Torque & Equilibrium

by Patrick Ford
219 views
3
Was this helpful ?
0
Hey, guys. So you may remember back when we talked about forces that we mentioned how if the forces of an object cancel, the object is in equilibrium. Well, now that we're talking about torques, there's a similar situation where you may have what we're gonna call rotational equilibrium. Let's check it out. So remember, if the Net force on an object if the Net force or the sum of all forces an object is zero, then the acceleration on that object is zero. And that's because of Newton's second law. F equals M A. If the Net force zero means that some of all forces equals zero and therefore the acceleration has to be zero. This situation is called equilibrium. But now that we know about torques, things are not gonna be that simple. It's gonna be a little more complicated. Okay, sometimes this this condition here of the Net force being zero is not sufficient for equilibrium. It waas before we knew about torques. So here's an example you imagine have a continuous bar that has uniformed mass distribution. So the bars center of mass happens right here, and at this point is where the weight force acts There's a little folk room here that holds the bar up and this is going to push back with a force of Newton normal. And this normal may even be equal to M. G. The problem is that even though the forces will cancel on the vertical, um, this MG produces a torque here torque of M. G. This is clockwise, so it's negative. But there is no torque due to the normal force. And that's because the normal force acts on the axis of rotation r equals zero. It acts on the axis of rotation. So the net torque will be this. So the network will be equal to this. And because I have a network, I will have a new Alfa. So even though the forces canceled, the torques don't cancel. So what happens is that this thing would tip over this way and fall. Okay, so we're gonna have a situation where we don't have equilibrium. Alright, this brings us to the fact that there has to be. Actually, there are two conditions that are necessary for a knob jek toe. Have what I'm gonna call complete equilibrium. It's not enough that the forces canceled the first condition is that the sum of all forces must be zero. And this gives us an acceleration of zero. And this is good old equilibrium. But now we're gonna call it linear equilibrium because they're going to be two types. Okay, the second condition is that the sum of all torques also has to be zero. And this will give us an Alfa that is zero. And we're gonna call this rotational a political. And if we have, both of them were gonna have what I call complete equilibrium. Okay, um, this topic is in most most textbooks refer to its static equilibrium and static refers to the fact that you're gonna be in a situation where there's no velocity and no linear velocity, no angular velocity. So not only do you have, um, complete equilibrium, but also, the object is not moving. Okay, Um, this is sometimes called also the equilibrium of rigid bodies, because we're going to deal with rigid bodies exclusively. So we're always gonna have these extended objects rather than a point mass. Okay, let's do I want to do a sort of introductory example here to talk about different situations where you may have one type of equilibrium, but not the other or both. Or neither. All right, so let's check it out. So here it says, a light bar, which is this gray horizontal bar, is free to rotate about a perpendicular access through its center. So the bar here's the axis of rotation. The bar can spend either this way or this way, but the middle is fixed, right? Um and then it says the bar is not attached, so it is free to move horizontally, vertically. What that means is, for example, in this first situation, 32 forces air pushing this thing up. So the bar, if the bar has two forces pulling it up and it's not fixed in the middle, the bar would actually do this. Okay, It's only fixed in that it could Onley rotate in the middle, but somehow it could actually move up and down. All right, All these little arrows have the same magnitude. And if you see a double arrow, which we see here, um, this just means double the magnitude. Okay, we wanna know. Do we have linear equilibrium in rotation equilibrium? So check this out here. We don't have linear equilibrium because both of these forces mean that the net force will be going up. Okay, The forces air both pushing up, so the bar has to move up. However, we do have rotational equilibrium. And that's because this force here causes the torque. This way. I'm gonna call this Torque one in this force caused the torque this way. Torque to, um, those two torques air going opposite direction. This one is counter clockwise. Positive. This one is clockwise negative. And they have the same magnitude. The reason we have the same magnitude. Let me bring back the torque equation. Is torque equals f R sine of data? The forces are the same in both sides. Notice that they're both the same distance from the axis of rotation R one and R two. Okay, they're both too little sticks away. And they both make an angle of 90 degrees. So same for same distance. Same angle. The torques are the same. They will cancel each other house. What about here? Here you have two forces. One pushing up, the other one pushing down. They will cancel each other and we will have linear equilibrium. However, we're not going to have rotation equilibrium. Because both of these cause a talk about the middle that has the same direction. So this is torque one, which is counterclockwise negative and torque to which is counter clockwise, negative as well. So there will be a net torque. There will be a net torque. That is negative. Here. There was a net force that is positive going up. All right, so what about here? Here again? We have two forces cancel each other, so we have linear equilibrium, But we're not going to have rotation of equilibrium. We didn't have rotation equilibrium here. We're not gonna have rotation equilibrium here. Why? Well, because even though these forces air going in different directions, the torques are difference in magnitude. Check it out. So this guy is going this way. Torque one. This is counterclockwise, so it's positive. And this guy is producing a torque. That is, um that is I'm sorry. This guy's actually to the left of the dot so the torque will be this way. Talk of two is negative. Okay, so if you imagine a bar, right, if you push this way, it's going to be positive. And if you push this way, it's going to be negative rotation, but t one is farther away. So our one is greater than our two. Here's our one. And here's are to therefore Tark. One has a greater magnitude than torque to so talk one winds in the bar ends up spinning this way. So no rotational equilibrium. What about here? So I can say that there's two forces up and two forces down, so the forces will cancel each other out. And I have a linear equilibrium. What about rotational equilibrium? Well, guy, let's call this 11234 I hope you'll see that one in four will cancel. Um, Torque one goes this way, which is negative. Torque four goes this way, which is positive. They're both the same distance from the axis, Same angles, everything. So these to cancel these two guys also are opposite to each other, but they're gonna have the same magnitude because the same distance so torque to looks like this again, you're to the left of the axis pushing it down. So it's gonna cause it to spin like this, and Torque three is gonna go the other way. So I hope you see how three one in four canceled, and two and three cancel each other as well. So here, actually both rotational in linear equilibrium. What about this problem here? Here, the two forces cancel each other. I have linear equilibrium. Um, what about rotation equilibrium? I only have one on each side. They're both causing torques in different directions. One to torque one Talk to torque. One is clockwise negative. Talking to his counterclockwise. Positive. They perfectly cancel each other out because the ours are the same. They're pushing at the same distance from the axis of rotation. I have both. Equilibrium is here is Well, what about here? Here, All my forces air going up. So I will have a net force that is going up. Let's call that positive. So there is no linear equilibrium. But I do have rotational equilibrium because here I have Let's call this distance here two. And let's call this distance one. You can see how this distance is double. So the first talk over here torque one, which is these two arrows talk One would be two f are and talk to would be, um f to our right. So the guy on the left has double the force. But the guy on the right has doubled the distance. So these two end up canceling each other. Okay? They end up canceling each other, and I have rotation equilibrium as well. All right, so this is just a bunch of Let me get out of the way here. Um, this is a bunch of difference. Uh, situations for you, toe. Sort of get an understanding of when you would have linear and when you have rotational equilibrium. Alright. So that's it for this one. Let's keep going.