[Music] hey welcome back everybody today we're talking about centroids okay we're skipping ahead a little bit but i think this makes logical sense to cover next what is a centroid okay that's a weird word i've never heard that before i've heard of like center of gravity is that what you're talking about well a centroid can be more than just the center of gravity okay or the center of weight okay it can be the center of mass the center of area the center of volume the center of pressure okay so it's a lot more than what we think of just center of gravity okay but it is a location in space okay it is like if i had a map the centroid would be the location where if i put my way i like to think about it if i put my finger on that point i could balance that whole shape on the end of my finger right so centroid is a location therefore it has x and y coordinate points okay of course in 3d it would have a z coordinate point but it's not just any old coordinate point it's the point where i could balance the whole thing on the end of my finger okay so we put a little extra something on there just to denote that and that are these little bars above these and cleverly you're never going to guess what we call these this is called x bar and y bar okay all right and so this denotes the x coordinate of that point where the center of gravity the center of weight the center mass is this is the y coordinate where that point is of course like i said in 3d we'll also have a z bar okay so we can have any of those now in this chapter there's a couple things number one in the very back of your book okay i've got a book right here in the very back of your book there's a table okay right here you can see it here okay and that table is called the geometric properties of line and area elements and it basically has the equations the formulas or the location of the centroids for a lot of common shapes common shapes like circles rectangles triangles parabolas all kinds of different shapes okay so this is a table that i typically will put in the on the back of the test so that the students can refer to this but we'll also be deriving where these come from in this chapter so in this chapter there is there is a new equation right so far all of the equations i understood at one time said i'm trying to memorize the equations in statics how are you going to memorize them when we just make them all up every time okay but there is an equation for this chapter it may be our first equation of the semester and it goes like this x bar is equal to the sum of the x sub i times a sub i divided by the sum of a sub i okay the little sub i just means how many elements do you have typically we'll have some kind of crazy shape we may divide it up into a whole bunch of different parts and if i have seven different parts then that would be i'd have an a1 an a2 an a3 a4 a5 right and the sum means add all those up and we will certainly do this in some example problems where this makes absolute sense to you okay and of course y bar is going to be equal to the sum of y sub i a sub i divided by the sum of a sub i okay now the nice thing about this equation is is that we can substitute lots of things into it okay so the same equation could be the sum of x sub i v sub i divided by the sum of v sub i okay and same down here over the sum of the v sub i so what does that mean now instead of being area right the center of area maybe you had a 2d area now it's the center of volume okay what else could you substitute there well you could do x sub i m sub i the center of mass okay you could put a p in there for a pressure um you could put a w for weight right you can do all kinds of substitutions in this equation and it works exactly the same so this is kind of the central equation for this chapter and the other one is of course z bar is the sum of the z sub i a sub i divided by sum of a sub i okay so what is a centroid how where are these where these equations come from they're actually little moment equations so let's say that i have my book here okay if i take my book and i try and balance it on my hand here right oh okay when i balance my book on my hand there is a force on one half of the book okay let's say it's this right there okay so there's a force on this half of the book that's trying to rotate my hand that way and that force comes from the weight of the book it's trying to rotate it that way okay on the other side of my hand all right is another force over here all right my little arrows stand up arrows right it's a is a force on this side of my hand that's trying to rotate it in the opposite direction so when my hand gets right in the middle these two forces here are balanced that's the moment that's the centroid that allows us to tell where the centroid is is when we balance that moment okay and that's where these equations here come from okay so again if i if i want to find the centroid of this if i know where it is i should be able to put my finger at that location and then that thing balances on my finger ah i'm going to use two fingers right so now my finger is at the centroid of that book okay and it balances so let's talk about some other shapes okay what if we talk about my little fisherman here can you see my little fisherman i'm kind of getting up close to you okay i'm gonna put him on my finger okay he's made out of little nails he's got a lead fish over here you don't want to eat that fish because he's full of lead he might get lit poison okay so what's happening here okay the little guy has a mass over here which is trying to rotate him that way right the fish made a lid certainly has a mass it's trying to rotate him the other way but when both of those forces balance he'll sit there on his heels so where is the center where is the centroid of this system the centroid of the system is right here at at this at his feet okay so the center of gravity will always be right there in the middle when it comes to try and balance something okay so what happens if i move the fish that way i scoot the fish away from the little guy okay i'm gonna bend this fishing pole and now the fish is way over there okay so when i let him go what's going to happen okay remember the weight hasn't changed but the weight has gotten farther away so what has changed the moment has increased right that turning force so i'm going to let him go now watch what happens whoa he falls off my hand okay look what happens now so that system wants to ride itself it wants to be in equilibrium and so that moment was too big it was too far away so it actually rotated over here so now now the force is on this side the force times this distance is the same as this guy's center of mass times this distance right and the guy balances out but still the centroid is at his feet all right let me show you one more example to kind of prove this all right so what's the weirdest shape you can think of did you think of this shape the great state of texas the greatest state okay the great state of texas it's a pretty weird shape okay i put a dot right here where i think the centroid of this shape is i may have run this experience experiment before and i kind of know where it is but i want to show you a trick here okay i want to suspend i have some screws in my map here around all the way around and i can suspend this map by one of those screws and if i suspend it by one of those screws what happens to the centroid right it should go directly below it shouldn't it let's try and see what happens okay so here's what i have i've got a little lead weight here on a on a string and i have a little loopity-do in my string let's hook it to east texas over here gotta love east texas right and let's put our string in front of us and let's just hang this and see where it goes i can't see it because i'm behind the map is it through the dot did it go right through the dot okay i think it did okay what are you gonna try let's try let's try down here on the coast okay did you know that the ocean touches texas what that's true okay hanging straight down where did it go did it go through the dot again i hope i didn't prove myself wrong okay it's a little off to the side well it's also touching my map oh it's pretty good isn't it okay let's try it again let's go oh let's go over to big ben okay here's big bend oh my map is upside down in my hand so i only have one hand okay here's big bend over here oh come on luke cooperate with me okay it's not cooperating with me did you see it okay i got it i got it i got it there it goes okay so here's big bend okay now texas is all upside down here where does it go this time right through there doesn't it so no matter where you go the centroid is always directly below you and for those of you out there wondering just where the heck is the centroid of texas it's at mercury texas a little bitty town right north of brady texas there you go okay so this weird shape so what i should be able to do is put this on the end of my thumb there right right on that dot and i should be able to balance that shape on the centroid right so my thumb is right on that dot and look perfectly balanced so that's the location where that centroid is that's that x y x bar and y bar to balance that shape that's so cool all right so the last thing i want to tell you is that those equations in the back of the book we can we'll be able to derive those two this is the first chapter in statics where we're going to use the c word okay and that is this equation here x bar is equal to the integral of x d a over the integral of d a of course integral in the calculus language just means sum right so it's the same as this guy over here okay this one is just using little elements to add them up but this one over here when we have weird shapes that we don't uh we don't have this not the table in the back of our book we'll have to use the calculus method right to come up and solve for what is x bar what is that centroid and of course y bar is the integral of y d a over the integral of d a now this is pretty low level calculus that's the c word in case you're wondering pretty low level calculus stuff and i will show it to you in the next video where it makes absolute perfect sense to you so there's your introduction to what is a centroid get your map out and see if you can balance colorado not that fun it's just a rectangle okay so anyway all right i'll see you back here next time