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14. Torque & Rotational Dynamics

1

concept

15m

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Hey, guys. So in this video, we're gonna introduce the idea of torque, which is the rotational equivalent of force. Let's check it out. All right. So you can think of torque as a twist that a force gives on object around an axis of rotation. So here's the most classic example. If you have a door that is fixed around an axis here, this is the hinge of the door, which is also its axis of rotation, meaning the door is free to rotate around the axis. And if you push this way with a force F, it causes the door to spin. This way, I'm going to say to the door accelerates in that direction. He gains an Alfa. Okay, eso When you push on the door, it rotates around its hinges. So more generally speaking, when a force acts an object as it does here, away from its axes, it produces a torque on it. So let's talk about these two parts away from the axes. If you push on the door here, it doesn't actually cause spinning. I'm gonna say Alfa equals zero, right? If you try to push the door by pushing it, if you try to open the door by pushing the hinge. It doesn't spin. You have to be away from the axis of rotation, and then it produces a torque. So this is the idea that a force causes a torque which causes an acceleration. Okay, so what you're doing, you're not doing your not producing a torque. You're producing a force which then results in a torque, um, on that object. Okay, so now the other point is that a force may produce a torque. We already talked about how If you push here, it doesn't cause it to rotate, so you don't produce a torque. So force may or may not produce a torque. And a torque may or may not produce a rotation on angular acceleration. Andi, that's let's say if you're pushing this way, but then someone else, um, f two is pushing this way, and these to cancel out. In that case, you wouldn't really produce them. Okay, But the most important point I wanna make is that you have f produces a T which produces an Alfa. That's the sequence. Okay? And we just talked about this. Let's fill it in here. Similar to forces caused linear acceleration. Remember, some of all forces equals m A, um, some of all forces. As long as you have a net force, you're going to have an acceleration. It's the same thing with torques. Torques cause angular acceleration. We're not gonna talk about that just yet. Um, Alfa, we're gonna talk about this a little bit later, okay? Now, another difference between torque and force is that forces a straightforward number. If I tell you, we push with 10 that's the end of it. But for torques, torque depends on how hard you push it. Depends on how far you push and some other stuff. So we actually have an equation for torque. You don't have an equation for force. You're given a force. But for a torque if the calculated and it is f r sine of data f r sine of data where f is the force you push with its a vector are is a vector says here are is a vector from the axis of rotation to the point where the force is applied. Remember, little are in most rotation problems Has to do with distance from the center. It's the same thing here. Okay, now Theta theta is the angle between these two vectors. See these two vectors right here F and R theta is simply the angle between those two guys. Fatum glimpse f and R I meant to put enough and our and one thing I like to do is I like to think of like an arrow pointing towards these two guys here, right? And that's to remind me that in that equation of theta is the angle between the F and they are The two guys are hanging out next to the data. Okay, theaters The angle between these two guys now the unit for forces Newton's the Newton for the units for distance are is meters, so torque is measured in Newton Times meter. Okay, that's the units for torque. One last point I wanna make here is that when we to maximize the torque to get the most possible torque, another way to think about this is the way to apply the least amount of force and get the most amount of results. To be the most efficient with causing something to rotate is to apply the force as far as possible and perpendicular and apply the force perpendicular perpendicular to the our vector perpendicular means 90 degrees. It's got the little perpendicular symbol, um, to the our vector. So what does that mean? So let's draw the our vector real quick on this picture. Here are Vector is from the axis of rotation, which is here to the point where the force is applied, which is over here. This is the our vector. You want your force to make This is your force. You want your force to make 90 degrees with the vector, which in this case, it does. Um, that's how you get the maximum torque. The easiest way to rotate. Imagine if you're instead pushing the door this way, right, This would be a little bit weaker. You have to push harder to get the same rotation. So you want to push at an angle of 90 degrees? Thea other things that you wanna push as far as possible way um, from the axis of rotation. So you've all open doors before. If you try to open the door by pushing on it, let's say right here, right. I'm gonna make a little bit of a mess, but I'm gonna racist. Um, if you push right here. It's much harder to open the door here than to open the door at the end. In fact, that's why the door knob is on the opposite side of the hinge. Because that's what you're supposed to push Its easiest. Okay, so and you can relate that directly to the equation. Right? So if you want the torque to be a Zayas possible, you obviously wanna push as much as hard as you can as far away our asses distance from the axis A so far away as you can, and you want to maximize sign of data. Now, let me remind you that sign in co sign fluctuates between negative one and one, right, So it looks like that. So the greatest possible value of sign you can have is one. Now, where does this happen? This happens when data is zero. I'm sorry. When data is 90 sign of 90 is one. That's why if you look at the equation, that's why you get the greatest possible value for torque. Okay, So again, your torque is Max. When you push us far away from the edge is possible. And when you push perpendicular making a 90 degree angle with the art vector. All right, Now, let's do an example. We're going to use three steps to font to solve all of these torque questions. We're gonna draw the our vector. You're gonna figure out what your data is, then you're gonna plug numbers into an equation, okay? And here I have five different forces just to show you all the different variations. So you push on, you push or pull on a 3 m wide door. So the length of the store here, let me. Actually just right length. It was 3 m with 10 Newtons and a bunch of different ways. So all of these forces air 10. We want to calculate the torque that each force produces on this door. And then the rest is just explaining that F one of four and five all act on the edge of the door right here. In other words, they act at a distance of three from the axis. This guy is halfway through the door, and this guy's at the Hinch. So I'm gonna bright in all this information here. I'm gonna say this is 1.5 m. The rest here is 1.5 as well. So the whole thing is three. What else? It says that F five is directed 60 degrees below the X axis. So if you do this, this guy is 60 degrees right there. So I got all the information. Let's do this Torque one. Remember, the equation is just f one R one sign of data one. Okay, the forces 10. But I gotta figure out the r. And I got to figure out the sign. So the first thing you do hear the three steps, you know, draw the our vector. Okay, so the our vector is from the point where the axes is to the point where the force happens, the force happens right here. So they are vector for force One. Looks like this. Okay, so I'm going to say that this is my are one. And how long is our one? It's 3 m, so 3 m. Okay, Now sign is Thea. Angle is the angle between r and F. Okay. F one is this way. You actually draw this down here. F one is this way, and our one is this way. The angle between these two guys is simply 90 degrees. I'm gonna put it here. And remember, the sign of nineties one actually is. You remember that? That's gonna make your life easier. But if you forget just checking a calculator real quick. So this is gonna be 10 times. Three times one, which is 30. And the unit is Newton Meter 30 Newton meter. Cool. We're gonna do this four more times. S o t to again F two R to sign of data to the forces 10 I'd like to leave spaces for are in Fada. So what do you think the r is here, Right? So what would the our vector look like? I want you to draw that real quick and tell me how long that our vector is. And I hope you're thinking that it is 2 m. Sorry. 1.5 m, because it's just half right. So are too. Looks like this. This is our two. I'm gonna draw it down here so I don't make a huge mess. Actually. Put it here as well. Our two looks like that, okay? And the force is F two right here, and the angle between them is also 90 degrees. Okay, now we'll talk a little bit more about angles because there's some places where it might get confusing. But for now, we're just gonna keep going. If you multiply everything. The answer is 15 Newton meter. So right away, you see how when you pushed farther, which was in the first situation, you got bigger torque than when you pushed closer to the edge. All right, let's keep going. What about torque three F three, which is 10 are sign of data. What do you think the our vector looks like here? And what do you think the length of the vector is? So I hope you're thinking about this. You're thinking Well, if you push it the hinge, it doesn't really move at all. So the torque should be zero. There is no you're not producing anything that causes acceleration, not cause an acceleration. And that's correct. And that's because the our vector is going to have a length of zero. Okay, the R is the distance between you can't even draw it. It would look like this, right? That this is like your are three. It's a dot because it's from the axis of rotation, which is here, where the force acts, which is also here. So there's no distance because it's the same two points. Okay, in that case, it doesn't even matter what the sign is. You can't even do technically draw sign because there's no arrows to, um, there's no arrows for you to figure out. The angle between them is eso doesn't matter. This is just zero. And that's because you act on the axis whenever you act in the access torque is zero and the story. Okay, Now what about torque four? It's 10 are sign of data. What do you think the our vector hit here is so And how long is it? Right? So the our vector for 44 acts over here. So the our vector is the same one as this guy here. This is our one. And this is the same as our four. The length of this factor is the entire length of the door, which is three. So far. So good. What about the angle, right? What do you think? You put for the angle. And do you think there's a torque here? Eso The second question might be easier to answer if you pull on the door in the direction of before the door doesn't move side. The door doesn't open or close right? The door doesn't spin. So you should expect that the torque is zero and it is actually zero. But the reason why that happens in terms of the equation, you can see this in equation. It's because you are vector. Looks like this. This is our four. And this is your force four. Okay, the angle between these two guys is zero degrees. They're both going the same direction. Right. Um, if you got this wrong and you thought that the angles 1 80 that's okay. You get the same answer, but the angle zero degrees. And that's because you can put them sort of side by side on. Do you can see how the angle zero. Okay, so the angle here is zero degrees. And if you do sign of zero, the answer is zero. All right. In the last won t five it's at an angle. The forces 10. How long is he our vector? And what is the angle that we use so again T five is at a distance off three. So this vector here the long arrow is the Same for all all three of these guys. Um, what about the angle? The angle has to be. Be careful here, Right? A lot of torque problems who give you an angle, And people would just sort of blindly put us 60 here. And sometimes that's wrong. They do that on purpose. So you gotta watch out for that. You should actually assume that they're trying to trick you and and make sure that you're using the correct angle. Okay, So how do you know this? And I'm gonna draw this. I'm gonna draw. Are five right here. This is where you slow down. Make sure you do this correctly, and I'm gonna draw f five here. And here is the 60. Is that the angle between those two? And the technique that I like to use is to make it where the two arrows point from the same dots. I wanna have something like this, f and R, because then it's easy to confirm that this is in fact, the angle now to do this, what I do is I shift the arrow around. So for example, cities are five right here. I'm going to move it. I'm going to move it this way. I haven't changed the direction of the r five. I'm just shifting it from pointing into the dots to coming out of the dock. It's the same thing. But the nice thing about this is that it becomes very easy to see that this angle is in fact, the one between F five and are five because I have the two vectors like this, and it's just the angle between them. Cool. So sometimes you're gonna do a lot of the shifting, um, to make sure that the angle that they gave you was the correct angle. Okay. S O, for example, if they had said 30 here, it wouldn't be the correct angle. The correct angle is instead the angle between the R and D F, which is 60. So that's good. Sign of 60 goes here. Um, the And then if you multiply this whole thing, you get 26. I'm rounding in 26 Newton meters as the answer. Cool. It's really important. You know how to calculate torques. You should make sure you understood this, and you can do all all five of these things on your own. All right, this is just the beginning. We're gonna have tons more, some more stuff. But you've got to make sure you know how to calculate basic talks. Let's keep going.

2

example

4m

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Hey, guys, let's check out this example where we asked to find the torque produced by a force with two different angles here. So here we have a fish catching your bait. Um, your fishing pole is at an angle of 50 degrees above the X axis. So let's say you are like this holding onto your fishing pole that has its 3 m long 2 kg, and it makes an angle off 50. Right there. We wanna calculate the torque that's produced on your fishing pole about an axis of rotation on your hands. In other words, think that the fishing pole rotates about the hands right there. So that's the axis of rotation access right here. If the fish pulls on it with 40 Newton's, that's a force directed at 20 degrees below the X axis. So the line is here you're pulling from here. The fish is pulling at an angle. Direct that 20 degrees below the X axis. So here's a positive X axes 20 degrees below the X axis. Looks like this. So I'm gonna negative 20 over here, and the fish is pulling with a force of 40 Nuyens and we want to know How much does this force How much of a torque does this force produced on this axis? Right here. So we're gonna use my right. The torque equation. Torque equals f r sine of fada. I remember the steps. The first thing we're gonna do is writes the draw, our vector. Then we're gonna figure out what Data is, and then we're gonna plug it into the equation. Okay, So what is our our vector are vector is an arrow. It's a vector from the axis of rotation to the point where the force happens where the force is applied on the object. So the acts of rotations here, the force pulls right there were drawn arrow. This is our our vector. Okay, Now, the length of the our vector is 3 m, So that's what I'm gonna put here. So 40 three times three sign of theta now drawing the our vector is important. Actually, not so much so that you can figure out how long it is because you could have just looked at this. Poland said the polls 3 m long. That's the answer. What's really important about drawing the our vector is so that you can figure out what angle to use, which is the hardest part. Okay, you've got to make sure you're using the right angle. So here, should we use 50? Should we use 20 issue? We use negative 20. It actually turns out that in this problem, it's none of these options. It's a it's a different angle. It's a combination of the two. I want to remind you that data is the angle between f and R and to figure out which angle goes between them. The technique I like to use is to try to get F and are to be pointing from the same common point something like this. And then it's just a matter of finding this angle right here. And to do this, I'm going to shift f around or shift are around so that they start from the same point. So what I'm gonna do here is I'm going to push. I'm gonna shift my are up so that they both start from this point. Okay, so let me draw this again. I have 50. Draw this a little lower. I have 50 over here, and then I have 20 here. Okay. What I'm gonna do is I'm gonna get the our vector and push it over here so that I have are and f And then the angle I need is the angle between these two guys. Okay, Now you see here how There is a 50 between the our and the X axes. Right? Right here. There's a 50 between our and the X axis. So this angle here between our and the X axis is also a 50. So this 50 gets transferred over here and now, I hope you see that the total angle between R and F is actually 70 degrees. So you're supposed to add up those two guys. Okay. So this the entire purpose of this question was to look into how to solve questions with non trivial, um, fatal values, angle values and how to figure out the correct angle to use. Okay. To multiply all of this, you get 113 Newton's, um, meter, and that's your final answer. Cool. That's it for this one. Let me know if you have any questions. Let's keep going

3

example

2m

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Hey guys, on this torque example, we're trying to figure out what's the minimum force needed to produce a certain torque. Another way to think about this is how to maximize your torque with the load with the least amount of force how to be the most effective in causing something to spin. So it says here you must produce a torque of 100. Torque equals 100 to properly tighten a given bolt using a 20 centimeter ranch. 20 centimeter is the length of the ranch ranch. So we're gonna length equals 200.2 m. And I want to know what is the minimum force needed to do this. Now, this is just a play on the torch equation. So we're gonna write torque equals F R sine of fada. The torque value is fixed. It's 100. I want to know what is my force. And to do this, I have to figure out which values to plug in for our in fatum. Remember, if you want the most effectiveness with your torque with your effort, um you want the greatest possible force. But here we're trying to find the least amount of force, which means that I wanna have the most effective are in the most effective data. Okay, the most effective are the most effective data. Let me draw a little ranch here. It's terrible. Probably. Um So what you wanna do this is often what you wanna do to produce a torque is you wanna push as far away from the axis now the range the bolt is here. Right? Um so we're spinning like that. You wanna push us far away from the ranch? Think about this. If you got the ranch and you pull it here, it doesn't do as much if you pull the range all the way at the end all the way over here. That's why you grab it. Here, you produce the most torque. So you wanna be as far as possible from the axis of rotation. And that's where this 20 centimeter comes in. If the ranch is 20 centimeters long, you wanna be a sfar assed possible. We're gonna assume that we can push it at the very tip. Therefore, we're gonna use an r of point to also you. Not only do you wanna push all the way over here at the end, you wanna push with an angle of 90 degrees to the access. So if you push it here, I'm sorry to the our vector. If you push it here, you're our victories. Access to a point where the force is applied R f and you wanna make an angle of 90 degrees. Okay, so in all these questions where you want to maximize torque or minimize force you want you are to be as long as possible, which usually means the entire length of the thing. And you want your stated to be a Z biggest possible, which usually means a naval of 90 to give you a sign off one andan. That's it. So this gives us, um f 0.2 one. So force equals 100. Divided by point to the force. Here would be 500 Newtons. Alright, that's it for this one. Let me know if you have questions

4

Problem

You pull with a 100 N at the edge of a 25 cm long wrench, to tighten a bolt (gold), as shown. The angle shown is 53° . Calculate the torque your force produces on the wrench, about an axis perpendicular to it and through the bolt.

A

9.9 N•m

B

990 N•m

C

15 N•m

D

1500 N•m

E

20 N•m

F

2000 N•m

Additional resources for Intro to Torque

PRACTICE PROBLEMS AND ACTIVITIES (6)

- One force acting on a machine part is F = (-5.00 N)i + (4.00 N)j. The vector from the origin to the point wher...
- A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long, and he exerts a 17.0-N force at the...
- Calculate the torque (magnitude and direction) about point O due to the force F in each of the cases sketched ...
- A metal bar is in the xy-plane with one end of the bar at the origin. A force F = 97.00 N)i + (-3.00 N)j is ap...
- A metal bar is in the xy-plane with one end of the bar at the origin. A force F = 97.00 N)i + (-3.00 N)j is ap...
- A machinist is using a wrench to loosen a nut. The wrench is 25.0 cm long, and he exerts a 17.0-N force at the...

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