentary football earlier in that stretch where you had the 71-yard punt by Stonehouse, Key Strip sack, and then Henry's touchdown. Is that the best example of how you guys could really... Well, you know, I mean, that's a great, another great example. The other great example is right after that is, you know, they're trying to get some yards there and try to kick a field goal, and we stop them four-cent of punt. We don't have any timeouts. They took our last timeout there with the injury. And, offensively, we, you know, we could turn that into three points. And it executed a down-down clock in 11 seconds, you know. Great execution there by our guys and a great kick by Nick. So, that's probably those two great examples. We just need more of them. Why did that timeout come off with, like, the incomplete pass? What does the rules say there to... It's just an injury timeout underneath two minutes.  So, once that came out, there's an injury timeout under two minutes. I have to talk to Rana. I think they're across the hall. One click of this year's play, would the defender keep his head up? Again, I saw one quick replay, and I didn't think that this, the launch door, you know, those are bang-bang plays. I didn't see a player that launched in a linear position. You know, I'm sure they saw the same thing. I'm not, I didn't see the replay other than quickly on the film. So, you know, having a huge debate with the way it was officiated? I'm not, what am I going to debate?  They throw flags. Whatever it is, we got to overcome it. We overcame it on third down. You know, we've had probably six third down conversions by penalty in the last two weeks. So, there's not much to debate. What motivation does this give going into the next Sunday's game? Motivation. I hope we're motivated every week. You know, it's professional football. I'd like to, while you closed out the first half, getting in position for a field goal. It was just fun to watch all those stuff that we talk about and cover and show and watch. You know, that'll be us on the Friday tape this week, you know, with the situational football well executed. And it was great. You know, they changed their look. They called timeout, changed their look, went back into a different play, different defense, had the sideline covered and will and the offense changed their play at the line of scrimmage, allowed for us to get inside and got Nick down there to get lined up for the field goal. What was the explanation that they gave you on the challenge because it looked like the ball was on the ground as he was going down? Didn't have a clear view. And, you know, I just thought on fourth down that it was probably worth the challenge. I saw what I saw. And, you know, they just said they didn't have a good look at it to overturn it. You went forward on fourth and one and you're on 30 early. We're trying to maybe create a spark and I'd like to see how the team respond to that. Well, it was great, you know, to be able to go down there and convert, you know, and turn those into points. Derek, did you do a better job today of being more efficient on his plays and leaning on that defense over the course of four-quarters? Derek? Yeah. I mean, we just got to, you know, block better. Like, I don't know, like efficient on his plays. I'm not sure what that means. He said last week that he didn't feel like he was being as efficient as he could on his plays. He'll come up here and he'll be able to ask him about his efficiency. I'm going to tell you that, you know, we got to block better and run better. But when there was some space that, you know, I mean, it was cool in the four-minute there. It was fantastic. You know, and then, again, we just got to get him started, you know, and not beat ourselves with negative yardage runs and, you know, whether they pressure and, you know, we have to be ready for it. But, you know, we hit him there on a run.  They hit him on the touchdown run, or they kind of pressured and moved spiked. And, you know, he went out the back door and then, you know, had the long one there at the start of four-minute. But then, again, we stalled and weren't able to finish up. So it's just a consistency, I think, probably with, you know, that.  You know, I know that.  I should come down next Sunday. What's the plan there with that divisional team? The plan? What's the game plan going into that? Hopefully, when I prepare, I'll know more of the game plan here as we work through. But knowing it'll be a physical contest in the division, you know, they were able to run the football on us last time. And we stalled in the red zone. And so those are things that, off the top of my head, that we'll need to get fixed in a, I would imagine, a closely contested divisional game. Do you think some doubt had crept into some guys? And was there a little bit of a sense of relief out of kind of knowing, hey, we can do this?  You know, again, we just put a lot into it. And I don't think there's doubt.  You got to eliminate the penalties, the X plays, uncover guys, the penalties, you know, third down that extend drives and, you know, because our good is good enough to win. I know that I've seen it and I believe in it.  Do you have some red zone today? I know you need to get there more, but you got the points when they were at home. Yeah.  So great play from the high red for, you know, Chig to come through and make a great catch and a great throw. Got us into the one and then we were able to punch it in. Mike, TK and Clinton was someone who started from the defensive line today for someone like him and some other younger guys on the offensive, the defensive side. How does this last kind of stretch prove as an opportunity for them to, you know, show what they're made up?  So yeah, these are good opportunities for him and for everybody. And we just, you know, like Colton Dow, for example, I mean, it's really, really cool to see him impact the game on special teams. And I know that he'll continue to work hard, you know, to carve out a role on offense when we need it. But I mean, that's a fantastic job from a young player, a young receiver that, you know, took it, you know, took his job and his role very seriously and knew the challenge that he had with, you know, their special teams, guys with Franklin and, you know, be able to go down there, come to balance, tackle inside the 10 on a punt, factoring on a kickoff. So those are all really, really cool, positive things. No, I'm just, you know, I'm talking and just making sure that we're good on some of the pick routes and stuff that we thought may come in that down a distance. You know, quarterback check to a slip screen there and, you know, guys look like guys were running to the football and I think Hook was able to secure to tackle. It's just a great effort right there. How did you manage to take people out of the game and was that one of the keys coming in? It was a key, you know, it was a key and, you know, we were able to put some guys around them on third down.    DPI, whatever it may be, came back and tried to hit him on a post. We got a sack and Shark came back and ran into him and Christian was square, standing right by the official and he let him go. So, again, as long as we're learning from things and understanding what we have to do, you know, it's whatever happens early in the game, it happens and it's over with and we got to move on and then, you know, keep playing and continue to factor. You can build on this. Yeah, just a win is a win and talk about how hard winning in this league is and those are ones, that's one where it kind of feels like you didn't, you know, but we got it and we got to keep building on it and keep rolling from there. I mean, it's the first time in a while, I feel like we got a win, so it's a good feeling. Do you guys require to build on it when you guys are in these situations and have some momentum? What do you need to put together here? I mean, just continuing the keys that we've been talking about these past few weeks and even today, like we did better probably in the first half, but in the second half, just examples of us just shooting ourselves in the foot and let me do a better job of just operating the offense and putting us in an easier spot so that we're not, you know, setting ourselves back. So a couple of third downs, just a couple of opportunities for us to extend drives, which I wasn't able to do for us today, but I mean, just got to just key on those moments and make sure that we can operate in those situations. Have you ever posted a read-out of how to get in and out of the huddle as, you know, one of the keys? How do you feel you guys did that? Yeah, I thought we did a good job of getting in the call quick, getting in the huddle quick and getting out quick, not so much the operational part of it though. Like I think just on me, like we had a, we had a Keynes issue on the last drive that put us in a bad spot, which I could have helped us by just getting things going a little faster. And for the most part, though, I thought we did a good job and we got to continue to continue to do that and use it as a weapon. Do you think any degree of doubt had crept in because you guys haven't played well? And is there some sense of relief or re-establishment that comes with this win?  We got to just get our mojo back. And it's tough, it's tough, it's tough these past few weeks looking around after games and just seeing the looks and dudes' eyes of dudes that we all know are dogs and just got guys go out there and have done this year in and year out and to just let's regain that confidence today, which I think getting the win alone was enough to do that. And hopefully we can keep that rolling and get these guys' confidence up going into Indy. How much did it help when you go through that on that first touchdown drive and you're kind of in a good rhythm and you're completely involved with different guys and getting them involved? How much did that kind of help change the mindset and get it rolling? Yeah, I mean, that was awesome. I mean, it was, well, 17 play drives and like that. That was, I think we ran more plays on the drive than we did in the second half. But it was, those are the types of drives that, you know, brings up the entire stadium, you know, defense offense, special teams, the fans. Like, when you're able to, you know, just capitulate down the field like that and just then finish it. Like, I mean, it's the best feeling in a ball to just finish a drive like that. I was surprised because the coach went for the 30 and I liked plays like that. Has that helped kind of get the juice going like that? I was so glad we did because, you know, we had messed up and I had done something that could have put us in a better spot if I hadn't on the third down or on the second down or the third down before the penalty. And then to get that close on that play, that's obviously a risky situation but I'm for it, obviously. Your third down numbers today weren't really all that good. What do you guys need to do better on the third down to convert one? Yeah, I mean, looking back for me, I think this was the first game where I had more than a couple, at least location issues in terms of just accuracy, which, you know, I'm just, just got to keep working and know that I can make those throws but couple third down opportunities where I had someone and I just missed the throw and then maybe one or two where I could have gone somewhere else with the ball but I mean, the location I got to just trust and I'm going to make those throws and they'll come next time. Jay, when Duncan's first start, how do you kind of notice him protecting you there? Yeah, I was on him all week just about getting his confidence up and I was his demeanor and just how he went out there and played. I was very happy for him and gave him a good hug after the games because it's this huge. Anyway, to explain, we'll be, you know, 4-0 at home now and still haven't found a way to win tomorrow. No, I guess we got good fans. It's a cleaner process though for you guys on offense. At home, isn't it? Yeah, it will definitely operationally just with the noise. But even today, like I feel like in terms of the operational mistakes we had this week compared to last week, it was pretty similar, even though we were at home, which is interesting to think about. But again, just got to hone in on those things. One by seven, but how important maybe those three or four and a half and how you executed that? Yeah, that was sweet. I mean, to get points there, to stall out on the two-minute drive before it and then to get another crack at it, which is some like 30 seconds left. Like that was cool to, you know, that they had the trust in us to go out there and try to make something happen. We had a couple plays dialed up and a couple guys who we weren't really thinking of that could be getting it, ended up getting it. The one right before half was nice because they pretty much knew what we were running and we still converted on it. So it's cool. Should I ask Michael, the spears I guess took a knee 15 yards, maybe behind the line of scrimmage. Was that him being smart, getting down or did you play, did you talk about, sorry? I guess right before the other half it took a knee on the spot. Oh, just, I mean, he doesn't need to be anywhere specifically in that formation. If we knows we're clocking it, he can just stay. So that's just him being a heads-up player. As you said, you know, when is the win, but is it a little frustrating when the second half you guys weren't able to get it? Yeah, I mean, we were stagnant all second half, which is tough. But I mean, we got to just focus on, watch the film and learn from it. But to have a first half like that and to play complimentary football with the defense and to have that great feeling at halftime to then roll out and just kind of be meh for the rest of the game, it's not acceptable. We got to be able to keep it rolling and, you know, keep putting our foot on gas and finish these types of games out. No, I literally heard it yesterday. Ryan was the first player. She was like, soups to the AMA. And I was like, I have no idea what you're talking about. But yeah, I guess now he's staying, but I mean, I got to talk to him. I'm out of the, out of the, you know, whatever on that one. Every time you turn around and hand the ball up to Derek Henry, he gets to the end zone. He's climbing up the board. He's now, we have 17. How cool is that to, you know, be associated with a guy that's continuing to do what he does? It's so cool. I mean, and he's one of the dudes I was talking about earlier without staying with getting our, getting our swagger back.  And when you get him, when you get him rolling, there's not many people that can stop him. So it was good to get him in the end zone a couple of times today, get his confidence up. I mean, I still think, you know, he's definitely hardened himself and knows that he could, he and we could have done a better job just running the ball offensively. But I mean, he's going to, this is going to motivate him to, you know, get us ready for next week and get those guys up front, rolling for him. Oh, step forward. Can you speak for Chick?  Yeah, shoot. I wish we would have gotten him that score. I think he was still in. I don't know why they didn't review that, but don't reach the ball next time, Chick. But that was, it was cool to get him, you know, some burn. And he's a guy that we got to get the ball in the hands or get the ball in his hands for. He's explosive. He's not many people at his position that can run and move the way he does. So understanding his strengths and finding ways to get him the ball is going to be important for us. It seems like when you guys have some of those negative plays to get you behind the down and distance, the play calling can go a little more conservative, some screen passes, some run plays. You obviously have a lot of confidence in yourself and want opportunities to take shots. How do you balance wanting those chances to throw the ball down the field while also trying to take, you know, what your coaches are giving you? Yeah, I'm just taking my coach in. I'm seeing the game and reading it out as we see fit. And I try to do a good job of knowing when, you know, you can go a little off schedule and see things and be confident, but not, you know, not reckless with types of throws. So, I mean, I'm just operating the game plan. And I wouldn't say off the top of my head that I could really say that I'd agree with you in that statement in terms of just the, I don't know if I forget the word you used, but yeah, I mean, love taking shots on the field, but you got to know when to. And today, there were some opportunities that I could have hit that I either missed the throw or didn't make the decision to. But got to learn from it because there are big plays out there for us to be made. I got a little bit of clock there near the end. I mean, didn't keep it the whole way, but the run-out time. Some of the clock, how important was that there? Yeah, just being there for a minute, trying to get a lot of time out of the clock as much as we can. And just trying to move the ball down field so we can keep the ball and keep it out of the hands. It was coming for a long time. What it means to you to cross the 9,000-yard rushing threshold for your career and to do it in a win? Extremely blessed. Thankful to be able to make it this far. Thankful for the players and coaches that my teammates helped me get to this point. And just blessed. And as a kid, wanting to be able to play this game as long as I can. So I'm just thankful. You moved up a couple spots in career rushing touchdowns. You've now won three times. Marcia and Litz, how much does that mean? To be able to get into the top 20 at number 17 overall? Yeah, it's a blessing. Grew up watching Marcia and admire his game, the way he played. And just a blessing to be able to get to that list and get to where I'm at. And just thankful for the people that helped me. I can assume you were, I guess, with Ajay when you took that big hit on their sideline and it seemed like you came to just defense a little bit. Were you not liking what you're hearing as far as the chatter or what was going on over there? Yeah, I just wanted to make sure that he was okay. He took a big hit. I've just seen a lot of guys over there chatting, wanting to see what the promotion was about and checking on him. And my wife's going to come to his defense in that situation.  Bounce back, Matt, come back and play. Yeah, definitely. You know, I think you've seen that when he was at Tulane. You know, he's tough. He works hard. And, you know, he took a shot right there, but glad he was okay to be able to get back in the game. That's like right where we're going for in 4-1 early and did that provide some kind of spark early for you guys? You know, I think Coach was just trying to be aggressive. He gets into a drive, get a spark going on offense so we can end the points. We were all for it and excited that we were able to convert it. A couple guys said it's kind of inevitable for some degree of doubt to creep in when things aren't going well and for confidence to rebound with a win like today. You feel like that kind of an accurate assessment and how much does today kind of bring you guys back to a good feeling about yourself? Yeah, I mean, it's always positive for a lot of good things to build off of when you get a win.  Play it hard today. Try to do whatever they can to help us win. Glad we were able to get in the win column. It's been a little while. And, you know, just build off that momentum, going into next weekend in the vision game. I thought you'd try to keep guys up when things aren't going well. But did you feel guys getting down as things were not so good? I think guys were just hungry to get something going and to make something happen. When stuff wasn't happening, I don't think no guys had doubt. I think guys were just ready to get it going just so, you know, we can move the ball downfield and both sides of the ball, everybody making plays. How do you constantly play your end guys' ears, pushing them through the week? Do you think they're different this week, sitting three and seven, needing to, you know, make the most of this last seven games? No, I think guys were just ready to win. I think that was just the main focus. It's getting the win column. And that's, you know, been the whole, if it's just all week, let's just win the game. Have a great week of practice and let it translate on Sunday and then go out there and execute and play the way we want to play. You know, the two rookies you have blocking for you on the offensive line, how do you feel like this last stretch of the season can be a good moment for them to kind of prove themselves and also get you guys success? Oh, yeah, I mean, just a hell of an opportunity for those guys to get in and show what they can do and keep building as the season goes on. And then, you know, just also building for the future and getting in position and doing whatever they can to help us win, held opportunity for them and excited to see what Jaylin does and all the other rookies to get an opportunity. How do you like their performance today? I thought they played great. I'm gonna focus on what I did and how I played. I mean, I thought those guys tried to do the best they can to help us win and that's all you can hope for. How did you feel like you did? You talked earlier in the week about trying to be more effective with your touches. Did you feel some more effectiveness in your carries this week? No, I feel like I could have done a little bit better today. But just watch the film and grow forward, grow from it and just try to be better and try to work on it as this week comes up and just try to go out there and let it translate. But I could have been better today. Two for two in the red zone. Was it as simple as being able to run the football? I think it was just getting in there. We get down there and getting in there. And I'm thankful that I was able to help us and put some points on the board. But, you know, we were two for two, so that's good. How big of a moment was for you guys to bounce on that immediately? Yeah, that was cool. The defense got a turnover and then two plays later, we were able to score. So, you know, you definitely want to capitalize on things like that. I was glad we were able to. Our boy yesterday used to stick out guys like Darren Banks after a game like that. I haven't talked to Banks in a while, but glad we were able to win.   So, being able to do it, so I'm happy. Thanks, Derrick. Thanks, Derrick.


Hi, I am Dr. Raj Shekhar C. Jaka, consultant surgical oncologist and robotic surgeon, Manipal Hospital, Whitefield.  Fortunately, thyroid cancer is one of the curable cancers. There are four varieties of thyroid cancer. One is papillary carcinoma and the second one is follicular carcinoma. These two are very common cancers and majority of the times they are curable. The third one is medullary thyroid carcinoma and the fourth one is anaplastic carcinoma. Medullary, if treated early, yes, it is also curable. But if it is late, means if there is a spread, then the cure becomes difficult. Anaplastic carcinoma is a very aggressive cancer. So majority of the times it is not curable. But fortunately, majority of the people come with papillary or follicular carcinoma which are curable. And here, we don't have treatments which are having complications like we don't have chemotherapy, we don't have radiotherapy. It is only surgery followed by radioactive iodine treatment. So the primary treatment of thyroid cancer is surgery. I always say the first treatment is the best treatment. So first time surgery is the best chance to get cured. So a proper surgery where we remove entire thyroid gland and if at all there are enlarged lymph nodes or probable risk lymph nodes, we remove those lymph nodes. We say neck dissection. Either it is central compartment neck dissection or sometimes if there is a spread to the lateral neck, then we do lateral neck dissection. So we remove complete disease from the neck and then over a period of three weeks we wait. And if at all we find that the pathology report says it is little advanced, in that case we ask you to get a radioactive iodine treatment. This is also very simple treatment which is just a tablet to be taken. So after this, you will be put on one tablet which is a replacement of thyroid hormone and also it will act as a suppressor to the redevelopment of a cancer. So in that way, hardly you are taking any other medication which really causes any complications or side effects. This thyroid hormone is as natural as possible. We usually don't do PET CT scan like any other cancer. Here we do radioactive iodine scan after the surgery. So the scan will not be done before surgery. The scan will be done after the surgery to know whether there is any residual disease in the body and that will be removed by or killed by the radioactive iodine treatment. It is more common in females. So I would urge any male or female who sees a swelling in the throat that is in front of the throat here. This is the right place where thyroid gland can get enlarged. So if it is enlarged, please come to a doctor, the qualified surgical oncologist and get it checked. Any thyroid swelling is not always cancerous. Majority of the times it can be non-cancerous and some of the patients I have heard saying that they have done a blood test, thyroid was normal. Thyroid was normal doesn't say that structure of the thyroid was normal. It is the function of the thyroid which will be evaluated through the blood test. The cancer will not be shown in the blood test. Blood test can only tell whether thyroid function is normal or not. And in cancer, majority of the times function will be normal. In spite of that, there can be cancer. So that is why blood test is totally different, thyroid function test is totally different and an ultrasound scan and an FNAC, fine needle aspiration cytology. These are the two things will conclusively tell us whether it is a cancerous or not and guide us further to treat you better. We have a fantastic infrastructure in Manipal Hospital Whitefield here to treat for this thyroid cancer and we have the most modern technology that is a robotic surgery. Whenever it is feasible to operate through robot, we will be using that Da Vinthi robot and we will be operating. Basically this robot is used for a people who doesn't want a scar over here. So from the axilla or some other approach, we go into the neck and dissect it so that there is a scarless, we say scarless surgery or scarless neck. It is basically for the cosmetic reason. So from the high end technology to the basic technology, we have everything at Manipal Hospital Whitefield to take care of you.


Outright rejection is more likely than outright acceptance. But unless there are serious flaws in the study design, in the data analysis, interpretation of results, or the writing itself, the most common reason for rejection is that the article was judged either not of interest to the readership or not within the scope of the editorial policies.


All right, if you remember now, looking at rigid body motions, specifically we're working on the kinetics of rigid body motion and as such, we had to break it into a couple parts. So we actually can break this into, well, we're going to break it into three parts. The first part we're on, but it's also broken into three parts, so the first part that we're working on now is the F equals MA, part of the kinetics, not just acceleration, but it's how we cause that acceleration, but we also now are including in there this general business of we're not just strictly accelerating things, but we're also rotating things. This we're also breaking into three parts. We did the first part Monday and that was pure translational motion, however, for the first time really, well, I guess for the first time really in the work we've done over the last couple of years the actual size of the object really mattered in the whole part. We did some things with some friction business where we wanted to see if an object would tip over if we pushed it and then we cared about how big it was, but for the most part those were acceleration problems, those weren't dynamics problems, so that required us to pay attention to the size of the body, and of course that had to do with the moment of inertia of the object. Remember, however, in the pure translational motion there was a bit of a trap and it came with this particular equation here or its alternate form where instead of summing the moments about the center of gravity sometimes it made more sense to sum the moments about some other point, just as it had in statics all those times. Sometimes problems get a little bit easier if you pick a different moment about which to make your sums. Sometimes the forces disappear from the equation, so the solution is just easier. If you do that, don't forget that these subscripts, you can use any form of this equation as long as these subscripts match each other, and that's where students sometimes get into some trouble, because for translational motion naturally there's no angular acceleration. If there was, then we have a rotational component, and we don't when we're talking about pure translation. In that case, then this alpha equals zero, but this one does not, because remember that we had to add on to it. This is actually an angular momentum term, and now this alpha here equals zero, this term left over does not equal zero, which shows that this alpha is not equal zero. As much as you may be tempted to make it zero, it's not. It's a very easy trap to fall into, which is my subtle way of saying I've fallen into it, so I've found over the years it's just best to really make a belabor point of being careful with this. Today we're going to look at pure rotation as part of the problems, and in this case now there will be some angular acceleration, and this doesn't become as much of a problem, because alpha isn't zero in any of the parts of it, so you're not going to get trapped with the difference between these two things. If we need to sum them all at a different place, we'll just do it. It is still true that this part must be in here if we're not summing about the center of gravity, but we'll just add it in, and we'll do a couple problems with that, and hopefully it'll make some good sense. And then Monday we're going to put these two back together to get the general motion of rigid bodies, general motion being both translation and rotation at the same time. Again, the simplest example of just that type of motion is a car tire. As the car moves along, the tire rotates and translates, then we've got to put those two motions together to get a good feel of what we're doing. Alright, before we go any further I want to make sure that you know how to get this moment of inertia when needed. So there's five ways in this class to get the moment of inertia. Now remember if you need the moment of inertia about another point, use the parallel axis theorem. But five ways to get the moment of inertia for these problems. Not in any particular order they are one. It might just simply be given. You'll notice in some of the problems when you need moment of inertia, it just simply says so right on there what the moment of inertia is. Pay attention to the moment of inertia about which point or axis. Two, you might need to look it up. If we're working with a regular shaped body, then it's just in the tables and you've seen those before. We have 2D tables. Those are tables I think it's 3D and 3D tables 3D objects is table 40. So if it's just an ordinary regular shape, go there for the moment of inertia in particular simple shapes. If it's neither one of those, you might need to calculate it. And that can take two forms. One you might need to integrate it directly right from the definition of the moment of inertia. Integral of bar squared dm. That will work for semi regular solids because you're going to need the functional relationship between the shapes to do the integration. But as we established in the earlier class this morning we still love integration as much as anything we ever do. So that's a possibility. It's also a possibility to remember to take various breakthroughs from the tables and combine them into a compound shape to get the moment of inertia you might need. Again this is no particular order. It's just maybe the thought, the steps you go through until you find it. You can also get it from the definition of the radius of gyration which is the, we use the little letter K. It also is about a specific axis. So again these two subscripts must match each other. You can have the radius of gyration about a different axis just as you can have the moment of inertia about a different axis. This is an artificial construct. It's as if we've taken whatever object we're talking about and we know whatever its moment of inertia is. We move all of the mass out to some distance from the center of rotation K, the radius of gyration and it's as if we put all of the mass right there. So if we've got some wheel of some kind and it's got a rim on it that might be thicker than the inner part or even could be a spoke wheel of some kind. Whatever its moment of inertia is, we can get an idea and a relationship between that and the radius of gyration. If we imagine an artificial object with all of the mass located at this distance, the radius of gyration, those two things, if they have the same moment of inertia then the radius of gyration is the place where all the mass would be concentrated in an artificially infinitesimally thin ring at that distance. It's just another way of representing what the radius of gyration is, what the moment of inertia is and then the fifth way. Anybody know what that one might be? No, that's not on there. Asking me it's not on there. Actually it might be worth it on tests. It's really a type of test to forget to put something in. I never do that but I've heard of it in the past. There are problems where you just don't need it. You might think you do but as you work farther through the problem you'll find out that it later cancels. If you remember we had some problems over the years where we needed the mass but the mass ends up cancelling and we don't need it in the end anyway. So there might be some problems where you think you need it but you don't and you might not need that and you might not know that until you get a little farther. So if you're doing a problem and you don't have the moment of inertia and you think you need it, take the problem a little bit farther and it may just simply cancel out and you're fine without it. Then you don't have to bother me. Which at this time of year you do it your own peril. That is on the tape. Alright, so we're ready to look at a couple of rotation problems. Alright, the general setup of what we're doing can illustrate with a simple picture. We've got something that can rotate in this case what's called a physical pendulum. It'll have some place where its center of gravity exists and it may have some angular acceleration and it may have some angular velocity. I'll just arbitrarily choose them in that direction mostly because when we do so that establishes our tangential and normal directions if we so choose to use those. The tangential direction is always in the direction of the motion at the moment as shown by the omega in that direction and then the normal direction of motion, normal coordinate direction is always towards the center of rotation. So that just simply sets up our general method. Some of the problems, those coordinate directions happen to fall in the typical direction of the horizontal vertical direction we're used to so often and so you can just go ahead and use the IJK there if you want. But we've got a couple things that we can use for our typical two dimensional problems as this one is. We know that if the forces are unbalanced on it in any way then the center of gravity will accelerate. So for our purposes that's two equations because that's in the x and the y direction that we could do that and then we can sum the moments about some point and it may be the point g it might not be and for our two dimensional problems and that's one equation. If we're doing three dimensional problems then there are six of these available to us. Oops, I need the m in there. That makes more sense. There we go. And these are our kinetics equations of rigid body motions. And then of course we've got any possibility, any available kinematics equations that we can bring in as needed. For example the acceleration of the center of gravity will be relative to the acceleration of point a if point a is the pivot plus the acceleration of g relative to a. But of course in pure rotational motion like we're looking at today that pivot point doesn't accelerate. It will when we look at general motion on Monday but today it doesn't. We're looking at just pure rotation. So the center of rotation itself does not accelerate. So we can count that as, well let's see, we can do more with it. Remember then that that is r alpha in the tangential direction plus r omega squared in the normal direction. And typically for us that, well depending on the problem I could count as either one or two equations. And then the other kinematic piece we can bring in has to do with the velocity which is also dependent upon the velocity of point a and the velocity of g relative to a just as we looked at the week before. But in pure rotational motion point a is not moving. So we've got then just the piece r omega in the tangential direction because there is no normal component to the velocity of g. g cannot move anywhere closer to point a or farther away from point a because it's a rigid body. So there's two more equations there and the kinematics that we can bring in and any others that might be appropriate. But that pretty much covers it all for the ones we want to use. So that's the general setup of where we're going today. We'll do a problem now to help illustrate it. So imagine we've got a slender rigid bar at that instant. It happens to have a slender rigid bar already some angular velocity to it. It's 3 meters long, has a mass of 20 kilograms and just at that instant as it's rotating past the horizontal a moment is applied about this point rather than certainly rather than g because g is at the center. The type of thing you do, you know this is some kind of hatch coming down and you need to apply a breaking moment. You'd apply one in the opposite direction and it might not occur or tell some particular point in the motion. This one we happen to have it as an aiding moment but could be virtually anything as we go through this. So we'll give that a magnitude of 60 Newton meters and omega zero we'll say at that instant is 5 radians per second. And so we want to find two things in particular. One, once it passes that point and once it has the moment applied what's the acceleration now and what's the reaction at a? We'll find that it's a little bit different than we might have thought if we'd approached this with the nae and the te we would have had doing it in statics. Two things we generally need to do. Generally to find forces we need a free body diagram and to find something about the particular motion we'll need a kinetic diagram. You're welcome if you're careful enough to do these on the same diagram but just be careful with it. If you're drawing acceleration vectors on a free body diagram make sure that they don't look like force vectors and you then put them into the particular solution. So free body diagram of course we have r that has some particular weight to it. We don't take that as an unknown since we know what the mass is. There's also some reaction at a so we'll break that into components. Let's establish a coordinate system. We'll just use normal and tangential coordinates tangential down because that's the way it happens to be moving at the time. That will give us a positive on some of the velocities and accelerations we need. And the normal is always towards the center of rotation. So that will be a in the tangential direction and then a in the normal direction as well. Is that it? A free body diagram we also have to draw any applied moments. So we have this applied moment ma as well. Anything that's any of the kinetics that's affecting the kinematics of the object goes on the free body diagram. Alright then the kinetic diagram we'll put what we expect to be the result of any of these forces and moments as applied. So naturally expect the kinetic that the center of gravity will accelerate and certainly at that point we expect it to celebrate. Well that's our tangential direction so let's just leave it as that. We will sum the forces in the moments to determine what makes up that bit. We also expect some angular acceleration of course but we're trying to find that not only is it falling past that point and would probably be accelerating in some way and I believe that was a physics one problem that we did. It also has this moment applied to it at this instant that will contribute to the acceleration. What else? There's one other piece and this is the piece that's easier of any of them to forget and it's also the piece that will show us that this normal component on the reaction is not zero. Certainly our temptation would be say oh this has got to be zero but it's not. These are not static problems. David? Centripetal acceleration. The centripetal acceleration and we'll put that as acceleration of the center of gravity in the normal direction and that's always defined as straight towards the center. So what we've drawn in the free body diagram causes what we expect to see in the kinetic diagram itself and so we do what we've always done when we're looking for accelerations and reactions the length of some of the forces and the moment. So some of the forces in the normal direction I guess tell us what that normal acceleration is and then we'll do the same thing in the tangential direction and we'll do the same thing with the moments. So this part, this is, well we've done this kind of thing before so let's see. A and that's in the positive direction of how we've established, well because the normal coordinate is always towards the center of rotation. A t of course doesn't figure, ma isn't a force anyway and so that's all there is but again that's not equal to zero because that's the centripetal, that's a result of the centripetal acceleration. And that we know to be, just like it always is, r omega squared. And so I think that's I think we've got all those pieces so we can actually calculate this one. Mass of the objects 20 kilograms. What do we mean by r? Is that the whole length of the bar or this distance out to the center? You said something, John? Yes, it's the distance from the center of rotation to the center of gravity because it's the center of gravity we're talking about right here is the acceleration so that's how we have to peg it. So that r will be 1.5 meters omega. At that instant remember it has an angular velocity of 5 radians per second. So that will give us, let's see, kilogram meters per second squared. Remember when we use the kinematics and the kinetics the radians disappear as a unit. So the units work out and that comes out to be 750. Is that right? 750. Well that one was easy, just kind of a warm up. You know, get the juices flowing here now. So we're having a tangential direction. Some forces, if they don't balance and we certainly wouldn't expect them to we'll have some tangential acceleration. So let's see. Positive is down, that's the tangential direction so it's w minus a t. Those are the two vertical forces. Remember the moment's not a force so it doesn't contribute to this. And that's going to cause the center of gravity to have some tangential acceleration but that tangential acceleration is going to be r alpha. We have m and r but we don't have alpha. In fact that's one of the things we're supposed to find. So there's the first appearance of r unknown. So we've got to keep going. We have two unknowns. Alpha and a t are in here. We need to figure out more. We'll sum the moments about the center of gravity and that should give us some idea of what the angular acceleration is. We'll sum about the center of gravity because for a slender rod that's easy to find. It's one of those ones that's in the book. IG equals 112 ML square and we have all those. We know the mass is 20 kilograms and we know the length is three meters so we can figure that out. As requested this morning we want me to do the hardest stuff so you don't have to. So this is 15 kilogram in our square. Alright so let's set up. We'll take the, we expect the acceleration to be clockwise so we'll take that as our positive direction. It doesn't hurt to indicate that just so you don't mess up. So sum the moments about G. MA is a moment on the entire piece so it goes in there just like that. It's all we need to do with it. AN goes through G so it has no moment that it contributes. AT has a positive direction to it which would be, its moment would be ATR and W goes right through G so we don't put it in there either. So that's good. We haven't introduced any more unknowns. I think we only have two unknowns anyway so we're coming up on the two equations so we need to solve it. And that's going to be equal IG alpha and we know what, we know what I is and we're just looking to find alpha. So there we go. There's two equations, two unknowns. Two unknowns alpha and AT are two equations that, two equations we just came up with. So as you go through these be careful what we're doing here in dynamic is different than what we are doing in statics. Don't try to apply your statics equations unless you know for sure that in a particular direction there's no acceleration. It would be very, very tempting to have this point said AN equals zero. But it doesn't. As we saw, it's not only not zero. It's, we're 750 atoms. Alright, two equations, two unknowns. Then we can solve it. It's just algebra left so I won't belabor that. We have alpha 5.9 radians per second squared and we only need the tangential component of a reaction force because we already have the normal component and this is 19 N. So in this particular situation it's that normal component you might have thought was zero that is significantly larger than is the tangential component. That, at least to me, was not at all intuitive. So you've got to trust the equations. You've got to be careful with them. You've got to avoid the temptation to just start knocking some stuff off as zero because it should be. It will make the problem easier. As calling a bunch of stuff that's not zero, zero always makes the problem easier. It just also makes the problem wrong. So be careful with it. Alright, questions on that before I clean up? Pretty straightforward. Just be careful. Take your time through it. Make sure you get your minus signs right. Make sure you don't miss any of the parts but that's always been the case. Certainly ever since you started with me in Physics 1 we've been watching minus signs and making sure we're doing the complete problem. So any questions before I clean up? And we do an alternate solution to this? Okay, alternate solution. Well, there's no easier way to have found that A n. So we're not going to find an alternate solution to that. That only appeared in one equation and so there's only going to be one way to solve for it and that's summing the forces in the normal direction that we already did. But we have an alternate possibility. So, alternate solution. Let's sum the moments, not about G but about A. We have two forces that go through A. One of them is unknown. So let's sum the moments about A and see if that isn't a little bit easier. Remember though that we need the moment of inertia about A. But for a slender rod that's no big deal. That's one of the ones in the book. You might even remember that for a slender rod it's just four times what it was about its center of gravity. That's only true for a slender rod. It's not true for other pieces. It's just true for this for a slender rod. And by slender rod we mean that the diameter is much, much less than the length. So we're not even concerned with the diameter. Okay, so let's give this maybe a try. Let's see. MA is still there. That's just one of the givens. That's the kinetic set of driving the whole problem. And W causes a moment in the same direction. It's moment arm is R. They happen to be perfectly, perfectly a moment so that's easy. And that's all there is. The other two forces go through the point A. So it's then I, A, alpha. And the only unknown in that problem is alpha. And so it is a little bit easier to solve for. Much to R relief you get the same thing as you did before. Still though you need to solve for the other component of the force. So you have to then sum the forces in the tangential direction and actually get the tangential component then with the force there. So it doesn't cut down on the number of equations but it does mean that each of the equations independently solves for one of the unknowns and you don't have to use a system of equations. So it's a little bit easier in that regard. Questions there because we need an alternate, alternate solution. Just when you were hoping things were easier to decide about, they aren't necessarily. So we'll have an alternate, alternate solution. And can't do anything with that but what we can do is just look at the summation a little bit differently. It could be that we didn't have something as simple as a slender rod to use and thus maybe it would be a pain to find the moment of inertia about a different point. So we can go back and do it about the original point, the center of gravity. But we need to take into account the fact that that's not a point where we're summing the moments themselves. So you have to put it together like that. And taking the clockwise direction as positive as before because that's where our alpha goes. That looks like ma plus wr just like before because we're still summing the moments about a just as we did before. It's just now things are a little bit different because we use ig instead of ia. The mass is the mass of the bar. Ag with respect to point a is r alpha. So we only have one unknown in this equation but it does appear twice. And then d is the distance between the two points which for us is that r. That's ag. Specifically ag in the tangential direction. Then r is just that part d. So the numbers come out to be exactly the same as they were before. I have a question for you. This ag, I've got a bt taken care of. How do we handle agn? Because that is not zero. That took care of agt, not agn.  I'm working way harder than I am. It's my chalk sponsor. Chalk does come off the board. Oh come on. I don't generate chalk does do I? I'm placing it there for a couple million bucks. Alright. What did you decide? How do we handle the normal component here? Because this is not specific to either direction. This is just the moments about a and the kinematics they cause. Who's got an idea? He's great over a million bucks. What? He's great over r. Well we can certainly do, we can find that. It's r omega squared. We have both of those. So I could certainly put it in. In fact, just for the sake of argument, I will. So there's r omega squared. That is ag in the normal direction. That's normal direction. David? I was thinking it didn't count because it was parallel to d. No. You're close. But you need to be even more specific than that. What is d for this component? It's not r. It's zero because this component goes straight through the point a and it's the minimum distance between that acceleration and the point of interest that is d. So it has a d on it, but for that component of the acceleration, it's zero. So I told you this was going to be advanced physics one. Some of these things are pretty subtle. You need to trust the physics, trust the equations, and be careful about jumping to conclusions just because some things might fit. They don't necessarily. So one equation, one unknown alpha, it just happens in two places so it's a little bit more complex. It also has this potential trap in it. But that's why it's the second alternate solution and not the first alternate solution. But you get the same answer. You get a little shell shock from the dynamics here. Yeah? Oh, let's see how we're doing. It just gets hard and hard. Now we'll do it in two pages, I promise. It need to be a bi-ish paper. Okay. Another problem. Harking back a little bit to physics one for you, those of you that had it, not those of you who had it with me. Those of you who had it with me. All right.  And we have a mass on it. All right. Starting from rest, find the angular acceleration of the pulley. All right. Some of the details, the radius of the pulley, 0.4 meters. The mass of the pulley, m sub p, we'll call it, 60 kilograms. And the radius of gyration. What this really tells you when we have that instead of the moment of inertia is that it's not a uniform density solid. Certainly it has mass distributed in different places. Maybe more at the rim or something. Maybe there's some spokes or something that helps it turn on the axle, whatever it is. Doesn't matter because that leads to it directly to the moment of inertia if that's what we need. And that is, it's a regular length. It's the radius at which we can concentrate all the mass and still have the same moment of inertia. And then the little mass there itself called the mass of the block is 20 kilograms. All right. So find the angular acceleration of the pulley. So I'll leave that to you a bit. We see what you can put together. Start with a free body diagram. And separately if you wish, or you can combine a magnetic diagram as your purpose.  There's no stop in Travis. Do not get between Travis and the solution. All right. So set up your free body diagram. Do not assume forces are zero because we just showed a problem where might have been very tempting to say certain forces were zero and they're not. And remember every force is caused by something real. There are no forces caused by the motion. The motion causes reporting can also have two fugues. So get just the right number of equations under the forces and the free body diagram and proceed. A few minutes on the free body diagrams. Some of you are getting them. Some of you are throwing in a couple extra just for fun. Yes, John, what are you doing? I'm throwing arrows in all over the place. Notice everybody's got the basis here for a free body diagram. Of course the pulley weighs something. So there's some force down. It's counter by reaction. We'll call it A. We'll call it AY. That thing's not gentlemen any longer so we might as well use XY ordinance. And then AX. I happen to pick in this direction of course the tension in the line. That's actually what's causing it to turn. That's the applied torque as far as the drums. And then we've got the mass with the same tension up and its own weight of a WP for the pulley and WB for the block. So that should be your free body diagram. Any major disputes? The kinetic diagram is the cause of those forces. It's going to cause the wheel, the pulley, to rotate. It's not going to cause the pulley to do anything else. That's all it's going to do is it's going to start to rotate. It's going to require some angular acceleration. Nothing else. The block will have some translational acceleration. And of course those two are related. Alpha where R is the radius of the pulley is going to be the acceleration of the block. How can I say that? That's a bit of an assumption but I talked about it the other day. Remember what I called that? That is only true if the rope is not slipping on the pulley. If that's slipping on the pulley, if the rope is slipping on the pulley not wrapped around well enough, then I can't say that. That's equivalent to saying that this point on the edge of the rim will have the same acceleration as the block at that instant. That point also has a normal component because it's in a circular path but that has nothing to do with the block. The block is not going to have a normal component to the acceleration because that point will go in but the contact point with the rope will not stay where it is. So now we've got enough of the pieces and you can now start to solve the problem. So we've got one, two, three, looks like four unknowns. However, two of them, the two reaction forces go away if we sum the moments about the center point. We're not asked for those so that would be a good place to start. Plus the radius and the gyration that we have is about the center. So I don't know if it's worth calling that A or calling it G. It doesn't matter. It's just semantics. The center of gravity are the same. Okay, pick the positive direction as the direction we expect the positive acceleration just makes things a little bit easier. Less hassle with minus signs. So the moments about point G or A, the center, is just the tension at a particular radius. And that is positive in the direction specified. And that will equal KG squared MP alpha. Two unknowns, one equation so we're not ready to solve yet. We've got to do something else. It'd be nice if the two equations didn't bring any more unknowns into it. If we can do an equation without AX and AY, we'll be that one's closer to the solution. If we bring in that equation as A and X, Y, we've got to keep going because we've got it in an unknown. Right now we have two unknowns. If we can keep it at that one more equation, we'll do it. Well, we sum the forces in the vertical direction, that brings AY into it. What? R alpha is equal to AB, where AB is the absolute block times acceleration of gravity. Well, if we sum the forces on the block, it'll give us that acceleration, but that acceleration we know to be directly related to alpha. So technically we would bring in another unknown, but it's very easily taken out, perhaps our simplest equations. So sum the forces in the Y direction on the block will cause the block to accelerate. And we've got T up, WB down. We've got to cause the block to have an acceleration that we know to be related to R alpha. So there's a second equation with just as many unknowns as before. So it's a matter of a high school algebra to solve that, and you'll have to do that because I didn't write it down. Oh, wait, there it is. 11.3 for the angular acceleration. We should guess ahead of time how it's kind of an office pool on what these numbers should be, something like a little bit of cash before we head out for summer vacation. Alright, there's another way we could have solved this one, and it eliminates an unknown by taking the block and the pulley as a single system, and not separating them like we did here. We didn't need to find T, so it may not have been the most prudent thing to do to bring it into the problem. Still, we need a free body diagram. So we've got the weight of the pulley there, and the weight of the block there, and the two reaction forces, neither of which we were asked for, and we didn't bother finding them the first time. Since we don't want those, it makes sense again to sum the moments about that point, where this is of the entire system. We can make it a little bit simpler if we use I, G. Remember point G and point A are the same anyway, but we need to add on the effect of the acceleration of the block itself. We don't know the moment of inertia of that whole system with respect to point A, because the block itself changes that, but we can split that out as a separate part then. And now we don't need to know the tension, nor do we need to know the reaction forces. So it comes out to be very similar to what it was before. None of those have anything to do with it. We have w, b at a distance r. It's the only moment applied about point A, and then this is KG squared m alpha, mp alpha, and I had that over there design. That's just the moment of inertia from the radius of gyration. And this is, by the way, that's the block, and it's the point of it being off center. So that's mvr, alpha, or this is the radius of the pulley, that again is our no slip condition. And then d is the radius of the pulley yet again. That little piece is the acceleration of the block, and this little piece is d, the distance of the, actually the line of action of that acceleration with respect to the point picked, which is point A. Again, it's a trade off, there's only one unknown in there, just the alpha. Everything else is given. However, getting to that point is a lot more subtle than I think this was. That's a lot more straightforward. So, maybe it's an academic exercise for you, which one you want to do, but it's your choice. Even if you're not going to use something, I think it's helpful to see how it works, even without using it. So, oh, I think we need to get out of class question at the end of the weekend. What? Yeah, but then I got to add lid for 12 minutes, and that's not pretty. We've all seen the kind of things I say when I'm winging it up. So, here's a problem for you. It was actually a quiz problem from a couple of years ago. Pulling a crate up a ramp using a winch, and the details are, like, it's 20 degrees. It is coefficient of kinetic friction between the block and the slope is .4. The block is 45 kilograms, and the winch is radius of 15 centimeters, a moment of inertia about the pulley part, 4 kilograms, 5 meters per second, I'm sorry, for meter squared, kilogram meter squared, and the moment applied by the motor to the pulley is 50 Newton meters. That's the motor turning the pulley itself. So, not only does the motor have to haul up the mass, it also has to get the pulley itself started. Let's see, find the, well, just set up the equations. Finding the acceleration or something is just a matter of solving the system of equations. That's just down front. So let's get the physics set up. Set up the equations, however you choose, and then then it's just a matter of, I've got the equations right, then it's just algebra after that, and that's not all we're here for. So, free body diagram. The kinetic diagram's pretty simple in some of these problems. It might not be worth it. It could be worth it in more complex problems. It might be worth it, not worth it in these, especially in small, some of you draw. Okay, so set up the free body diagram. Set up the equations. You will need all three forces in the x direction and y. It probably makes sense to incline them with the slope. If you're looking for the acceleration of the crate, if you incline your axes with the unknown, it usually just makes the solution a little bit simpler. But you can do it any way you want with it. Free body diagram of the crate. Now, some of that is statics. The dynamics is just the t, a little bit of w down, and the f down. But when you sum the forces in the different directions, it comes out in the wash anyway, that's how we get the normal force that we need for the friction. And then for the winch, we've got the tension. We've got reactions, and it's weight, but the weight wasn't given, and the reactions weren't asked for. So all we're really concerned with is the dynamics of the situation, which is just the tension that's causing some angular acceleration. Oh, it's got the moment on it too, of course. But all that's going to result is that alpha. So we can wrap everything up with some of the forces in the x, y direction on the block, and some of the moments on the winch should be able to relate all of those. So I already said that the box is a crater, whatever, coffin. Not up, we're talking about murder history. We're talking about murder history. We're talking about patient stand-hold. I'm going to find out until later, while north of here, so it's a long drive. The mass? Nope. Well, yeah, it does, but not given, because it exists. But those are static forces. This is the dynamic force that causes the acceleration. So we don't need the mass on the pulley, the mass of the pulley. If you sum the moments about the center of gravity of the pulley, those don't come into it, because they go right through there. So I'm not concerned. If you sum the moments about g on the pulley, all that's in there is t and m, and n is given t we don't have, but that's what's going to cause the acceleration of the block up the table. So we'll be able to find that. We're overseeing the acceleration of the block. So just set up the three equations, and we'll see what we've got if we all compare. On the block, we sum the forces in the x direction. That's what's going to cause the acceleration we're looking for. That'll have in it the friction force. We need the normal force before we can solve for the friction force. So we'll also have to sum the forces in the y direction, but it is in a static balance in the y direction. So the dynamic balance in the x direction, but not the y direction. So that'll give us n, which will give us f. So we'll have two unknowns, the tension in the line and the acceleration of the block. And then the tension in the line we can solve with the third equation. Sum the moments about g for the pulley. That way we don't care what the weight of the pulley is. We don't care what the reaction forces are supporting the pulley on the motor. That will leave us with three unknowns. From this one, we're going to have t up, f down, which is mu n minus w sine 20. Is that right? And that's all the x components. And so that's going to cause the block to accelerate in the x direction. We're looking for that. So that should be the equation. And then we also need, of course, that f equals mu k n, which we find in the y direction. So that's mu k w cosine 20. The sum of the forces in the y direction is solely going to find the normal force for us that we need for the friction. So how we do it? t is unknown. f will take as known because it's just a matter of filling all those little pieces in. And the acceleration of the block is unknown. So some of the moments about the center of the pulley taking the positive direction to be the direction of the acceleration. Just, you don't have to, but it makes things a little easier. So we've got the tension kind of turn up one way. The moment exerted by the motor trying to turn it the other way. And the difference between those two is going to cause, actually I do the acceleration in the wrong direction. So that's a matter of coming out now as a minus sign so we knew we suffered during the wrong direction. So now how many unknowns? The tension, the acceleration, the tension's already there. R's given, M's given. Alpha's not given. Three unknowns, two equations. This doesn't count as a second equation because we can use that just to find n. So essentially two equations, three unknowns. Not treating that friction force as an unknown. Just because it's a matter of filling stuff in. What's the third equation? Forty-five seconds into your weekend. Third equation. Who's going to rescue the class and send us all home? Travis? The no-slip condition. Assuming the cable's not slipping on the pulley. There's your last equation and now it's just a matter of the algebra to solve it. So there we go. We'll do general motion on Monday.