Welcome to module 28 of Applications in Engineering Mechanics. We're going to extend our engineering mechanics knowledge to two new real world applications. One involving flat, flexible flight belts and one involving the belt. And so here's the flat, flexible flat belts situation, we got a flat drum, fixed drum and we've got a belt that's going over top that drum, a flat belt. we can also use this situation for a cable or a rope. We've got a large pulling tension on one side and we've got a smaller resisting tension on the other side and so the direction of impending slip is towards the large pulling tension. So, the friction in this situation between the belt and the drum is going to resisted that impending motion, and we see that we have got an angle of surface contact which we are going to call belta over which the cable or belt or rope contacts the drum. And so here are my my parameters Tl and Ts are the belt tensions, Tl being greater than Ts. And here's the relationship that we can develop for Tl and Ts. And this is developed in any standard engineering statics textbook, or you might even find this development online. So let's look at this situation with an application to a real world problem. Here's a gentleman who's holding up an engine. he, so, he is exerting the resisting tension on this side, and we've got the pulling tension in the the larger force in the engine itself. if I just wrapped that rope over the three once, or what we can refer to as the fixed drum. We have a surface contact angle of pi or 180 degrees, but in this case, we're going to see what's the difference if we go ahead and wrap it? You may have some practical experience with this when you're, and you may Wrap it, and you wonder, I wonder how much I've gained by wrapping it. Well, in this case, let's go ahead and wrap it an additional turn. So we're going 360 degrees plus another 180 degrees, or 2 pi plus another pi, or 3 pi surface contact. And so if I put those values in for a coefficient of friction a typical coefficient of friction of point 3. We find out that in the case with just a half a wrap, if the man was able to resist with the small tension of let's say 100 pounds. The large tension or the, the amount that he could hold up would be 2.57 times that 100 pounds or 257 pounds so quite, quite an increase. But look here if, if, if, if we rapid once now we have a contact angle of 3 pie if we put that in Now the gentleman would be able to hold up 1,690 pounds. And so, that's an increase by wrapping of, of, of, a tension ratio of 5.6 times. So you can get a lot of benefit from that. And here's another practical application, where we, we use that situation, where we, we, on a guitar string, we wrap The the string several times to hold the large tension. Okay, let's go on now to V belts. With a V-belt, instead of a flat, flexible drum, and a flat belt, what we have is a, a a v type situation. The equation that relates TL to T small, or T sub L to Ts, is slightly different because it includes this angle 2 delta of the, of the V, and 2 delta is typically between 30 and 38 degrees. And so let's look at a demo of this situation. Here's a pulley you can see that it has a V type shape. And the angle is somewhere between 30 and 38 degrees. And then V belt seats right into that pulley. And then you can have a tension on one side, and the other. And so a very real world situation that I'm sure you've, you've encountered. So, let's look at the advantage of that. We're going to look at what the increase in fiction is with a V-belt vs a flat belt. So here are my relationships for the V-belt, the flat, flexible, flat belt and the V-belt. So let's put in some typical values, so we'll use U of 0.26. we'll use two delta, the V-belt angle of typically 30 degrees, anywhere from 30 to 38 degrees, and we'll say that beta is pi. And so if I put those values in, I see that if I had a small tension resisting of 100 pounds, the increase on a flat, flexible belt would be 226 pounds, or an increase of 2.26. However, if I use a V belt now we're able to increase the tension by 23.5 times. And so you're able to hold or, or, or, or resist 2350 pounds with, with 100 pounds. And so you see that there's an increase in the V-belt over the flexible flat belt situation by more than ten times. And so that's, that's quite substantial. And so next time we'll come back and we'll Actually do a problem which involves these belt friction situations.