Anderson Video - Torque Cross Product

Professor Anderson
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>> Hello class. Professor Anderson here. Let's continue our discussion about torque. And we're going to introduce something new. It's called the cross product. So let's go back to this idea that we are going to open the door. There's our door. And we're going to apply a force F to that door. Now what we said last time was this is probably not the ideal force that you would use to open the door. You know that when you are pushing on a door you want to push perpendicularly to the door. But let's say you do apply this sort of force. What is the torque that is exerted on the door? Well to do that we need to give the door a length. Let's call it r. And now let's redraw it right here. r is a displacement vector, OK? It goes from the axis of rotation to the end of the door. The force F is of course a vector. It's applied at the end of the door. This is where the hinges are, so that's the axis of rotation. And now we need to figure out what is the torque that is applied to the door and what we're going to introduce is this idea of a cross product. This is what we mean by this cross in the middle. This is not an x. This is a cross and it's the cross product, OK? So to define the cross product we need to define something else in this picture; let's call it this angle right here phi which is the angle between the line of r and the line of F. And if we do that then this cross product becomes the following: it's the magnitude of r times the magnitude of F times the sine of the angle between them and then there is a direction associated with it. This is the definition of the cross product. So let's write that over here and we will talk about some of those terms. The first one, r, is the magnitude of r. It's the length of the door. F is of course the magnitude of F. How hard are you pulling on that door? Phi is the angle between those two, OK? Which is this angle right here. Namely if I continued r and I looked at the angle between r and F, that would be the angle Phi. And then n hat is the direction of the torque. And the direction of torque is always perpendicular to r and F, which means in this case it's either into the glass or out of the glass. How do you determine whether it's into the glass or out of the glass? The way you do it is with something called the right-hand rule, OK? And this right-rule works for your right hand. Don't use your left hand. It's not called the left-hand rule, OK? It's called the right-hand rule and what it says is the following: if I take the direction of r and I put my fingers straight and then I take the direction of F and I curl my fingers into that direction then n hat is the direction of your thumb, OK? And let's see if we can try this for this particular case. So everybody out in the audience raise your right hand, OK? And now if you look at the screen over there and I look at this we can do it together. Put your fingers straight into the direction of r, OK? So that should be to your right. And now curl them slightly into the direction of F. Which way is your thumb going? Is it out of the screen or is it into the screen? I'm asking you guys. What do you think? >> Out. >> Out. It's out of the screen, OK? And so in this case the direction of torque is out of the glass, all right? Now this takes some practice once you get to more complicated geometries and certainly in your math courses you will deal with this idea of the cross product and the right-hand rule even more extensively. But there's another way to think about it which involves a saying that you probably already know. Let's talk about torque on a bolt and we're going to relate this to the direction of the torque that we just talked about. So pretend you have a Phillips head screw and your Phillips head screw looks like that, right? It has a cross on the end of it and you take your screwdriver and you stick it into that cross and you can tighten it or you can loosen it. And there's an expression that tells you about whether you're going to tighten it or whether you're going to loosen it. Does anybody know that expression? >> Yes John, what's the expression? >> Righty tighty, lefty loosey. >> Righty tighty, lefty loosey, OK? Righty tighty, lefty loosey. What does that mean? That means if I turn this thing to the right it gets tighter, OK? If I turn it to the right it gets tighter. If I turn it to the left it gets looser. Now let's see if we can make sense of this in terms of which way the bolt is actually going to move. So let's say this is my bolt in a piece of wood. I stick my screwdriver in and I turn it to the right. Does it go into the wood, or does it come out of the wood? Does it go into the wood, or does it come out of the wood? >> Into the wood. >> Into the wood. >> Into the wood. >> Into the wood. It goes into the page. When you go the other way it comes out of the page. And this is another manifestation of the right-hand rule, OK? How do I see that? Well let's blow up our bolt a little bit. There's our bolt. Here is our radius r. And if I tighten it to the right I am going to apply a force like that, OK? So everybody hold up your right hand and put your fingers -look at the screen over there- put your fingers in the direction of r, OK? Straight up. And now curl your fingers into the direction of F, OK? Which way is your thumb going? >> Into the page. >> Into the page, OK? This is going to be into the page. Did everybody see that? If you didn't see that you might have held up your left hand, so make sure you have you right hand up, OK? So that rotation to the right is exactly the same as we were talking about with the bolt, it's going to screw into the page. Right tighty, lefty loosey. Now, anybody know what type of screws are on our particular bolt? Anybody know the name for those screws, those threads that are on the bolt? I'll give you a hint. What? >> Teeth? >> John, did you have a comment? >> Teeth? >> OK. Yes, they're like teeth, right? They're sort of like teeth on a gear, but they have a particular orientation to their helix pattern, OK? And that bolt is called a right-handed bolt or right-handed threads. And for bolts that are right-handed this applies; righty tighty, lefty loosey. For bolts that are left-handed it's exactly opposite, OK? And this turns out to be something like 99% of all bolts are right-handed. If you go to Home Depot and you buy a bolt or a wood screw or anything like that, it's going to have right-handed threads on it, which means that this applies; righty tighty, left loosey. But not all of them, only 99% of them. I'm just making that number up, but it's a lot of them, OK? There is one place where you find left-handed threads, in which case this is exactly reversed. It becomes right loosey, lefty tighty. They should really just come up with a new acronym, a new moniker for that. Anybody know where you find a left-handed bolt? You have all experienced this left-handed bolt somewhere in your life. You might even have a left-handed bolt at home or in your garage. Can you pass the mic over to Christian? Let's have a chat with Christian over here. >> OK. >> One of the Christians. There's three Christians in here right now, so I'll just say, "Christian" and it could be anybody. Christian, where might I find a left-handed bolt? Any ideas? >> You're telling me it's somewhere in the garage? >> It's somewhere in the garage. >> Is it in the car somewhere in the garage? >> OK, close. Different vehicle. >> The motor for your garage opener? >> OK, close. >> Lawnmower? >> What's that? >> Lawnmower? I'll just keep spitting out ideas here. I have no clue. >> OK. What if I rode to school on a certain sort of very efficient device? >> Bicycle? >> Bicycle. >> Oh, my first guess. >> You said skateboard or roller skates? Blades? People blade anymore? Bicycle, right? Everybody has a bicycle probably or you've had one in your life. On your bicycle there is a left-handed bolt. Where is that left-handed bolt? >> Oh, uh. Isn't it when you tighten your seat down? >> OK, no. >> Handlebars? The wheels? >> No. >> Handlebars? >> Not on the handlebars. Hand the mic back to -what's your name? >> Emilio. >> Emilio, where might I find a left-handed bolt? >> On the wheels? >> OK. Not on the wheels. >> Where you place the wheels to the... >> OK. Not on the wheels. Not on the axel of the wheels. But there's something else that spins on your bike. What is it? >> The crank? >> The gear? >> The crank, right? The crank. Where the pedals attach to that piece of metal that you spin with your feet, right? And then there's a gear, there's a chain and so forth, right? That crank, one of those is left-handed, one of those is right-handed. It turns out that the pedal on the right is actually left-handed threads, OK? Everything else on the bike is pretty much right-handed except for that one pedal is left-handed. And if you've ever done bike repair, when you screw in that pedal, you have to screw it the other way. Why? Why would they put a left-handed thread just on that one part of the bike? Emilio, why? >> I'm guessing because when you push on your leg on the crank it's going -uh, I'm not sure. >> OK. When you push on the pedal, right? And you're trying to move the bicycle, you don't want your pedal to fall off, right? You would like that thing to tighten up as you pedal rather than loosen up. And so one of those pedals is right-handed because as you pedal it tightens up, but the other one is exactly the opposite, it has to be left-handed. If you instead made that right-handed, as you pedaled it could work itself loose. And this in fact happens sometimes. If you just pedal your bicycle backwards for a very long time, occasionally those pedals fall off, OK? And that's because the thread is loosening up, not tightening up as you pedal, all right? We learn enough about bike repair here?
>> Hello class. Professor Anderson here. Let's continue our discussion about torque. And we're going to introduce something new. It's called the cross product. So let's go back to this idea that we are going to open the door. There's our door. And we're going to apply a force F to that door. Now what we said last time was this is probably not the ideal force that you would use to open the door. You know that when you are pushing on a door you want to push perpendicularly to the door. But let's say you do apply this sort of force. What is the torque that is exerted on the door? Well to do that we need to give the door a length. Let's call it r. And now let's redraw it right here. r is a displacement vector, OK? It goes from the axis of rotation to the end of the door. The force F is of course a vector. It's applied at the end of the door. This is where the hinges are, so that's the axis of rotation. And now we need to figure out what is the torque that is applied to the door and what we're going to introduce is this idea of a cross product. This is what we mean by this cross in the middle. This is not an x. This is a cross and it's the cross product, OK? So to define the cross product we need to define something else in this picture; let's call it this angle right here phi which is the angle between the line of r and the line of F. And if we do that then this cross product becomes the following: it's the magnitude of r times the magnitude of F times the sine of the angle between them and then there is a direction associated with it. This is the definition of the cross product. So let's write that over here and we will talk about some of those terms. The first one, r, is the magnitude of r. It's the length of the door. F is of course the magnitude of F. How hard are you pulling on that door? Phi is the angle between those two, OK? Which is this angle right here. Namely if I continued r and I looked at the angle between r and F, that would be the angle Phi. And then n hat is the direction of the torque. And the direction of torque is always perpendicular to r and F, which means in this case it's either into the glass or out of the glass. How do you determine whether it's into the glass or out of the glass? The way you do it is with something called the right-hand rule, OK? And this right-rule works for your right hand. Don't use your left hand. It's not called the left-hand rule, OK? It's called the right-hand rule and what it says is the following: if I take the direction of r and I put my fingers straight and then I take the direction of F and I curl my fingers into that direction then n hat is the direction of your thumb, OK? And let's see if we can try this for this particular case. So everybody out in the audience raise your right hand, OK? And now if you look at the screen over there and I look at this we can do it together. Put your fingers straight into the direction of r, OK? So that should be to your right. And now curl them slightly into the direction of F. Which way is your thumb going? Is it out of the screen or is it into the screen? I'm asking you guys. What do you think? >> Out. >> Out. It's out of the screen, OK? And so in this case the direction of torque is out of the glass, all right? Now this takes some practice once you get to more complicated geometries and certainly in your math courses you will deal with this idea of the cross product and the right-hand rule even more extensively. But there's another way to think about it which involves a saying that you probably already know. Let's talk about torque on a bolt and we're going to relate this to the direction of the torque that we just talked about. So pretend you have a Phillips head screw and your Phillips head screw looks like that, right? It has a cross on the end of it and you take your screwdriver and you stick it into that cross and you can tighten it or you can loosen it. And there's an expression that tells you about whether you're going to tighten it or whether you're going to loosen it. Does anybody know that expression? >> Yes John, what's the expression? >> Righty tighty, lefty loosey. >> Righty tighty, lefty loosey, OK? Righty tighty, lefty loosey. What does that mean? That means if I turn this thing to the right it gets tighter, OK? If I turn it to the right it gets tighter. If I turn it to the left it gets looser. Now let's see if we can make sense of this in terms of which way the bolt is actually going to move. So let's say this is my bolt in a piece of wood. I stick my screwdriver in and I turn it to the right. Does it go into the wood, or does it come out of the wood? Does it go into the wood, or does it come out of the wood? >> Into the wood. >> Into the wood. >> Into the wood. >> Into the wood. It goes into the page. When you go the other way it comes out of the page. And this is another manifestation of the right-hand rule, OK? How do I see that? Well let's blow up our bolt a little bit. There's our bolt. Here is our radius r. And if I tighten it to the right I am going to apply a force like that, OK? So everybody hold up your right hand and put your fingers -look at the screen over there- put your fingers in the direction of r, OK? Straight up. And now curl your fingers into the direction of F, OK? Which way is your thumb going? >> Into the page. >> Into the page, OK? This is going to be into the page. Did everybody see that? If you didn't see that you might have held up your left hand, so make sure you have you right hand up, OK? So that rotation to the right is exactly the same as we were talking about with the bolt, it's going to screw into the page. Right tighty, lefty loosey. Now, anybody know what type of screws are on our particular bolt? Anybody know the name for those screws, those threads that are on the bolt? I'll give you a hint. What? >> Teeth? >> John, did you have a comment? >> Teeth? >> OK. Yes, they're like teeth, right? They're sort of like teeth on a gear, but they have a particular orientation to their helix pattern, OK? And that bolt is called a right-handed bolt or right-handed threads. And for bolts that are right-handed this applies; righty tighty, lefty loosey. For bolts that are left-handed it's exactly opposite, OK? And this turns out to be something like 99% of all bolts are right-handed. If you go to Home Depot and you buy a bolt or a wood screw or anything like that, it's going to have right-handed threads on it, which means that this applies; righty tighty, left loosey. But not all of them, only 99% of them. I'm just making that number up, but it's a lot of them, OK? There is one place where you find left-handed threads, in which case this is exactly reversed. It becomes right loosey, lefty tighty. They should really just come up with a new acronym, a new moniker for that. Anybody know where you find a left-handed bolt? You have all experienced this left-handed bolt somewhere in your life. You might even have a left-handed bolt at home or in your garage. Can you pass the mic over to Christian? Let's have a chat with Christian over here. >> OK. >> One of the Christians. There's three Christians in here right now, so I'll just say, "Christian" and it could be anybody. Christian, where might I find a left-handed bolt? Any ideas? >> You're telling me it's somewhere in the garage? >> It's somewhere in the garage. >> Is it in the car somewhere in the garage? >> OK, close. Different vehicle. >> The motor for your garage opener? >> OK, close. >> Lawnmower? >> What's that? >> Lawnmower? I'll just keep spitting out ideas here. I have no clue. >> OK. What if I rode to school on a certain sort of very efficient device? >> Bicycle? >> Bicycle. >> Oh, my first guess. >> You said skateboard or roller skates? Blades? People blade anymore? Bicycle, right? Everybody has a bicycle probably or you've had one in your life. On your bicycle there is a left-handed bolt. Where is that left-handed bolt? >> Oh, uh. Isn't it when you tighten your seat down? >> OK, no. >> Handlebars? The wheels? >> No. >> Handlebars? >> Not on the handlebars. Hand the mic back to -what's your name? >> Emilio. >> Emilio, where might I find a left-handed bolt? >> On the wheels? >> OK. Not on the wheels. >> Where you place the wheels to the... >> OK. Not on the wheels. Not on the axel of the wheels. But there's something else that spins on your bike. What is it? >> The crank? >> The gear? >> The crank, right? The crank. Where the pedals attach to that piece of metal that you spin with your feet, right? And then there's a gear, there's a chain and so forth, right? That crank, one of those is left-handed, one of those is right-handed. It turns out that the pedal on the right is actually left-handed threads, OK? Everything else on the bike is pretty much right-handed except for that one pedal is left-handed. And if you've ever done bike repair, when you screw in that pedal, you have to screw it the other way. Why? Why would they put a left-handed thread just on that one part of the bike? Emilio, why? >> I'm guessing because when you push on your leg on the crank it's going -uh, I'm not sure. >> OK. When you push on the pedal, right? And you're trying to move the bicycle, you don't want your pedal to fall off, right? You would like that thing to tighten up as you pedal rather than loosen up. And so one of those pedals is right-handed because as you pedal it tightens up, but the other one is exactly the opposite, it has to be left-handed. If you instead made that right-handed, as you pedaled it could work itself loose. And this in fact happens sometimes. If you just pedal your bicycle backwards for a very long time, occasionally those pedals fall off, OK? And that's because the thread is loosening up, not tightening up as you pedal, all right? We learn enough about bike repair here?