 Hi everyone. In this video I want to walk through the analysis procedure for looking at a belt or belts in a pulley system. So I already mentioned that belts can come in two different variations, flat and V being two common variations, and that it's often the smaller pulley that we are concerned about when it comes to slippage. And really slippage is what we're thinking about when we say, you know, how much torque can we transmit through this system? So basic analysis, we have our pulley, which is rotating around its center, and we have the belt coming off on either side. And that belt is going to have tensions P1 and P2, which are on either side of the belt. The angle of wrap or the angle of contact between the belt and the pulley will call phi, and then we'll go from there. So basically if we have this as our free body diagram, our torque that we want to be able to transmit through this system is equal to the difference between the tensions multiplied by the radius. So that's going to be our torque balance around that pulley. And if we would go ahead and carry out the analysis, what we would end up finding is that it looks a lot like the analysis for a band break. Indeed, it comes out effectively the same. So we get something like this, E equal to mu phi, or E to the power of mu phi, where mu is our friction coefficient. And again, phi is this angle of wrap of where the belt makes contact with the pulley. Now, one thing that we would probably want to take into consideration in this case as a difference between this and band breaks is that in a band break, we have a rotating mass and our band is stationary and the friction between those two cause the breaking feature. In a pulley system, we have a rotating mass on the pulley, but then our belt, of course, is traveling around and rotating itself. And what that means is that the mass of the belt itself is going to have inertia and cause a change in the tension in the belt. So because the belt is not massless, we have to take into account that change in tension. So we'll introduce a variable called P sub C, which is going to be a centrifugal force or a centrifugal tension in the belt. And it's basically an inertial tension where it is equal to the mass per length multiplied by the velocity squared. So here we're talking about, you know, kilograms per meter multiplied by meters squared over second squared. So if you cancel out that meter, you get a kilogram meter per second squared, which is a Newton. So we can see that our units work out if we convert this to rotational velocity, we have our mass here, and we can write it in terms of omega squared, r squared. So substituting those in for linear velocity and using the rotational velocity instead. So this is that centrifugal force or tension in the belt. And if we make use of this, we can substitute it into our prior equation and we would get something that looks like this. So we've factored in the centrifugal force and it acts as like a reduction on the overall tension being supplied by the P1 and P2 on either side. When we look at V belts, so this is for flat belts, when we look at V belts, the situation doesn't really change all that much. And I'm not going to derive the equation in this scenario, but we get something that looks very similar, where we have factored in now the sign of this angle beta. And beta is related to the V belt itself. So if the V belt has a shape that looks like this, the angle beta is basically just the angle that defines the geometry of the V.