Anderson Video - Angular Momentum and Cross Product

Professor Anderson
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>> Hello class, Professor Anderson here. We're talking about angular momentum today. And let's see what we know about momentum and what we don't know about angular momentum. So we're all familiar with this term of momentum, right? We've talked about how linear momentum, p is just m times v. But what is angular momentum? As you might expect this has to do with rotational motion. So angular momentum we actually write with an L. And L is equal to r cross p. So let's identify what these different terms are. This is our new angular momentum term, and again for some reason we write it with an L because we've run out of other letters, r is some sort of distance. And in fact it is the distance from the axis of rotation to the object. P is the linear momentum of that object. What about this funny looking x thing? This is actually called the cross product. And the cross product is a mathematical tool for figuring out how to combine two vectors and still maintain a vector quantity over on the left hand side. Remember when we talked about the dot product the end result was a scaler, right, two vectors dotted together would give us a scaler. But now the cross product is in fact a vector quantity and that's why the L has a vector on top of it over on the left side. So this is what we need to understand and let's identify first what the cross product is and then how to apply it to various angular momentum problems. Ok what is this cross product that we're talking about? The cross product is the following, let's say I have vector A and it looks like this and I have vector B and it looks like that. And I want to figure out the cross product between these two, not the dot product. The way you figure it out is you put them together, A and B, and then you figure out what the angle is between them and we're going to call that angle phi. Before when we did the dot product we called it theta so in this case we'll call it phi. The cross product A cross B is defined as the following, it's magnitude of A times magnitude of B times the sine of the angle between them, remember when we did the dot product we had cosine and now it's the sine of the angle between them but there's one more thing that's really important here which is cross product is a vector quantity. This that we've written so far is just a magnitude. And so we need to identify a direction. And so for the direction we do an n hat right there. This is of course the magnitude of vector A. This is the magnitude of vector B. This is the angle between them. And this is the resulting direction which we need to figure out from something called the right hand rule. So how do we figure out this direction? The direction n hat we follow what's called the right hand rule, ok? If I have vector A and vector B I can figure out the direction of the cross product if I'm very careful about following the right hand rule. Now this is a little tricky with the learning glass because my right hand is looking like left hand to you. So I in fact have to use my left hand. You'll notice that the ring is on this hand. This is really my left hand even though it looks like my right hand to you guys. The right hand rule is the following. If I have A cross B and that's going to equal some vector C, the way you figure out the direction is the following, A you do your fingers straight, B is you curl your fingers, C is the direction of your thumb. And it's called the right hand rule because you have to do it with your right hand. So let's see if we can figure it out for this simple example that we just talked about. Ok if we just have two vectors, and I'm going to change the order slightly, let's say we have vector A pointing to the right and we have vector B pointing up. What is A cross B? A cross B is going to be magnitude of A, magnitude of B times the sine of the angle between them with a particular direction. But the angle between these two is a right angle and so we get sine of 90 degrees and so we just get AB n hat. Sine of 90 degrees is of course 1. Now how do we figure out the direction for this thing? Ok which way is it going? Well what we said over here was if I do my fingers straight in the direction of A and then I do my fingers curled in the direction of B that should be the direction for C which is given by my thumb. So let's try it. Ok we have vector A going to the right, fingers straight in that direction. And then I want to curl my fingers towards B which would be up. My thumb is now the direction of C, the direction of the cross product, which is of course, according to how we just did it, out of the screen. Ok so try this at home. Sitting there in front of your computer screen, try it. Put your fingers straight in the direction of A, put your curled fingers up towards the direction of B and what you should get is a thumb coming out of the computer screen right towards you. Ok and so this is going to be A, B and the direction is out of the computer screen. Alright let's take a look at that and make sure it works for everybody at home. We're going to move on to the next topic but if you're having trouble with the right hand rule, definitely come see me in office hours. Cheers.
>> Hello class, Professor Anderson here. We're talking about angular momentum today. And let's see what we know about momentum and what we don't know about angular momentum. So we're all familiar with this term of momentum, right? We've talked about how linear momentum, p is just m times v. But what is angular momentum? As you might expect this has to do with rotational motion. So angular momentum we actually write with an L. And L is equal to r cross p. So let's identify what these different terms are. This is our new angular momentum term, and again for some reason we write it with an L because we've run out of other letters, r is some sort of distance. And in fact it is the distance from the axis of rotation to the object. P is the linear momentum of that object. What about this funny looking x thing? This is actually called the cross product. And the cross product is a mathematical tool for figuring out how to combine two vectors and still maintain a vector quantity over on the left hand side. Remember when we talked about the dot product the end result was a scaler, right, two vectors dotted together would give us a scaler. But now the cross product is in fact a vector quantity and that's why the L has a vector on top of it over on the left side. So this is what we need to understand and let's identify first what the cross product is and then how to apply it to various angular momentum problems. Ok what is this cross product that we're talking about? The cross product is the following, let's say I have vector A and it looks like this and I have vector B and it looks like that. And I want to figure out the cross product between these two, not the dot product. The way you figure it out is you put them together, A and B, and then you figure out what the angle is between them and we're going to call that angle phi. Before when we did the dot product we called it theta so in this case we'll call it phi. The cross product A cross B is defined as the following, it's magnitude of A times magnitude of B times the sine of the angle between them, remember when we did the dot product we had cosine and now it's the sine of the angle between them but there's one more thing that's really important here which is cross product is a vector quantity. This that we've written so far is just a magnitude. And so we need to identify a direction. And so for the direction we do an n hat right there. This is of course the magnitude of vector A. This is the magnitude of vector B. This is the angle between them. And this is the resulting direction which we need to figure out from something called the right hand rule. So how do we figure out this direction? The direction n hat we follow what's called the right hand rule, ok? If I have vector A and vector B I can figure out the direction of the cross product if I'm very careful about following the right hand rule. Now this is a little tricky with the learning glass because my right hand is looking like left hand to you. So I in fact have to use my left hand. You'll notice that the ring is on this hand. This is really my left hand even though it looks like my right hand to you guys. The right hand rule is the following. If I have A cross B and that's going to equal some vector C, the way you figure out the direction is the following, A you do your fingers straight, B is you curl your fingers, C is the direction of your thumb. And it's called the right hand rule because you have to do it with your right hand. So let's see if we can figure it out for this simple example that we just talked about. Ok if we just have two vectors, and I'm going to change the order slightly, let's say we have vector A pointing to the right and we have vector B pointing up. What is A cross B? A cross B is going to be magnitude of A, magnitude of B times the sine of the angle between them with a particular direction. But the angle between these two is a right angle and so we get sine of 90 degrees and so we just get AB n hat. Sine of 90 degrees is of course 1. Now how do we figure out the direction for this thing? Ok which way is it going? Well what we said over here was if I do my fingers straight in the direction of A and then I do my fingers curled in the direction of B that should be the direction for C which is given by my thumb. So let's try it. Ok we have vector A going to the right, fingers straight in that direction. And then I want to curl my fingers towards B which would be up. My thumb is now the direction of C, the direction of the cross product, which is of course, according to how we just did it, out of the screen. Ok so try this at home. Sitting there in front of your computer screen, try it. Put your fingers straight in the direction of A, put your curled fingers up towards the direction of B and what you should get is a thumb coming out of the computer screen right towards you. Ok and so this is going to be A, B and the direction is out of the computer screen. Alright let's take a look at that and make sure it works for everybody at home. We're going to move on to the next topic but if you're having trouble with the right hand rule, definitely come see me in office hours. Cheers.