Torque, moments, and angular momentum
Introduction to torque
Welcome to the presentation on torque. So, if you watched the presentation on the center of mass, which you should have, you might have gotten a little bit of a glancing view of what torque is. And now we'll do some more in detail. So in general, from the center of mass video, we learned, if this is a ruler and this is the ruler's center of mass. And if I were to apply force at the center of mass, I would accelerate the whole ruler in the direction of the force. If I have the force applying at the center of mass there, the whole ruler would accelerate in that direction. And we'd figure it out by taking the force we're applying to it and dividing by the mass of the ruler. And in that center of mass video, I imply-- well, what happens if the force is applied here? Away from the center of mass? Well, in this situation, the object, assuming it's a free floating object on the Space Shuttle or something, it will rotate around the center of mass. And that's also true, if we didn't use the center of mass, but instead we fixed the point. Let's say we had another ruler. Although it has less height than the previous one. Instead of worrying about its center of mass, let's say that it's just fixed at a point here. Let's say it's fixed here. So if this could be the hand of a clock, and it's nailed down to the back of the clock right there. So if we were trying to rotate it, it would always rotate around this point. And the same thing would happen. If I were to apply a force at this point, maybe I could break the nail off the back of the clock, or something, but I won't rotate this needle or this ruler, or whatever you want to call it. But if I would apply a force here, I would rotate the ruler around the pivot point. And this force that's applied a distance away from the pivot point, or we could say from the axis of rotation, or the center of mass. That's called torque. And torque, the letter for torque is this Greek, I think that's tau, it's a curvy T. And torque is defined as force times distance. And what force and what distance is it? It's the force that's perpendicular to the object. I guess you could say to the distance vector. If this is the distance vector-- let me do it in a different color. If this is the distance vector, the component of the force is perpendicular to this distance vector. And this is torque. And so what are its units? Well, force is newtons, and distance is meters, so this is newton meters. And you're saying, hey Sal, newtons times meters, force times distance, that looks an awful lot like work. And it's very important to realize that this isn't work, and that's why we won't call this joules. Because in work, what are we doing? We are translating an object. If this is an object, and I'm applying a force, I'm taking the force over the distance in the same direction as the force. Here the distance and the force are parallel to each other. You could say the distance vector and the force vector are in the same direction. Of course, that's translational. The whole object is just moving. It's not rotating or anything. In the situation of torque, let me switch colors. The distance vector, this is the distance from the fulcrum or the pivot point of the center of mass, to where I'm applying the force. This distance vector is perpendicular to the force that's being applied. So torque and work are fundamentally two different things, even though their units are the same. And this is a little bit of notational. This distance is often called the moment arm distance. And I don't know where that came from. Maybe one of you all can write me a message saying where it did come from. And often in some of your physics classes they'll often call torque as a moment. But we'll deal with the term torque. And that's more fun, because eventually we can understand concepts like torque horsepower in cars. So let's do a little bit of math, hopefully I've given you a little bit of intuition. So let's say I had this ruler. And let's say that this is its pivot point right here. So it would rotate around that point. It's nailed to the wall or something. And let's say that I apply a force-- Let's say the moment arm distance. So let's say this distance, let me do it in different color. Let's say that this distance right here is 10 meters. And I were to apply a force of 5 newtons perpendicular to the distance vector, or to dimension of the moment arm, you could view it either way. So torque is pretty easy in this situation. Torque is going to be equal to the force, 5 newtons, times the distance, 10. So it would be 50 newton meters. And you're probably saying, well, Sal, how do I know if this torque is going to be positive or negative? And this is where there's just a general arbitrary convention in physics. And it's good to know. If you're rotating clockwise torque is negative. Let me go the other way. If you were rotating counterclockwise, like we were in this example, rotating counterclockwise, the opposite direction of which a clock would move in. Torque is positive. And if you rotate clockwise the other way, torque is negative. So clockwise is negative. And I'm not going to go into the whole cross product and the linear algebra of torque right now, because I think that's a little bit beyond the scope. But we'll do that once we do more mathematically intensive physics. But, so, good enough. There's a torque of 50 newton meters. And that's all of the torque that is acting on this object . So it's going to rotate in this direction. And we don't have the tools yet to figure out how quickly it will rotate. But we know it will rotate. And that's vaguely useful. But what if I said that the object is not rotating? And that I have another force acting here? And let's say that that force is-- I don't know, let me make up something, that's 5 meters to the left of the pivot point. If I were tell you that this object does not rotate. So if I tell you that the object is not rotating, that means the net torque on this ruler must be 0, because it's not-- its rate of change of rotation is not changing. I should be a little exact. If I'm applying some force here, and still not rotating, then we know that the net torque on this object is 0. So what is the force being applied here? Well, what is the net torque? Well, it's this torque, which we already figured out. It's going in the clockwise direction. So it's 5-- Let me do it in a brighter color. 5 times 10. And then the net torque. The sum of all the torques have to be equal to 0. So what's this torque? So let's call this f. This is the force. So, plus-- Well, this force is acting in what direction? Clockwise or counterclockwise? Well, it's acting in the clockwise direction. This force wants to make the ruler rotate this way. So this is actually going to be a negative torque. So let's say, put a negative number here times f, times its moment arm distance, times 5, and all of this has to equal 0. The net torque is 0, because the object's rate of change of rotation isn't changing, or if it started off not rotating, it's still not rotating. So here we get 50 minus 5 f is equal to 0. That's 50 is equal to 5 f. f is equal to 10. If we follow the units all the way through, we would get that f is equal to 10 newtons. So that's interesting. I applied double the force at half the distance. And it offsetted half the force at twice the distance. And that should all connect, or start to connect, with what we talked about with mechanical advantage. You could view it the other way. Let's say these are people applying these forces. Say this guy over here is applying 10 newtons. He's much stronger. He's twice as strong as this guy over here. But because this guy is twice as far away from the pivot point, he balances the other guy. So you can kind of view it as this guy having some mechanical advantage or having a mechanical advantage of 2. And watch the mechanical advantage videos if that confuses you a little bit. But this is where to torque is useful. Because if an object's rate of rotation is not changing, you know that the net torque on that object is 0. And you can solve for the forces or the distances. I'm about to run out of time, so I will see you in the next video.