Hey, guys. So in this video, we're going to start talking about torques acting on discs, on disc-like objects such as pulleys or cylinders. This is important because it's going to be a key part of more elaborate problems we're going to have to solve later on. So let's check it out. Alright. So let's say you have a disc or a pulley, and you're pulling on it like this with the force of f, which then causes it to spin like this. Now what matters, as it says here, what matters is r, which is the distance from the axis to the force, not the radius. In this particular example here, little r is the same as the radius because the rope pulls on the disc at the edge of the disc. But let's say you were pulling with another force right here, F2. In this case, the force doesn't pull from the edge, so what matters is not the radius of the disc, but the distance, in this case, r2 is not the radius. So what always matters is the distance; most of the time, the distance will be the radius of the disc, but sometimes it won't be.

Let's do an example here. So 2 masses, m1 and m2. m1 is 4. Let's put it here. m2 is 5, are connected by a light string which passes through the edge of a solid cylinder. The cylinder has mass m3 equals 10. And radius, remember radius is big R. Little r is distance. The system is free to rotate about an axis that is perpendicular to the cylinder. To the cylinder and through its center. So basically, the cylinder spins around its central axis. We want to know what is the net torque produced on the cylinder when you release the blocks.

The net torque, torque_{net}, is the sum of all torques and we want to know the net torque on the cylinder. So we must figure out how many torques act on the cylinder. Remember, a force may produce a torque. First, I want to show you how there is no torque due to m3g and due to tension because they act on the axis of rotation. So the torque of t, right, the torque of any force is F r sin of Theta. In this case, F is t. So the torque of t is t r sin of Theta. But the tension is pulling. The tension is pulling from the middle here. It's holding the cylinder from the middle. So this r is 0. Now this would have been 0 even if the tension was somehow holding it up here. The problem with this part is that the tension pulls it up. Let me draw it over here. Tension let's say tension was going this way. You have to draw the r vector from the middle to the point where the force happens, the r vector. And these two arrows are both going in the same direction, which means that the angle between them is 0 degrees, which means that here you would plug in sine of 0, which is 0. So whether the tension pulls in the middle or if it pushes in the edge, it doesn't matter. They make the same. They have the same angle with each other here. So this whole thing would be 0. The torque due to mg is for sure in the middle. So it's mg0, and it also makes an angle of 0 degrees. So both of these factors don't actually produce any torque. So the only forces that would produce a torque are m1 and m2. The only forces that cause a torque are forces that could cause it to potentially spin, and that's what you get with m1 that's trying to do this to the disc, and m2 that's trying to do this. So torque 1 and torque 2.

So if you imagine a disc, if you pull from the edge of the disc, it would roll from either side. Cool. So let's now calculate the torque due to these two forces. So I'm going to call this Torque_{1}, which is force, which is m1g r sin of theta. So what's the r vector for m1g? Well, it's acting m1 is acting all the way at the edge of the disk because m1, the cable for m1 is passing through the outside, the edge of the disk. So the r vector is exactly the radius. So r1 is the radius. And by the way, that's the same thing that happens with m2g. R2 is also the radius because both of these guys are all the way at the edges. The angle between these guys, both of them is 90 degrees. So look how the r vector and the force make an angle of 90, and the r vector over here and its respective force makes an angle of 90 degrees. K? So both of these guys, we're going to have that the distance is the entire radius. Boom. Boom. And the angle is 90 degrees. So this obviously becomes a 1. And now I just have to multiply the numbers. The last thing you got to do is also figure out the direction. Is it positive or negative, the direction of the rotation? The M1g is trying to do this. This is if you do a complete sort of spin with your hand, you see that this is counterclockwise, so it's positive. This one is in the direction of the clock, so it's clockwise, so it's negative. Alright? So torque_{1} is going to be positive. M1 is 4. G, we're going to round to 10, and the radius of this thing is 3. So this is going to be 120 Newton meter. And then for Torque_{2}, negative, the mass is 5, gravity rounding to 10, and the radius is 3 meters. So this is going to be negative 150 Newton meter. And when you add the whole thing, the sum of all torques will be 120 positive plus 150 negative. So the net torque is going to be negative 30 Newton meter. And that's it for this one. So hopefully, this made sense. Let me know if you have any questions. Let's keep going.