Professor Anderson

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>> One concept that we've been talking about a bit is this idea of the center of mass. But we need to be careful in defining what exactly we mean by the center of mass. So anytime you have some object, you can always draw a coordinate system wherever you like. If that was my coordinate system, where would the center of mass be for this object? I don't know but let's pick. Let's say somewhere about there. That's a good enough guess. How would I calculate exactly where that point is? Well, for a blob like this, it's maybe a little bit tricky. But let's pretend that it's not a blob for a second. Let's say it is something like this. Pretend it's a square with masses m on each side. And now I can draw my coordinate system wherever I like. So why don't we draw the coordinate system right through the center? If this thing is a square and all the corners have mass M on it, you already know the answer to this. Where is the center of mass? What's your name in the back over there? >> (student speaking) Travis. >> Travis, where is the center of mass? >> (student speaking) In the center. >> In the center, right? It's got it in the name -- in the center. That's it. How do I write down an equation that tells me that? I do the following. X position of the center of mass is the following, M1, X1, plus M2, X2, plus dot, dot, dot. And I divide by the total mass, M1 plus M2 plus dot, dot, dot. I can rewrite this as a summation. It's the sum of M sub i, X sub i divided by capital M, the total mass of the system. And I can, of course, do the same thing for the Y center of mass. Y center of mass is going to be the sum over i, M sub i, Y sub i divided by the total mass. All right, with that information, let's calculate it for this particular example and see if Travis was right. Here's our object. We said that the X position of the center of mass is the sum of M sub i, X sub i divided by the total mass of the system. So let's just start from right here. And let's say that each side of this square is L. So we're going to have M1, X1 plus M2, X2 plus M3, X3, plus M4, X4. We have four particles and we're going to divide by the total mass. And let's just for kicks, let's start right here. Okay, what is the X position of that particle? Travis, what do you think? What's the X position of this particle down here? >> (student speaking) I don't think I understand the X position portion of it? >> Yeah, what would be the X location of this particle? It would be this position right here which is if this is the origin -- that would be negative L over 2, okay. So negative L over 2 for that first one and the first one we also said is just mass M so we'll take out that subscript. This is number one. This is number two. This is number three and this is number four. M2 which is also there's the mass M, that also has an X position of negative L over 2. M3 has an X position of positive L over two. X4 is also positive L over 2. And we're going to divide this thing by the total mass of the system which is just four m. And now you see what happens, right? We have a negative there that cancels with the positive there. We have a negative there that cancels with the positive there. And we get Xcm is equal to zero. So on our coordinate system, the X position of the center of mass is equal to zero. And by symmetry, you're going to get Ycm also equal to zero. Okay, so we were right. The center of mass is right in the middle of the system. And it is centered on our coordinate system that we drew. So back to our complicated blob. We have the rock, something like that. And we need to figure out how to find the center of mass of that rock, okay. How do you do it? Well, what we said last time was if we have discrete particles, we can write X center of mass is the sum over i M sub i, X sub i divided by the total mass. But we don't have individual particles anymore. We have a continuous distribution of matter. And so we need to change this thing into an integral. How do you do that? M becomes dm. The summation becomes an integral. R is what we call X. The bottom is still total mass. So let's call this one -- oops. Let's write it now with an integral. It's Xdm divided by the total mass. X is the X position of that little mass element dm. Y is going to look like this. Okay and if you have three dimensions, you can write Z center of mass as well. So if you know what the functional form is of your blob then you can plug it into the integral and you can calculate it. But let's say you don't know what it is. Let's say I give you a big rock and I ask you, "Where is the center of mass?" How do you find it? Let's say I hand you a rock. And I want you to tell me where the center of mass is of that rock. How would you do it? I'm asking you guys. What do you think? How would you find the center of mass of this rock if I hand it to you? Somebody hand the mic to Nasim. Nasim, what do you think? How would I find the center of mass of this rock if I handed you this rock? >> (student speaking) I'm not totally sure but -- maybe try like balancing it in some way? >> Okay. >> (student speaking) On something? >> Excellent. So why don't we take that rock and let's put it on the table? And if I put it on the table and I can get it to balance then I can draw a line straight down through that intersection point between the rock and the table. And the center of mass has to lie along that line. And now, let's do it again. We'll take the rock and we'll rotate it until it's standing up on one edge. Okay and so it's going to look something like this. I had taken my rock and I had rotated it up and now there is a new line that goes straight through that intersection point. And I can redraw my old line on this new picture. And wherever those two lines meet, that is the center of mass. That's how you figure out the center of mass of some complicated object like a rock, okay. And so when you go down to the harbor down in San Diego and you see people balancing those rocks on top of each other. The way you balance those rocks on top of each other is you make sure that that point is directly underneath the center of mass. And if it's not, then it's going to tilt one way and if it's off the other way, it'll tilt the other way. And so you stand there and you very carefully hold it and adjust it. Tilt it left and right until you get that center of mass directly above that contact point and then it will balance, okay. And it's hard to do especially if you have a big, complicated object like this and a little tiny point on the bottom. It's tough to get that point just right but there's people that do it all the time and they're very skilled at it. Question, can you hand him the mike? What's your name? >> (student speaking) Eric. >> Eric, is there a -- >> Be careful of the mic. Hold on a second. >> (student speaking) Okay. >> Yeah, Eric. You got a question? >> (student speaking) Yes. So the balance point is also the center of mass? Is that basically what you're saying? >> What I'm saying is the balance point, this point, is directly beneath the center of mass of the object. >> Oh, okay. >> All right. >> That makes sense. >> So wherever that contact point is, it's directly beneath the center of mass. So we did it twice. We put it on one side and then we put it on another. And we figured out where they intersect and that's where the center of mass is. >> (student speaking) Okay. >> All right? And now you kind of can see what's going to happen, right? If the rock tilts slightly to the right, gravity acting on that center of mass is going to torque the object and cause it to fall to the right. If it's slightly to the left then it's going to torque it and cause it to fall to the left. Yeah, so this is a great exercise of art, and physics, and balancing. It's really very tricky to do but pretty remarkable when you see it in action.

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>> One concept that we've been talking about a bit is this idea of the center of mass. But we need to be careful in defining what exactly we mean by the center of mass. So anytime you have some object, you can always draw a coordinate system wherever you like. If that was my coordinate system, where would the center of mass be for this object? I don't know but let's pick. Let's say somewhere about there. That's a good enough guess. How would I calculate exactly where that point is? Well, for a blob like this, it's maybe a little bit tricky. But let's pretend that it's not a blob for a second. Let's say it is something like this. Pretend it's a square with masses m on each side. And now I can draw my coordinate system wherever I like. So why don't we draw the coordinate system right through the center? If this thing is a square and all the corners have mass M on it, you already know the answer to this. Where is the center of mass? What's your name in the back over there? >> (student speaking) Travis. >> Travis, where is the center of mass? >> (student speaking) In the center. >> In the center, right? It's got it in the name -- in the center. That's it. How do I write down an equation that tells me that? I do the following. X position of the center of mass is the following, M1, X1, plus M2, X2, plus dot, dot, dot. And I divide by the total mass, M1 plus M2 plus dot, dot, dot. I can rewrite this as a summation. It's the sum of M sub i, X sub i divided by capital M, the total mass of the system. And I can, of course, do the same thing for the Y center of mass. Y center of mass is going to be the sum over i, M sub i, Y sub i divided by the total mass. All right, with that information, let's calculate it for this particular example and see if Travis was right. Here's our object. We said that the X position of the center of mass is the sum of M sub i, X sub i divided by the total mass of the system. So let's just start from right here. And let's say that each side of this square is L. So we're going to have M1, X1 plus M2, X2 plus M3, X3, plus M4, X4. We have four particles and we're going to divide by the total mass. And let's just for kicks, let's start right here. Okay, what is the X position of that particle? Travis, what do you think? What's the X position of this particle down here? >> (student speaking) I don't think I understand the X position portion of it? >> Yeah, what would be the X location of this particle? It would be this position right here which is if this is the origin -- that would be negative L over 2, okay. So negative L over 2 for that first one and the first one we also said is just mass M so we'll take out that subscript. This is number one. This is number two. This is number three and this is number four. M2 which is also there's the mass M, that also has an X position of negative L over 2. M3 has an X position of positive L over two. X4 is also positive L over 2. And we're going to divide this thing by the total mass of the system which is just four m. And now you see what happens, right? We have a negative there that cancels with the positive there. We have a negative there that cancels with the positive there. And we get Xcm is equal to zero. So on our coordinate system, the X position of the center of mass is equal to zero. And by symmetry, you're going to get Ycm also equal to zero. Okay, so we were right. The center of mass is right in the middle of the system. And it is centered on our coordinate system that we drew. So back to our complicated blob. We have the rock, something like that. And we need to figure out how to find the center of mass of that rock, okay. How do you do it? Well, what we said last time was if we have discrete particles, we can write X center of mass is the sum over i M sub i, X sub i divided by the total mass. But we don't have individual particles anymore. We have a continuous distribution of matter. And so we need to change this thing into an integral. How do you do that? M becomes dm. The summation becomes an integral. R is what we call X. The bottom is still total mass. So let's call this one -- oops. Let's write it now with an integral. It's Xdm divided by the total mass. X is the X position of that little mass element dm. Y is going to look like this. Okay and if you have three dimensions, you can write Z center of mass as well. So if you know what the functional form is of your blob then you can plug it into the integral and you can calculate it. But let's say you don't know what it is. Let's say I give you a big rock and I ask you, "Where is the center of mass?" How do you find it? Let's say I hand you a rock. And I want you to tell me where the center of mass is of that rock. How would you do it? I'm asking you guys. What do you think? How would you find the center of mass of this rock if I hand it to you? Somebody hand the mic to Nasim. Nasim, what do you think? How would I find the center of mass of this rock if I handed you this rock? >> (student speaking) I'm not totally sure but -- maybe try like balancing it in some way? >> Okay. >> (student speaking) On something? >> Excellent. So why don't we take that rock and let's put it on the table? And if I put it on the table and I can get it to balance then I can draw a line straight down through that intersection point between the rock and the table. And the center of mass has to lie along that line. And now, let's do it again. We'll take the rock and we'll rotate it until it's standing up on one edge. Okay and so it's going to look something like this. I had taken my rock and I had rotated it up and now there is a new line that goes straight through that intersection point. And I can redraw my old line on this new picture. And wherever those two lines meet, that is the center of mass. That's how you figure out the center of mass of some complicated object like a rock, okay. And so when you go down to the harbor down in San Diego and you see people balancing those rocks on top of each other. The way you balance those rocks on top of each other is you make sure that that point is directly underneath the center of mass. And if it's not, then it's going to tilt one way and if it's off the other way, it'll tilt the other way. And so you stand there and you very carefully hold it and adjust it. Tilt it left and right until you get that center of mass directly above that contact point and then it will balance, okay. And it's hard to do especially if you have a big, complicated object like this and a little tiny point on the bottom. It's tough to get that point just right but there's people that do it all the time and they're very skilled at it. Question, can you hand him the mike? What's your name? >> (student speaking) Eric. >> Eric, is there a -- >> Be careful of the mic. Hold on a second. >> (student speaking) Okay. >> Yeah, Eric. You got a question? >> (student speaking) Yes. So the balance point is also the center of mass? Is that basically what you're saying? >> What I'm saying is the balance point, this point, is directly beneath the center of mass of the object. >> Oh, okay. >> All right. >> That makes sense. >> So wherever that contact point is, it's directly beneath the center of mass. So we did it twice. We put it on one side and then we put it on another. And we figured out where they intersect and that's where the center of mass is. >> (student speaking) Okay. >> All right? And now you kind of can see what's going to happen, right? If the rock tilts slightly to the right, gravity acting on that center of mass is going to torque the object and cause it to fall to the right. If it's slightly to the left then it's going to torque it and cause it to fall to the left. Yeah, so this is a great exercise of art, and physics, and balancing. It's really very tricky to do but pretty remarkable when you see it in action.