Hey, guys. So now that you know how to calculate torque in a bunch of different situations, we're gonna look into what happens as a result of a torque, which is that you get an acceleration. So let's check it out. Alright. So we're gonna look into torque with acceleration, not just calculate torque, but also maybe calculate the acceleration that results from it. And a combination of these two items, when you have an acceleration as a result of a torque, that situation in physics is called rotational dynamics. Okay? Statics typically refers to situations where there is no velocity. Right? So you're at rest, and dynamics means that there's going to be velocity or acceleration. So you already know that when a force causes rotation, it produces a torque or even when a force tries to cause rotation, it produces a torque. You can think of torque as the rotational equivalence of force. You know this as well. Now what I want to do is, since they're equivalent to each other, I want to do a compare and contrast between forces and torque. So force, remember, causes linear acceleration. And the relationship between force and acceleration "a" is described by this equation, by the sum of all forces equals ma, which we call Newton's second law. This is Newton's second law. Right? Now torque is very similar. Instead of causing linear acceleration "a", torque causes angular or rotational acceleration, same thing, alpha. Right? And the relationship between torque and what it causes, which is "alpha", is very similar to the relationship between "f" and "a". In fact, it's the same equation, except we're just going to switch the variables to their angular equivalence. So, instead of the sum of all forces, we're going to sum of all torques. Remember, the rotation equivalent of mass is not mass, what matters is your moments of inertia. So I'm gonna put an "I" here. And instead of "a", in rotation, we're gonna have "alpha". K. So these two equations are basically the same thing. Just one is with linear variables, the other one is with rotational variables. In fact, this is also Newton's second law, the rotational equivalence or the rotational version of Newton's second law. So you may see your professor or your textbook call this Newton's second law. And the idea is that both of these guys are Newton's second law. Alright? So the quantity of inertia, which is how much resistance you have to acceleration, to linear acceleration, is given by the letter "m", by mass. So mass is the amount of resistance to change, the amount of inertia you have, and the amount of resistance to alpha is not "m", but it is "I". Right? And the last point is that when you have force and acceleration, you have this branch of physics called linear dynamics, which in the past, we may have called it just dynamics because there's only one type. But if you have torque and alpha instead, you have what's called rotational dynamics. That's just the name. It doesn't really matter. But in case you hear these words, you know what's up. Alright. So that's the difference. So basically, you might remember doing a bunch of f=ma. Now you're gonna do a bunch of τ=Iα. And in fact, in some cases, you're going to do 2 of them combined. So let's do an example here, see how this stuff works.
Alright. So here it says I have a solid disc. Solid discs, remember, the shape of the disc, so I can stop and write at the moment of inertia is half mr squared. Solid disc of mass 100 and radius 2, so I already get those numbers, m equals 100, r equals 2. It's free to rotate, so it can spin around this, a fixed axis. So it can rotate around the axis, but the axis doesn't move. The disc is fixed in place and it can only spin in place. The axis is perpendicular to the disc. That means that if you have the face of the disc, the axis points this way, which just means the disc is going to spin around itself, and it's frictionless. Okay? You push tangentially on the disc. If you push tangentially on the disc, it looks like this, right? Like sort of at the edge of the disk, with a constant force of 50. So let me write this here, f equals 50 newtons. We want to derive an expression for the angular acceleration that this experiences. So part a, we want to find alpha, that's angular acceleration. And for part, we want to derive an expression. And for part b, we want to then calculate that. So find an expression and then calculate just means plugging all the numbers. Okay? So how do we do this? Well, I'm giving you a force, and I'm asking for an alpha. Back in the day, if I give you a force and ask you for an "a", you would use f=ma. But here, I'm giving you a force but asking for an alpha. So instead, you're going to use Στ=Iα and that's because you're looking for alpha. Okay. Now the only force that causes this torque that produces a torque on this disc is this force here. So the only torque we have is going to be the torque of "f". Now that torque, that force is producing a torque that's trying to spin this thing this way, which is a clockwise torque, so it's negative. So I'm going to put a little negative in front. K. The moment of inertia is half mr^2. I'm going to go ahead and write this here, and I'm going to leave alpha known because that's what we're looking for. So now I have to expand torque of f, which is why it's important to know how to calculate a bunch of different torques. So the torque of any force f is fr sine of theta, where remember r is the distance from the axis and theta is the angle between f and r. Here, my r vector looks like this. And the length of the r vector is the entire radius of the wheel. And that's because you're pushing all the way at the edge of the wheel alright so this is going to be the force you're applying Little r is the radius and then sine of theta. The angle between the force theta is the angle between the force and the r vector. The angle between those 2 is this right here, which is 90. So I get sine of 90. That's what you get on left side. Let's rewrite the right side again here and you get this. This becomes sine of 90 becomes 1. This r cancels with 1 of the 2 r's on the other side and we're ready to go. I'm solving for alpha. I'm going to move everything to the other side so alpha is by itself. So I'm going to get negative, negative 2 f divided by mr. That's going to be our alpha. Okay. I'm getting notice I'm getting a negative acceleration to make sense. It's going this way. So the acceleration should be negative. So this is part a. For part b, all we're doing is plugging in the numbers. So that's easy, negative 2 f. The force is 50. The mass is 100 and the radius is 2. So this is going to be negative 0.5 radians per second squared. Cool? So that's it. That's it for this one. Hopefully, it made sense. Let me know if you have any questions. Let's keep going.
