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In Section 8.5 we calculated the center of mass by considering objects composed of a finite number of point masses or objects that, by symmetry, could be represented by a finite number of point masses. For a solid object whose mass distribution does not allow for a simple determination of the center of mass by symmetry, the sums of Eqs. $(8.28)$ must be generalized to integrals

$$x_{\mathrm{em}}=\frac{1}{M} \int x d m \quad y_{\mathrm{em}}=\frac{1}{M} \int y d m$$

where $x$ and $y$ are the coordinates of the small piece of the object that has mass $d m$ . The integration is over the whole of the object. Consider a thin rod of length $L,$ mass $M,$ and cross-sectional area A. Let the origin of the coordinates be at the left end of the rod and the positive $x$ -axis lie along the rod. (a) If the density $\rho=M / V$ of show that the $x$ -coordinate of the center of mass of the rod is at its geometrical center. (b) If the density of the object varies linearly with $x-$ that is, $\rho=\alpha x$ , where $\alpha$ is a positive constant - calculate the $x$ -coordinate of the rod's center of mass.

(a) $L / 2$

(b) 2$L / 3$

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Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

Hope College

University of Winnipeg

{'transcript': "once again welcome to a new problem. This time we have to deal with the scent, the scent off mus. So we're given a problem and it involves the center must remember if you think about a geometrical object like a rectangle, the scent of moss in terms ofthe the X and Y co ordinates will be at its geometric center. So this is the X coordinate. This's the Y. Coordinate. Uh, this is thie. Why sent off? Muss on This is the extent of moss. So that's kind of the deal behind the center of mass if you're dealing with regular ships. But sometimes, though in general, if you have a ship that's not regular, you could still get the scent off Mars. So maybe if you have a solid, um, object like a broad like that, you could still get the center mass. But this time you're going to sum up the into girls. These are the formulas that have given for the scent of moss and y. Scent of muss equals to one of them into goal. Why D m remember M is thie, but the total total muscle, the object, the M stands for the infinite terse in war, the mass of the object. So you slice a small piece and then you call that the end. So that's kind of the information given X and what are the coordinates of the object. So think about a teen rod like this. So we have a like a super like it's It's a super small pain, you know, super small pain like this. So that's the Y axis. This is your ex arkus and then you have a super small, uh, ten road. Maybe it should start at the origin, like right? They're very keen. I'm just expanding it to make sure you could see it are the men happens to be l and then the muss is him, and it has a cross sectional area. So if you are looking at this side like that, the cross sectional area will be secular, and it's gonna be a we want to find. This is but a we want to find the ah, you know, if the density raw is given by the total mass with total volume off the object, the uniform object, we want to show that the ex sent off muss is but the geometric geometric center. So another way of saying it is, You know, if you do your computations, right, the accent of mass on the geometric center will be a coincident, and you conceive the land. His L, then Thie Geometric center has to be l off two. And we're hypothesizing that that's also going to be the center of Mars. So they want us to show that it's equivalent to L over too learning. But be, um we're told that if the density of the object no is proportional to the land so, you know, think of an arbitrary object of Lend X home As you move along, you know, this is X distance was saying that there's an hour far which is provides a constant off proportionality to x x. The density of this object is related to good. The length of the object through a constant called out for where Alfa is a positive constants are off is greater than zero. Uh, so we want to get the X coordinate X coordinate off the center of mass. So that's, uh, the information that you've given, and this is what we want. So we want to find one approved that the center of Mass is the same as the geometric center. L over, too. And also want to find out the center of mass the X coordinate off the scent of moss. If this relationship holds, no, I'm going to start the problem in the next page and will say that are given given that the density off the road rows usually massive volume. Uh, and you know, they think about a secular road like that. This is Dean, Len. This is the area. And so the volume is area times land. So we can change this formula to Mars over a l um and that's the whole thing. But if it's infinite decimal, So you split or you slice it, you slice it like this, you find out. But that mass is D E M. As opposed to are the total mass, which is him. So the total mass is M, But then if you slice it, you get an infinite testable mus Um, so I think from the testimony level, if this relationship is true, you get to see that the mass becomes raw times a l but at the infinite tests, more level, we don't have l anymore. But this distance right here of becomes, uh, this distance becomes DX so instead ofthe l actually, we should do it in there, the side so we don't confuse it with the the mass itself. So on this side, we should have said that the infinite decimal length home this infinite as more land right here that's Dean X and then the infinite personal masses to him, the whole masses capital and want to see those relationships and lets the whole, which is the capital and or capital l and then the infinite testable, which is the AM in terms of mass of the X in terms ofthe a displacement or your home or distance or length. So if this is true, if chemicals to Brody l you could see why it makes sense to say the m e cause to grow a are the l. But we're calling the L D. X So roll, eh? No, the X and the reason why we're using the exes. Because if you go back to the formulas that were given, we have accent. Why? So we want to be consistent in that Instead of using Elsa, we'LL just switch it to the ex so over d m becomes Roy the ex like that. But then roll is a movie a l Then replace that. Multiply that by JD X thes to cancel out So we get em over l the ex member. Our focus is on the infinite decimal, the him and the reason why we have to change this to the because you want to do some type of integration. If this is in terms ofthe M and this is in terms ofthe X, it's impossible to do a valid integration. So you wanna have them everything with the same variable so you can you can run the integration. So of the Seine Tomas in the X becomes given this form in the one over m into go well, ex d m. Then we started doing the replacements. We have one of them integral x d m is em over l d x so em over hell. But the X it helps us to cancel all these two. You can pull out the l, and then now we have integral of x RDX. Then we wanna violet is a constant. You do the integration. So this is, um, X squared over to but remember, uh, if you can recall our broad had a length off l If you go back, you see that this land was hell. Okay, so then way means that, you know, was starting from the origin up until a position X rather l So going back means that you have to include that Those limits of integration the limits of integration from zero up until l So those are the same ones we're goingto have right here on. Then we'LL go back. Remember these problems. You always have to think about the whole unit and the infinitely small unit. And then you replace all your valuables to be consistent as long as they have Constance so sent off muss sent off muss equals to one of our health into role Or rather, will really did the interval, so we don't have to repeat that. Uh, this one is. It's a constant so it can stay outside. You have X squared over two zero two l excluded with two zero tl. And this is decent because now, if you plug in, uh, the l we get l squared over to and then minus zero because if you plug into into exit, Just get zero. So you know, we don't have to happen, So this is zero X and so we have one of the l a. Times l squared over to one of the elves. Cancel out. So then our center of muss becomes equality. L over too, which is exactly the same as the geometric center. So the center ofthe muss and geometric No center r coincident, they happened to be coincident. And that's what the wanted just proven. That's just put a off the problem. You know, that's the first part of the problem. This is but a wanted us to Teo. See if the center of mass in the geometric center of the same we've done that in the next stage will start in. But the where they want us to compute computes the center ofthe, not computers. Calculate thie X coordinate off the center of Mass. Given that the density roll varies with axe like that, that's the information we were given but gonna use. Some of the information we've picked up are from before, which is this one. The M is ro adie X, the chemicals to Roy the ex. But then you can see what role is Raoh is Alfa X So Alfa X, the X alfa and they happened to be Constance. Okay, How fine He happens to be Constance if if the road itself if you think about the body itself And this piece is infinite Asmal the m in terms of muss if you want to get the entire mass M we have to use into girls we have to use integral So we'LL say to get them we have to do the integral of GM on DH Then it's time to go from zero t l because this point is the beginning And this is the end. Okay, so from zero up until l so em becomes number one to get X X coordinate off the center of mass. But before we go to that, we have to compute them and change it to values which are easy and consistent to deal with. So this is into go from zero to l. The M is how far x a the ex like that because that's that's what our GM is. And so we have Alfa ex Adx Alfa and a constant so you can pull them out zero x t x and then do the into Gobi off after a x the X ray of X squared over two from zero to l. The constant is outside so we could make it out to plug in those numbers into the integral get health squared over to. And that's a new relationship for the mass m. So m he calls to Alfa, eh? Health squared over two. And you thinking, you know, why is that helpful? It is helpful because if you could haul the center of mass is given by that relationship. And now we do have a value for him, which is off of a L squared over two into girl care from zero to L then x d l you know, it's like you be m someone that's like, I don't know how to grow on something but the dent that the Yemen physics is the infinite decimals. Gm So alfa, um, roll a DX So the Emmys rule, eh? Uh, DX, that's what we're gonna have here instead of this d m. We have X times roll. He Oh, I'm going to have ro adie x. Okay. If you go back, you see so a t x This one his d m. So we're doing the appropriate replacements on. Then we end up having we flip it so to Alfa a hell squared. Okay, uh, into goal from zero to l roll X, the X These two is cancel out this way and that A cancel out the is cancelled out. And then we also hard another ex. So, uh, if this is roll adx, this is roll a d x d becomes way have to We have to do more replacements. So going back, going back right here we see that No, see that we need to find a body in a different value off. There's a relationship between always that ways. That right here, right here role is a ex. You know, Roy, he's a X so we can say to over Alfa hell squared into law from zero to hell. Role is Alfa Axe, not ex Alfa X Times and other X. The X So this role just too public. But there was alpha X linearly dependent. The office cancel out. So we have two over l squared X squared integral from zero to L d X. Now we could do then to go going to do that next stage. So xn toph muss becomes he called to go back two over l squared, which is a constant. And then we do the integral X cubed off a three. Then this is from zero to l. So we have two over l squared l Cube off three zero plugged in just gives you zero, so nothing else happens there. Uh, this l squared will cancel all the elves except one. So final answer for the centre of muss. Um, for the X What we call the X coordinate off percent of mass. In this case, given that relationship becomes too well over three too well off, three. So you know real quick we've got to see that were given to formulas for the X and the wife. For the most part, we're going to use the X. I want to find the accent off months to see if it's Emma's, the geometric center, and then well, given a relationship row equals to Alfa X and want to see what's the X coordinates off the scent of musk. We use the relationships of roll, which is massive volume. We change that in terms ofthe area. And the purpose is to make sure that the GM has a specific relationship with the X in the area. And then we do the replacements. We end up having a replacement for for the GM, which you can see there's AM over L. D x who plugged that in right there? Theorems counsel, you pull out the one of a l. You do the interval. You end up getting, ah, the ex center of mass being a lover too, which is some of the geometric center, as you can see right there. And then finally, we need to find the ex According it, given that Alfa Ro is off over X. So this time we change things in terms ofthe M which is the Thomas. And so were you. Plug that in into this formula for accent off must to eliminate the M and write everything in terms ofthe area and length because that's what we're dealing with. We're dealing with the cross sectional area and the land on. Then we simplify that we get the ex sent off Mars being twelve three. Hope you enjoyed the problem, Phil. Food to silently questions or comments and have a wonderful day. Okay, Bye."}

California State Polytechnic University, Pomona