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0:00
[MUSIC PLAYING]
0:06
JEFF HANSON: Hey, friends. We finally made it past chapter 3. We're on to chapter 4. We're talking about statics on a rigid body. And you say, what is a rigid body? I'll show you what a rigid-- oh man, don't show off too much, get you all worked up there. No, what a rigid body is-- in the last chapter, we were just talking about statics on a particle, a single point where we had all-- the forces always went through one single point. But in this chapter, we have-- what is a rigid body?
0:40
A rigid body could be anything. It could be a person. It could be a book. It could be a car. It could be your desk. It could be anything that's not a single point. Now, why do we call it a rigid body? Because it's statics, and it's a little magical in statics. In statics, are stuff doesn't bend. It doesn't deform. When it heats up, it doesn't get bigger. We save all that stuff for solids, for deformable bodies, but in statics, every time we have a load on something,
1:09
we just assume that it doesn't bend. It doesn't deform, and therefore, it is rigid. It's really not what happens in the real world, but for analyzing forces, that's what you have to assume. And then when you get to solids, we'll show you how those forces do indeed cause those deformations, and we'll talk about those deformations. So we know what a rigid body is. It's anything other than a particle. So let's say we have-- this is our rigid body, just a beam.
1:38
Now, in our particle situation over here, what had to be true to have equilibrium? Well, we had to have the sum of the forces, and the x had to equal 0. That's an x. Sum of the force in the y-- where'd my F go-- had to equal 0, and then in 3D, of course, the sum of the force in the z had to equal 0. So if we have this beam and we had two forces, then let's say we have this, 10 pounds, and here's 10 pounds.
2:11
That is that beam and equilibrium. Well, sum of the force of the x-- there aren't any, so yeah, that's 0. Sum of the force in the y-- I got 10 down, and I got 10 up. So that's 0. So yeah, well, I don't really think that's an equilibrium because if you talk about a point in the middle-- this is that little wind-up propeller you had, a little rubber band propeller on your balsa wood airplane. If I push like that, you know that this thing is going to rotate.
2:41
Now, that is the biggest difference in chapter 4. We didn't have any rotation before, but now we do. So we have these things. Ready? Torque, moment or a couple--
3:01
and these words, they all mean the same thing. They are a tendency to cause rotation.
3:18
So if your friend says, hey, man, I was watching that Ford commercial on TV last night, and they were talking about torque. What is torque? Tell them, hey, just a moment, and they'll wait for you. Get it? Just a moment. Never mind.
3:36
So these are some things that we need to know, these terms. They just are tendency to cause rotation. So for rigid bodies, those things still have to be true, but there must be something more. And that something more is we call moments, which is just a torque, just a rotation. The sum of the moments about some point must always-- or also, rather-- be equal to 0. Now, our moments-- this is like moment about point A. Our moment vectors-- well, do they have magnitude?
4:15
Yeah, like 20 foot pounds of torque or of moment. So it has magnitude. Does it have direction? Yes, this one is rotating counterclockwise or anticlockwise, and so moment is definitely a vector, definitely. And so in 3D world, we can also-- instead of just saying moment about a point, we can look at moment about individual axes. So we can have the sum of the moments about the x-axis has to be 0. Things that make me spin around the y has to be equal to 0, and things that make me spin around the z
4:50
has to be equal to 0. Do you know what this right here means? Do you know what that means? That means that we'll have six equations. We can solve for up to six unknowns. Aren't you excited? Not really. Yeah, and your TI-36 Pro, sorry, it won't do six equations, six unknowns. We're going to have to do some brutal "al-guh-bra" to get through this, but we can do it, can't we? So if you go back to physics, do you remember any equations for torque or moment or couples? What is a couple?
5:28
Well, a couple-- let me smear this all over the place. That's a couple right there. Couple-- two forces, equal magnitude, parallel lines of action, oh yeah, opposite direction. So two forces, equal magnitude, opposite direction, but on parallel lines of action.
6:12
That is a couple, and that causes a rotation. So some things that we remember from physics about moment or torque is maybe this. Torque is equal to force times distance. That's a good one. This is my favorite equation to use in to 2D. And what would you got to remember about this distance, though-- the distance is special. It is the perpendicular distance. It's perpendicular to the force always. The distance in this equation is always perpendicular to the force.
6:49
And then another equation that you might remember for moment, moment about some point is r cross F. Moment's on that side. I mean, vector's on this side. Vector's on that side. r cross F-- well, this is force, and that's distance. That's force. What the heck is that? Well, it's really distance, also.
7:14
Let me tell you what that is. Now, this force vector-- if you're in 3D, I would recommend at least while we're rookies at this, in 3D, boom, we're going to use r cross F. Now, what is F? F is a 3D force vector. Do we know how to write 3D force vectors? Yes. What are the three ways you can get them in? Tell me then. Go ahead.
7:39
You didn't say it. Blue triangles, directional cosines, or lambda hat, right? So yeah, we are all practiced up on how to do that, but I don't have any idea what to do with that right there. Well, actually, we've been doing this for a while. r is what we call a position vector, and it's just how you get to grandma's house. It's a distance from this point to that point. That's all r is, right? Remember, vector AB, we did B minus A. Well, if you just do B minus A that's r.
8:11
That's the r vector right there. Position vector is what it's called, position vector. Now, if y'all are taking notes on these videos, which you should be, I'm going to go back next video and ask you what a position vector is. And you're going to go, I don't remember. You got to remember what this is because the definition of this tells you how to draw your position vector, which is important to do r cross F equations. So a position vector-- a vector whose tail is at the point we wish to take
9:00
a moment-- (SINGING) we wish you to take a moment--
9:11
a vector whose tail-- that's vector, not a voctor-- vector whose tail is at the point we wish to take a moment about and whose tip is-- oh my goodness, I'm writing this big, underline, underline-- anywhere along the LOA, which is the line of action, of the force.
9:47
So a position vector-- a vector whose tail is at the point we wish to take the moment about and whose tip is anywhere along the line of action of the force. So the key about the position vector, which goes with this r cross F here-- the key about this equation is r does not have to be perpendicular to the F vector. This cross product here will take care of it for you. This one, if you use this equation, the distance has to be perpendicular distance to the force vector.
10:22
So if the force is in the x, then the distance has to be in the y. If the force is in the y, then the distance has to be in the x. But this one doesn't matter. As long as it starts at the point you're taking the moment about and it goes to the line of action to the force, this will take care of the rest for you. That's it. So what's next? I noticed that both of these equations have something that we haven't talked about yet, and that is, what is this? And what is that?
10:53
This time, we're not adding vectors.
11:02
We're multiplying vectors. So we need to review how the heck you do that. Let me erase the board. I'm going to just make up a couple of vectors--
11:17
3 i hat minus 7 j hat plus 6 k hat. Let's say there's one vector, and the other vector is-- I don't know-- 8 i hat plus 6 j hat minus 2 k hat. So there's two vectors. How do I multiply those two vectors together? Now, one thing you need to know-- when you cross two vectors together, remember, we're talking about cross product. Let's say this is vector a, and this is vector b. So if we want to take vector a and cross it with vector b, what do we get?
11:57
Well, we get some kind of vector that the cross product is. And what you're going to get is when you cross two products together, you can use the right-hand rule. So if I put my finger in the direction of one of the vectors and I curl my fingers in the direction of the second vector, I get that result vector. Now, what happens there is that what that tells me is any time you cross two vectors together, any two vectors, when you cross them together, you get a perpendicular vector.
12:30
So if I have two vectors like that, like that, if I cross them together, I get that perpendicular vector to that. So that's a handy trick we'll use in some problems in the book here to solve for homework, but let's review how to do this. This is Kramer's rule.
12:50
And you have an i hat, a j hat, and a k hat.
12:57
So the moment about some point is going to be a cross b. Now, I like to put them like this because it's a cross b. It's not b cross a. If you get these backwards, when you're doing r cross F, if you do F cross r and you get these two backwards, you get the correct vector, but all the signs are backwards, all of them. So that's a bad thing. So first line is this vector, which is 3, negative 7, and 6, and the next line is 8, 6, and negative 2. Now, if you're not a fan of doing this next step,
13:34
which I'm going to do manually, there's a video on my YouTube channel where the TI-36 Pro, our calculator, will do this for you. And there's some opportunity to mess this up with sine, so it'd be a good thing to learn just so you didn't make that error. But here's how you do this. When you do the cross product, we do i hat first. Then we do j hat. Then we do k hat. j hat is negative, so it's positive, negative, positive. And what you do is you cover up the i hat.
14:03
He doesn't get to play in his own party. And then we cross that and that, so it goes like this. Negative 7 times negative 2-- and I use lots of parentheses in brackets to distribute my negative signs there-- and then minus this cross 6 times 6. And that's your i hat. The next one-- so the next one, I cover up the j hat. Now, remember, this one's minus in the middle. So cover up the j hat, and I have this cross. So I have 3 times negative 2 minus 6 times 8, and there's my j hat.
14:49
And one more, so I'm back to plus again. Then it goes like this. Cover up the k, and what do I have left? This cross that. So 3 times 6 minus minus 7 times 8.
15:08
So let's see. There you go. Now all that's left to do is just simplify this, but be really careful with your signs. Negative 7 times negative 2 is a positive 14. 6 times 6 is 36. 3 times negative 2 is minus 6, and 6 times 8 is 48. And 3 times 6 is 18, and 7 times 8 is 42. But that's negative. So what do I have? 14 minus 36.
15:43
Just make sure I don't make a calculator mistake in my head. 14 minus 36 is minus 22, so minus 22 i hat. And this guy, minus 6 minus 48 more is minus 54. That's minus 54, but then there's a minus out there, which makes it a plus 54.
16:06
And then 18 minus a minus 42 turns into 18 plus 42. That's 60.
16:17
And there is your vector, your moment, moment vector for crossing those two vectors together. So do you remember how to do that? Because this chapter is all about multiplying vectors, where the other chapters that we've been covering have been about adding vectors, so a little bit different. So what I want to do now is work a little quick example problem using this technique, and then we'll be done. Here we go. We're going to try and find the moment for this problem,
16:49
so we got a stick stuck in the ground at an angle here, 30 degrees. This is 30 degrees, and it's got a 100-pound force on the end of it. So always, when you're talking about finding the moment at a point, think to yourself, that point is like a hinge point. So if I put my finger there and I grab that stick, it would rotate around that point. So find the moment due to that force. So here we go. Moment at point A is equal to r cross F. Number one,
17:23
I just need to write these two things in vector form. Now, F ought to be easy.
17:32
What do you think F is for this guy? No, that's not it. There you go, OK. So it's minus 100 pounds j hat. It's in the negative y direction. It doesn't have anything to do with this 30 over here. The force is completely in the y direction. That's it. It has no x component. That's it. So what is vector r? What is vector r? I forgot what vector r is. Do you remember?
18:09
Vector r is a position vector whose tail is at the point we wish to take the moment, which is point A, and whose head is anywhere along the line of action of the force, so how about there? I'll call that r1.
18:24
Oh, wait a second. Did you say anywhere along the line of action of the force? Well, here's the line of action of the force. So couldn't it start here and go there? We'll call that r2. The answer is yes. So here we go. We got vector r1, and we got vector r2. So we need to write those two guys in Cartesian form, and we'll check them out. So r1-- how do I get from here to there? Well, you got to go this much in the x and that much in the y, so r1 is 24 cos 30 i hat plus 24 sine 30 j hat.
19:14
That's how you go from there to there. How do you go from there to just there? Well, that's just the x component only. So this guy is just 24 cos 30, and that's it-- i hat. He has no j hat. OK OK? Well, let me get some numbers for those because I don't know what that is. 0.866 times 24 equals 20.78, so this is 20.78. And this is just sine of 30 is a half, so that's just 12, and this is 20.78. So let's build our little r cross F matrix, and let's see what the moment at point A is.
20:03
So the moment at point A is equal to-- here we go-- i hat, j hat, k hat. So r cross F-- so r is here, so r is 20.78, 12, and 0. I'm using r1. And F is 0, minus 100, and 0, so here we go. Now, what did I tell you? If you cross two vectors together, you get a perpendicular vector. So if I cross x and y vectors together, what should I always get? If I do this one and curl into that one, I should always get a k hat, always, OK? So I should get a k hat. Let's see.
20:51
Cover up the i, and you get 0 minus 0. OK cool. Cover up the j, 0 minus-- that's good, OK. Cover up the k, and I get 20.78 times minus 100, which is minus 2,078. And that is inch pounds, so inch pounds.
21:19
2,078 minus 12 times 0 minus 0. So there you go. That is the moment at point A, and that is k hat.
21:32
Now, what does the minus mean? I'm telling you this, OK?
21:39
Clockwise moments are negative.
21:51
Counterclockwise moments are positive. So don't you agree that if it's hinged here and I pull down on that stick that it's going to rotate that stick clockwise? That's all that negative right there is telling me, is that that moment is clockwise instead of counter. That's it. Brilliant. Now here we go. Now look here. The 20.78 and the 12-- who did all the work? You know what the 12 did? He was multiplied by 0. He didn't do anything. So wait a minute. So what about this?
22:35
Do it one more time, i hat, j hat, k hat, 0, minus 100, 0. And I'm going to use this time r2, which is 20.78, and 0 and 0. So cover up the i hat, 0 minus 0. Cover up the j hat, 0 minus 0. Cover up the k hat, 20.78 times 100 minus 0. So this one equals minus 2,078 inch pounds k hat. It's exactly the same. Remember what I told you. Our vector starts here and goes to the line of the action of the force. It goes anywhere on the line of action in the force,
23:17
so it doesn't matter if you go there or you go there or you go there or there or down there. It doesn't matter. Your matrix will automatically fix that for you. And if you'll notice, the only one it used was the 20.78, which is the perpendicular component. It ignored the parallel component. So the matrix automatically takes care of that for you. Doesn't matter where you take your r vector. As long as it goes from the point to the line of action, the math will fix itself.
23:52
So I hope that kind of gets us started on moments. We'll do some example problems, and we'll get really good at this. All right, see you next video. [AUDIO LOGO]
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