Hey, guys. So in this video, I want to show you how to calculate torque on a disc with forces in a bunch of different directions. Now, this picture looks sort of scary at first. What's going on with all these arrows? And you may not see something as complicated as this. I'm putting it all together so we can talk about it all at once, but you should know how to handle all these different situations individually or together. Let's check it out.

So here we have a composite disc, which means there are 2 different discs. You got the inner disc, which is the dark one here, and then the outer disc here. This allows you to have 2 different radii. They are free to rotate about a fixed axis perpendicular through their center. So basically, the discs can spin this way, right? All the forces listed here are 100 newtons, and there are 4 of them. So let's just write f1, actually, let's make this one f1, f1, f2, f3, and this one is going to be f4. They are all 100, and the angles are 37. The only angle here is 37; the rest are either flat in the x or y. The dotted lines are either exactly parallel or exactly perpendicular to each other. What does that mean? That just means that this line is the same as this line. They're parallel to each other, and these lines here make a 90-degree angle. So all these lines are making 90-degree angles with each other. Cool.

The inner disc has a radius of 3 meters. What that means is that this distance here is 3 meters, and the outer disc has a radius of 5 meters, which means that this entire distance over here is 5 meters. Okay?

We want to know the net torque produced on the disc about its central axis, and we're going to use plus or minus to indicate direction, clockwise, counterclockwise. So, the net torque is the same thing as the sum of all torques. There are 4 forces, so there are potentially 4 torques. Remember, a force may cause a torque. There are 4 forces. You could have as many as 4 torques, but some forces may not cause a torque.

Let's do one by one here. Torque 1, this guy is f1. Let's first leave a space for the sign, positive or negative, so we don't forget. f1rone sine ofθ1. And we'll calculate the sine a little later. The first thing we do is we draw an r vector, then we figure out θ, and then we work out the sine. Okay? So the r vector, is the distance, is the vector from the axis of rotation to the point where the force happens. In this case, the r vector for f1 is an arrow this way. This is r1. And r1 has a length of the outer disc, which is 5. Okay. And sine of θ. The angle that r1 makes with f1 is 90 degrees. So I'm going to put 90 here. Now in terms of sine, imagine you have a disc, and you're pushing this way on the disc. So you have a disc, and you're pulling like this, so the disc is going like this because you're pulling this direction. Right? You can also imagine as you're sort of stroking the disc this way, right? The disc is going to spin like that. So this is going to be a positive torque because it's causing the disc to spin in a counterclockwise direction of the unit circle. So it's positive. This is 1, and you just end up with positive 500 Newton meter. Alright.

So, torque 2: box f2r2sine of θ2. I know f2 is 100, but I got to figure out r and θ. So I'll leave those blank. F2 is right here, f2 acts in the middle of, acts on top of the axis of rotation. Therefore, the r2 will be 0. R2 equals 0, which means there is no torque at all. When you have something that pulls on the axis of rotation, it produces no torque because there is no r. And you can see from the equation that the whole thing becomes 0. So it doesn't matter what the angle is, and it doesn't matter what the sign is because you just have 0.

For torque 3, again, leave space for positive or negative f3r3sine ofθ3 and the force is 100. We got to figure out r and θ. If you look at f3, f3 acts on the edge of the outer disk. This is what the r3 vector looks like. Right. R3 right here. So r3 vector has a length of the outer radius, which is 5. But the problem is these two arrows make an angle of 180 degrees with each other, and the sine of 180 is 0. Okay. Sine of 180 is 0. So it doesn't matter what the sign is because, whether it's positive or negative, the direction, because the torque will be 0. Imagine the disc. And if you push directly towards the middle of the disc, you don't cause the disc to spin. The only way to cause the disc to spin is to either push sort of tangentially on the disc or to push at an angle. Right? So if you have a disc and you go like this, it's going to spin. But if you push like this on a disc, it doesn't spin.

Now let's do torque 4. This is the ugly one up here, and let's figure out what happened. So, box, I'm just going to jump straight into it. The f4 is a 100 radius sine ofθ4. So this one, we will slow down a little bit and be a little more careful. Notice that it's touching on the inner radius, the inner disc. So I'm just going to redraw just the inner disc. I'm going to write here that this has a radius of 3 because it's the inner one. Let me make this a little bigger. Radius equals 3. This force acts like this right there. This is the center. First thing we draw is the axis, the r vector. The r vector is from the axis to the point where the force happens. Notice that here, this would look like this. So this dotted line is just an extension of your R vector. And there's an angle of 37 degrees here. Okay. So you got to figure out which angle to use. First of all, the distance will be the entire radius of the inner circle of the inner disc. So it's 3. And what about the vector? So what about the angle? So what you could do is you get the r vector here, and you can extend it this way. Right? And to make it easier to notice that this is, in fact, the angle you should use. It's the angle between the two lines. So 37 is the correct angle. Okay. Remember, the angle given to you isn't always the one you're supposed to use. In fact, it's usually not the one you're supposed to use. But in this case, it turned out to be that way. What about the direction in which this thing will spin? So you should imagine that if you're pushing a disc like this, it's actually going to spin like this. One way that would make this easier is to think of this as a not as a push on the disc, but as a pull. You're essentially pulling the disc. I'm sort of redrawing this f4 over here, just kind of extending it down. You're pulling, you're pushing this way. You're causing it to go like that. Okay. So hopefully that makes sense. It's pretty visual, but hopefully, you can follow that. So this direction here is in the direction of the unit circle. It's counterclockwise, so it's positive. Okay. Positive. And if you multiply this whole thing, you get that torque 4 is positive 180 Newton meter.

To find the net torque, we just add everything up. I got 2 of them that were 0. So it's just the positive 500 and the positive 180, which gives you a positive 680 Newton meter. Okay? That's it for this one. Hopefully, it makes sense. Let me know if you have any questions, and let's keep going.