 Lesson two, rotational equilibrium. So recall, we said this, translational equilibrium is when the sum of all the forces on an object is zero. Symbolically, we said when the sum of all the forces is zero. And there was a fairly easy way to tell that an object was in translational equilibrium. It wasn't accelerating. Because if the force is zero, the acceleration has to be zero in this first law. It could be moving in a nice straight line at a steady speed, but it wasn't accelerating. The second type of equilibrium we're going to talk about is rotational equilibrium. So let's talk about torque. Torque is the strength of rotation that an object possesses. It's how much you can force something to spin or to rotate. I'd like to say it's rotational force except it's not a force because it's not measured in Newtons. So I'll call it rotational strength. How much you can get something that doesn't want to turn to rotate. And it depends on two factors. So torque depends on two things. It depends on the force that you apply, and it depends on the distance from the pivot. But there's a third idea that we have to keep in mind. So I'll be Tyler up here with one hand at the 50 centimeters, and we found that just by moving further from the pivot, Tyler, I could with one finger apply a lot of torque, but I'll apply the same force this way. Will that cause it to spin at all? So we have to add a second condition. The force and the distance have to be perpendicular to each other. If your force is in the same direction as the distance, no torque. Torque is defined as the perpendicular force multiplied by the distance is the force is from the pivot point or fulcrum point. It is a vector. It has direction. The equation now the symbol for torque is a curly T. It's actually a Greek letter tau. That's the symbol for torque. Why don't we use a capital letter T that's already taken by something else? By what? Next unit. And it's equal to force times distance. But what we say is this they have to be perpendicular to each other. So we add a little perpendicular symbol right there. And I think that's what it has on your formula sheet. Okay. What are the units? What's force measured in? Newtons? What's distance measured in? It's Newton meters. Now without that there, what's force times distance? It's work and it's joules. We're not talking about energy here. So we can't call this Newton meter a joule. We're going to have to call it a Newton meter because they are perpendicular to each other. Remember when I talked about energy I said the force had to be in the same direction as the distance. That's why I was so fussy then. Okay. Units, Newton meters, direction. Torque causes objects to rotate. So we're going to have two directions clockwise and counterclockwise. Except Justin that's way too much writing. CW, CCW. If you're not sure, if you get it mixed up, there's always gonna be a clock in my room. Look at it and figure out which way is clockwise and which way is counterclockwise. Please. How much torque would the following force exert on this beam? Okay. So the pivot point is right here and traditionally we'll use a little inverted triangle for the pivot point. First thing I would ask is, is there a distance? Check. Is there a force? Check. Are they perpendicular? Check. What if they weren't perpendicular? Components, torque is equal to force perpendicular times distance. 12.3 Newtons times, oh gotta make that meters. 0.89, 12.3 times 0.89. And I get 10.9, 10.9 Newton meters. There's my units. Direction. Gonna go this way around the pivot point. Usually in our final answer, we won't ask you for the direction. We'll just want the magnitude. But to get there, you're gonna have to be very careful with directions along the way. How much torque would the following force exert on the beam? Remember the word that I said a short little while ago? Components. I'm gonna break this up into perpendicular and parallel. So this is gonna be F perpendicular and this is gonna be F parallel. I need an angle. Hey, see the Z? So don't threaten that off. This angle here is 65 degrees. Which trig function will I use to figure out the perpendicular component of this 2.8? Okay, over here I'm gonna go sine of 65 equals perpendicular over 2.8. I guess that means that perpendicular is gonna be 2.8 sine 65. Okay, here we go. Torque is equal to perpendicular force times distance. And we said that was 2.8 sine 65 times 0.43. How much torque? You get 1.09. 1.09 units Newton meters. Direction clockwise. What about this? What does this do? The parallel? It doesn't do anything with torque because it's not in the perpendicular to the distance. So this brings us our second type of equilibrium. We said our first type was translational. All the forces equals 0. Our second type is rotational. When an object is not spinning, is not rotating, it is said to be in rotational equilibrium. In other words, what symbol is that? The sum of all the torques is 0. I'll be honest, I almost never use that. I prefer to use the sum of all the torques clockwise equals the sum of all the torques counterclockwise. When I first started teaching this about eight years ago, kids would get this mixed up all the time. And about two years in, I came up with something stupid that got rid of most of the dumb mistakes. I'm gonna add one more little thing. Look up. I actually don't write this either. I write the sum of all the torques clockwise and I draw a little curvy arrow that way equals the sum of all the torques counterclockwise and I draw a curvy arrow that way. And you know what? When I did that, Kellen, kids stopped getting clockwise and counterclockwise mixed up. I suggest for that little half second of time, it's the best bang for your buck you'll get this unit to avoid making dumb mistakes. I'm always gonna draw the little loop and clockwise. Sorry, for you it's backwards. Which way is clockwise? For you guys. Is this way? Is that right? Second thing we need to talk about, because with torques, we're gonna be dealing with meter sticks. Actually, we're gonna be dealing with beams, long pieces of steel or wood or something. Quite often, we need to talk about the center of mass or the center of gravity of an object. Y'all figured out where we are now, okay? So, here's my meter stick. Every molecule of wood has mass and therefore, it experiences a gravitational force. There's gravitational force there, there's gravitational force there. However, the real question we want to know is, if we're doing a free body diagram, where do we put MG? I don't want to put a million little MGs, one for each molecule. As it turns out, Justin, there is a point mathematically where you can put the force of gravity and pretend that all of the molecules mathematically are concentrated there. It's called the center of mass or the center of gravity. It's easy to find on a meter stick. Tyler, I'll borrow you again. Stand next to me facing that way this time. And I'd like you to hold both hands out with your fingers like this. And I'd like you to very slowly slide your fingers together. Where'd this stop? 50. Go wide again on the very end. Slowly slide them together again, but try and stop somewhere else. Try and stop somewhere else. Try and stop somewhere else. Try and stop somewhere else. Where'd this stop? 50. You can't. Easy way to find where the meter stick, center of mass is where it balances. Right there. That's how you know it's the center of mass. Everyone has a center of mass. Yours is roughly right around where your belly button is, where your navel is. That's about where most people's center of mass is. That's where you balance. In other words, if I want to knock you over, if I draw a line through here and right through here where that line crosses, if I get that spot past your feet, you have to step or you'll fall. Now balance works. Anything falls once its center of mass is past the base. So if I look at this Gordon, the center of mass is not only at the 50 centimeter mark right here, but it's also halfway down the middle. Like it's right dead center. This will stay balanced until that center of mass gets just past the base. Now it's no longer stable. So you see people balancing stuff on their chain or whatever. I had a student that could do that a couple of years ago. All he was doing was making sure that the center of mass stayed above the base. I showed you a video of the old man on the chair yesterday. Center of mass stays above the base and he's fine. He said love to practice. Okay. So for a nice sphere that's all uniform, center of mass or center of gravity is dead center. Car center of gravity is right about there. The center of gravity doesn't actually have to be inside the object for a glass. The center of mass, the center of gravity is in midair right about there. For a non uniform object like a baseball bat, which has more mass at one end. That's the sweet spot. Oh, yeah, that's the physics behind it. Force is what times what. That's where the mass can be said to be concentrated if you want to apply the maximum force to a baseball. That's the science behind the sweet spot. Cool. Okay. So this figure shows four objects with differing shapes. Notice the center of gravity or the center of mass can be outside the shape. It's quite possible. So. Oh, meter stick demonstration number two. I did that says this the center of mass for an object is the point where the mass of the object might be considered to be concentrated in order to do the arithmetic. For most situations, the center of mass is the same place at the center of gravity. You can almost use them interchangeably. The exception would be when the gravitational field over an object being considered as non uniform like a black hole. So I do have to be fussy and say, okay, this math wouldn't work near a black hole. Can we just agree that we're going to ignore that? And we're going to say center of gravity, center of mass, same thing for the most part. Okay. Where would you expect the center of mass of the earth moon system to be located? Well, the earth has a big mass. Moon has a small mass. The center of mass would be closer to the earth probably right about there. Sort of, you can think of that baseball bat. Right. So meter stick demonstration number two. Tyler already did that one. I was ahead of the gun, but you'll find easy way to find it on meter stick side, your fingers together. You can't stop anywhere else but at the center of mass. You're bored. Sometimes try it. The only way you can do it is if you do a really fast jerk, but now you're adding energy to the system and changing things. If we have both translational equilibrium, the sum of all the forces is zero. And rotational equilibrium, the sum of all the torques is zero. If something's not spinning and it's not accelerating, we have the two conditions required for static equilibrium. Static equilibrium says no torques, no force. Little note, you're going to be analyzing lots of diagrams in this unit. Dylan in general, if the diagram has a beam in it, torques. If it doesn't have a beam, but it has cables and ropes like last day with a sign hanging, equilibrium triangle. So what force F is needed to balance the beam in figure 1.31? So we're going to put a force F right there. And we want to create static equilibrium. Is there a beam? So we're going to start out by saying the sum of all the torques clockwise in that direction equals the sum of all the torques counterclockwise in this direction. Now my pivot point this time is in the middle here. So you want to ask yourself, what are the forces acting on it here? I have a force here. I have a force here. Which of those forces, or both of them maybe, would cause this to spin clockwise around the pivot? So now you're imagining this. Which of these would cause it to spin in this direction? I gave it a label. I use the letter F. That's clockwise, but we have to solve it with torques. Torques is what times what? What did we say about two minutes ago? Force times perpendicular distance but yet force. So clockwise F perpendicular times its distance from the pivot. Look carefully how far is it from the pivot in meters and don't say 175 and don't say 1.75. 0.75 you have to do a little arithmetic sometimes. Are there any other clockwise torques? Nope. Equals. What are the counterclockwise torques? That's this way, Ryan. Eight. Oh yeah, go ahead. Here's the pivot. How big is that distance? What distance is right there? Okay, you've got to be careful with these diagrams. This is a simple one, by the way. They're going to get way more complicated. Don't panic. Do a bit of arithmetic. Anyways, eight. Ryan, redeem yourself. What's this distance from the pivot? Centimeters? Nope. 0.05. Thank you, Ryan, for eventually redeeming yourself. Nice thing, though. After all that, that's a pretty good equation. How would I get the force by itself? That's a pretty good equation. How would I get the force by itself? Divide by 0.75 and I've got the perpendicular force by itself. The force is going to be 8 times 0.5 divided by 0.75. Usually, the tricky part is getting the equations. Once you get the equations in this unit, they're pretty good. 8 times 0.5 divided by 0.75. This is very similar to what Tyler and I were doing. You'll notice, Ryan, this force is closer to the pivot, which means since this guy's further away, less force will exert the same torque, just like I was able to cause yours to spin. In fact, because your distance was nearly 0, how much torque could you exert if torque was force times distance? Nearly 0, which is why with one finger, I could move the meter stick around. What did I say the answer was? 5.3. Units, what did we just find? Newtons. Example four. How many of you have been on a teeter-totter when you were a little kid? That's what this picture is. How many of you have been on a teeter-totter when you were a little kid, when your partner didn't weigh anywhere near the same as you? What do you figure out very quickly? How can you go still? Well, you got a new partner. You can move yourself forwards or backwards. You can change the distance and therefore change the torque and still come close to getting equilibrium as long as your masses are reasonably close. Yes, when I go on the teeter-totter with my little niece, there's no way I could get the mat, the torques, to be identical unless I was sitting right by the pivot. So, I end up doing all of the work, right? I have to jump up and then she goes down and I come back down and I got to jump up and then she goes, okay, fair enough. But when you're a kid, your masses are close enough that you clue in if I slide forward a little bit or slide back a little bit, I can do some of this. So, teeter-totter, how far from the pivot must the 64 Newton object be placed to balance the beam? Is there a beam? And we want equilibrium torques. The sum of all the torques clockwise equals the sum of all the torques counterclockwise that way. All right, what would, there's my pivot right here. What would cause it to spin in the clockwise direction if it could? 64 times its distance from the pivot. How far is it from the pivot? X. What would cause this to spin counterclockwise? 56 times its distance from the pivot, 0.42 meters. By the way, are these two forces fairly close together? I expect X, first of all, is going to be a little bit less because bigger force means less distance required, but it'll be close to that. If I get an answer of like 10, I've goofed somewhere. X is going to be 56 times 0.42 divided by 64. Let's go to meters, please. 0.37 meters. Like I said, pretty close. And 5 centimeters if you were kids, that you could certainly move back and forth and get the equilibrium. Big deal. Oh, here's the great thing about torques. Dylan, mathematically I can put the pivot point wherever I want to. Usually there'll be an obvious place to put it, but sometimes you'll have a choice and the choice where you want to put it because all you do is if you move the pivot around, it changes your distances. You have to adjust all those. Oh, and if you move it really far over, something that used to be clockwise might be now counterclockwise. You need to change those, but you can do some really good stuff like this next one says this. The force of gravity on that bridge is 9.6 times 10 to the fifth newtons right at the center of the bridge because we said that for a beam, we can represent its mass dead center. What upwards force must be exerted at end P and at end Q to support the bridge and the truck? If the force of gravity on the truck is 4.8 times 10 to the fourth newtons. We've just raised the bar, no pun intended, so we're going to draw a free body diagram now. Here's the beam. There's the bridge. What are the forces acting on the bridge? Get the obvious ones. Well, now they're not so obvious. Okay, let's look at my diagram. First of all, there's gravity on the bridge and where am I going to put that? Dead center. 9.6 times 10 to the fifth and there's also a truck right about here. 4.8 times 10 to the fourth and then I think there's two forces at P and at Q at either end. I think they're both up, but I'm not quite sure and here's the problem. How many forces don't I know? Two. Can I solve an equation with two variables in it? Not without going to systems of equations and I don't want to do that. This is why torque is so nice. I can put the pivot wherever I want to. I'm going to put the pivot at one end. What do you want to find first? Force P or force Q? Doesn't matter. Pick, Justin. P for pilgrim. Okay, we'll find force P. That means I'm going to put the pivot right here at force Q. Now, why is that so nice? Let's suppose force Q is pushing up. How far is force Q from the pivot? It's a trick question. Zero. How much torque does it exert then? It won't appear in my equation. If you have two unknown forces, put the pivot on one of them and it vanishes for torques. Now, here is force P. Is this bridge spinning? Which way are these two guys wanting to make the bridge spin if that was where the hinge was? You know what? Force P has to be counteracting that, doesn't it? So, not only does it help me get rid of one force, I can figure out for sure the direction of the other force. Back to my earlier question. Is this bridge spinning? No. That must mean the sum of all the torques clockwise equals the sum of all the torques counterclockwise. We put my pivot right there, mathematically. So, you have to use your Sesame Street imagination a little bit. What would cause this to spin clockwise? Which force or forces would cause it if it was free to rotate around that hinge spin clockwise? Force P. So, torque is force times its distance from the pivot. Look at your diagram. You have to do a little bit of thinking. How far is it from the pivot? Seventeen. You're going to have to do some thinking here. Seventeen. Are there any other clockwise torques? Are you looking for me? Okay. Oh yeah, I'm doing torques. I'm torquing about torque. I think that's it for clockwise. Counterclockwise. Now there's going to be two. This guy times its distance from the pivot. How far is it from the pivot? Again, you'll have to do a little bit of thinking. You're wanting to figure out this distance here. 8.5 plus this 4.8 times 10 to the fourth times its distance from the pivot. How would I get force P by itself? Do you see what I mean? The initial setup is tricky, but you almost always get pretty nice equations. How would I get the force P by itself? Divide by, and that's it. In fact, you can probably go straight to your calculator. How big is force P in Newton? 9.6 times, let's go to it, 9.6 times 10 to the fifth times 8.5 plus 4.8 times 10 to the fourth times 5 equals divided by 17. And I get 4.94 times 10 to the one, two, three, four, five. Newton's. Is force Q going to be exactly the same size? No, why not? The truck is closer to fact. Force Q, you think it's going to be bigger or smaller? It's taken a bit more of the weight of the truck. Now there's two ways I can find force Q. I can redo this question putting the pivot right there, and then these two would be clockwise with new distances from the pivot. This would be counterclockwise or I can label this as 4.94 times 10 to the fifth. And instead of asking myself, is this bridge spinning? I can ask myself, is this bridge falling? Is this bridge falling? That means all the downwards forces have to equal what? What we did yesterday, all of the upwards forces. So an easier way to find the mystery force is use torques when you have two unknowns and then use equilibrium. So if I want to find force Q, I would say this. Force P plus force Q has to cancel out 9.6 times 10 to the fifth plus 4.8 times 10 to the fourth. Force Q is going to be this plus this minus what? Force P, which I think I still have stored on my calculator. 9.6 times 10 to the fifth plus 4.8 times 10 to the fourth minus the previous answer which was still stored on my calculator. And I get 5.14 times 10 to the 1, 2, 3, 4, 5. 5.14 times 10 to the fifth nuisance. A little thought experiment for you. As this truck drives forward, what happens to force Q gets bigger or smaller or stays the same? I heard someone say bigger, I heard someone say smaller, someone convince me. Okay, when the truck is right there, these guys would be identical because they're both splitting both masses. So that means this guy gets a bit smaller and as he moves closer and closer to P, bigger, bigger, bigger, smaller, smaller, smaller, smaller until he gets all the way off. Okay, so those of you thinking about going into engineering, there's some load structure thinking. Turn the page. Diving board. Isn't that a great diagram? Diver. I drew that myself. I'm assuming most of you have stood on the end of a diving board. I don't think I ever looked at how they're structured. It says this, a diving board has a length of 6.5 meters and a mass of 52 kilograms. So the beam itself weighs 52. A diver of mass, 65 kilos, stands on one end of the board. Find the force exerted at location A and location B in order to maintain equilibrium. Okay, we'll label my diagram instead of doing a free body. What are the forces acting on this beam? Get the obvious one. The first one you want to get, Dylan, is the mass of the beam. How much is the beam weigh, Dylan? Where am I going to put that? Halfway along the beam. Yeah, right here. How much is the beam weigh, Dylan? 52. So I'm going to call this mass of the beam times g. What are their forces acting on here? Diver. I'll call this mass of the diver times g. Now, if that's all there was, which way would this diving board have to be accelerating downwards? I'm anticipating there's got to be at least one force up somewhere. And I'm also going to think in terms of torques. Okay, there is a force here and a force here. Put your pencils down because I'm going to make a mistake. Most kids, when they first see this, they go like this. Oh, force A and they go, oh, force B. But watch what happens if I now analyze this with torques. If I put my pivot right there, because I can mathematically, which way would that cause it to spin? Which way would that cause it to spin? Clockwise. Which way would that cause it to spin? It can't be in balance. This cannot be pointing upwards. Otherwise, it would have to be spinning. Force A, which is the most commonly mislabeled force on a diving board, is actually that way. I wonder if I got force B correct. Well, there's two ways that I can figure out that force B has to be pointing upwards. The first way is, do I have any upwards forces yet? Then this diving board would have to be falling into the ground. Does a diving board fall into the ground? So I need an upwards force. Or again, you could temporarily put your pivot there, mathematically. Which way does this cause it to spin that way? Which way does this cause it to spin that way? I need something to cause it to spin in this direction to cancel out the two that direction spins. So force B has to be in that direction. Does that make sense? Is that okay? Which way is this force? Down. Not torque. Which way is this force? Down. Which way? Which way? So this has to be falling into the ground. Are diving boards falling into the ground? It's got to be an upwards force. That's one way of getting it. But also if you look at it in terms of torques, if you temporarily put the pivot there, which would make this force vanish for torques, these are spinning it clockwise. You need something to balance it out counterclockwise, which would also have to be pointing up. So force B has got to be pointing upwards. How many forces don't I know? How many forces don't I know? Two. Is there a beam? I'm going to solve this with torques. I can put the pivot wherever I want to. What do you want to find first, Brendan? Force A or force B? It doesn't matter. Force B? So since I want to find force B, I want force A to vanish. I'm going to put the pivot temporarily right there. Why is that going to make force A vanish for torques? How far is force A from this pivot point? Zero. And torque is what times what? Force times distance. And if your distance is zero, what's your torque? So it will vanish. The sum of all the torques clockwise equals the sum of all the torques counterclockwise. Pivot's there, so imagine Meaghan that it's on a hinge right there. Is your imagination that it's free to spin right there? What would cause it to spin clockwise in this direction? The mass of the beam times g times its distance from the pivot. So now I've got to figure out how far is that distance. What's the total length of the diving board? 6.5 and I said that for a beam we always put this at center of mass halfway along the beam. Are there any other clockwise torques? Ah, the diver. The mass of the diver times g times its distance from the pivot. Pivot, how far? 6.5. Any other clockwise torques? Nope, equals. Counterclockwise torques times its distance from the pivot. How far? Oh, 1.8. Do I know the mass of the beam? Check. G9.8. Do I know the mass of the diver? Check. Oh, how would I get the FB by itself? Okay, FB is going to be mass of the beam, which was what, Dylan the Lesser? Thank you, I knew you'd come through this time. Times 9.8 times 3.25 plus mass of the diver, which was what, Dylan the Grader? Times 9.8 times 6.5 divided by 1.8. Go do it. What do you get? I get 3,220. Yep. Units, force. Now let's find force A. Again, we could find it two ways. We could say force B equals that plus that plus that. We can get there subtracting. Let's just for practice because I can see a few of your little glassy eyes. Let's find it using torques. So, if I use torques, I'll really quickly draw the diagram. Boom, boom, boom. Now we're going to put the pivot right here. What are the forces that we have? We have force A, mass of the beam G, mass of the diver times G. The sum of all the torques clockwise equals the sum of all the torques counterclockwise in that direction. Now, Jordan, we put the pivot here. So, you want to ask clockwise if this could spin right around here, which of these forces would cause it to spin in this direction? I just can't hear you. I'm sorry. Are you saying MBG, the mass of the beam times G? That's a D for diver. Same diagram as up above. See, I think both of, I think this would cause it to spin and this would cause it to spin. It's going to be two clockwise torques. So, let's do this one. First one here is mass of the beam times G, times its distance from the pivot. Uh-oh. I need that distance there. Let's see. How far from here to here? Look up. How far from here to here? 6.5, question says that. How far from here to here? 3.25, we put it at the center of mass. How far from here to here? 1.8. So, how big is this? It's going to be 6.25 minus that, minus that, or 3.25 minus 1.8. What'd you get? 1.45, is that what you said? Plus the mass of the diver times G, mass of the diver times G, times its distance from the pivot. How far from here to there? I think 6.5 minus 1.8 gives me that. 4.7. Jordan, are there any other clockwise torques? Nope. Equals. Counterclockwise, if that's my pivot, what could cause this to spin in this direction? Oh, this guy would, wouldn't he? Force A times its distance from the pivot. Oh, how far is it from the pivot? 1.8. Do I know the mass of the beam? 52 times 9.8 times 1.45 plus mass of the diver, 65 times 9.8 times 4.7. How would I get F A by itself? I'm going to do that right here. Divide by 1.8. That equals F A. How big is force A? 2074, the way else? 2074, Newton's.