justin miller oxford physics here and we are going to look at some more physics stuff so what we want to discuss and understand at this point is what is known as center of mass or sometimes interchangeably center of gravity we're going to stick with center of mass right now and we can talk about the difference between the two um that it is possible all right so center of mass what is it center of mass is ultimately like an average value of the mass of an object weighted by its mass so that sounds a little confusing but ultimately it's a matter of breaking an object into individual mass elements and saying what is the position of the average of all those elements weighted by the mass of the individual elements themselves so the center of mass is a position and we can ultimately trace it to a dimensional coordinate system in which there is an x position a y position and a z position we'll mostly just stick with x and y two dimensionalize things but really it's just a matter of where our individual mass elements and what's the average value then so for a system of discrete masses we'll just say in two dimensions [Music] the center of mass is given by [Music] xcm this is going to be the x position of the center of mass is equal to one over and total times the sum from i equals one to n of m sub i x sub i so what do we have here we've got the individual mass elements and their individual x positions that we're adding them all up and then we're dividing that by the total mass and that gives us what the x position of the center of mass is position [Music] along the x-axis of the systems i don't want to say object necessarily but systems [Music] and the total is the total mass of the system m sub i and sub i is really our individual discrete mass elements at positions x sub i along the x axis i shouldn't necessarily stay along the x axis yeah x-axis maybe not the best thing to put in there but along the x-axis that's okay positions excellent we'll just leave it at that then we can also have four y its position of the center of mass along the y axis is equal to 1 over the total mass times the sum from i equals one to n of m sub i y sub i so we got the same sort of thing except we're looking at the individual mass elements and their y positions to figure out what the center of mass position for the y position is all right so how does this all play out here right well if we could look at something sort of more one dimensional and think about a couple things here so it's kind of again the average value of the position of the masses weighted by the total mass of the system or weighted by the individual masses and as it turns out that as long as we're in a uniform gravitational field the center of mass and the center of gravity are defined the same way and it's nice to think about center of gravity because we can think of the center of gravity as the point at which gravity acts on the entire object an isolated single point even though it's distributed across the total object there is a position on an object that we can balance things out so i can generally find a balancing point on an object let's do something like this and say right now it's as if gravity is pulling down on this point right here with the force mg and i'm pushing it up on that point right there with the same amount of force and thus it's in equilibrium oh yeah truthfully gravity is acting on this whole thing um pulling it downward every little mass element but it's as if again that there's one singular point that i can idealize gravity acting on and that's uh the center of gravity so again with the gravitational force being uniform for the most part we can also say that that's the center of mass and it turns out that things want to rotate about their center of gravity or their center of mass in the uniform field so it's another way of finding it i could take this and spin it oh and what does it rotate about it rotates about its center why because the center of this object is where the center of mass actually is well what about something like a hammer here hammer where's the center of mass of this is at the center and well it kind of depends the orientation i could say that i could balance it like here and there's an axis that goes through it this way that defines the center of mass but i also could balance it well if it's not here in the center it's going to be over here so i'm going to try to get this here and it's about right there so we have is the center of mass along this direction is right around here right here why because the top's much more massive than this other side over here the head of the hammer is much more massive it's steel where the rest of it's um not steel anyways or at least it's hollow if i were to spin this where does it spin relative to it spins relative to its center of mass here this is where it wants to spin around so if i take it it spins or rotates about this point here let me try to be a little careful with this where you can kind of see that right it spins about here right that kind of goes for everything so i could start taking something like this and say well what happens if i spin this what do i think the center of mass is about the plane of this object here you probably say that's probably at the center and spin it and probably see his nose kind of stays in the same place because that's right about the geometric center of this and if i could i could maybe try to balance this about the center but that's it instead i spin it this way the center of mass cuts through the middle of it the intersection between those two points is where the actual center of mass of this object is so we've got something that cuts directly through the middle this way directly through the middle this way the true center of mass is where those two axes if you will intersect but it's all amount that all about the individual mass elements that comprise this object which are a bunch of atoms right where they're located relative to one another and that kind of gives us the center of mass thing here going on all right i could also take something like this rod again this i said hey the center of mass was right here what happens if i change the mass distribution of this rod what if i say i'm going to make this side towards the top much more massive than it is right now you can take a nice little mass here and fix it on the top here help up a nice little clamp so where's the center of mass now well it's not in the center anymore if i try to hold it right here it just wants to rotate if i'll go of it because well there's going to be a non-zero torque on it that's what we're getting to but anyways the new center of mass is right about there if i were to throw this up and spin it it would rotate about this point right here instead of the center as before why i've added another mass at another location and it shifts where the center of mass of the object is all right so we can do something kind of simple here and we'll get to the torque part that's anyways we went ahead and just said for instance we've got ourselves say some m1 is equal to three kilograms and some m2 is equal to five kilograms and this is at the position we'll just say zero zero meters and this is at the position of let's say three zero meters so these are point masses that are on the x-axis and kind of put those and say oh we've got one mass right here there's our m1 and then two three meters over here we've got m2 it's much more massive if these were hinged excuse me not hinged together but connected together with something that was massless or i just wanted to say hey where's the center of mass of this system we could go ahead and start applying what we have up here so if we take this and apply this xcm is equal to 1 over m total multiplied by the sum from i equals 1 to n of m sub i x sub i that's going to give us 1 over the total mass which is m1 plus m2 multiplied by the sum of these indexes here so we've got that's going to be multiplied by well we start with i equals 1 m1 x1 plus m2 x2 so if we come back up here this is x1 this is x2 those are the x positions of those individual masses so if we go ahead and throw this all in here we've got ourselves x1 is equal to 0 x2 is equal to 3 meters so we've got one over three kilograms plus five kilograms multiplied by three kilograms times zero meters plus five kilograms times three meters so hopefully we could go ahead and get this going pretty easily here and we've got ourselves 15 kilogram meters divided by eight kilograms which gives us 8 15 7 8 and 7 8. meter is the position of the center of mass along the x-axis so if we went ahead and cut this in half and say well this is three meters right here 1.5 meters would be about there 1 7 8 almost 2 meters so we could kind of say well one one half two maybe it's a little bit more over here xcm within feasibility there but there's the center of mass of this it's not just in the middle of it because it depends on the actual masses themselves and their positions it's pretty simple pretty straightforward figuring out the center of mass for a system of point masses or discrete objects at that all right so what we want to be able to do of this is sort of utilize some of the ideas here for when we're talking about torque because we take an object we can balance it on something like a fulcrum about its center of mass so this meter stick pretty nice continuous mass distribution may not be completely uniform but it's pretty close so i'm going to put this little uh pivot point at the geometric center of this at the 50 centimeter mark i put it in here and if i can get it at center of mass it'll stay balanced notice it doesn't stay balanced right now right it tilts that way so that means that this side is a little bit um too much a little bit too heavy so just slide it over a little bit that way see if we can get that the balance out that looks pretty good yeah that was really good actually so this point right here right where i have this little clamp in a sense this fulcrum is pushing up right there with the same amount of force that gravity's pulling this entire thing downward this is why it's an important aspect center of mass center of gravity because it's we can think of it as the one single point that gravity is acting on the entire object i push up on it at that point with the same amount of gravitational force we've got balance balance but what happens if we don't have this this wedge this fulcrum at the center of mass of the object well it changes things changes things the center of mass stays right here with respect to the object if i go ahead and push this over here the center of mass doesn't change regarding this object right here the center of mass still at the 50 centimeter mark what happens if i let go of it well we could think of gravity acting on this entire entire mass of this meter stick at this point right here we got an axis of rotation and we've got a point of application of a force that point of application is right here at the center of mass and the force is the gravitational force what does that constitute well it constitutes a torque so if i let go of this what's going to happen it accelerates downward because there's a net torque on it so that's uh something that's important to be able to consider in that we can have an object exerts a torque on itself due to its location of its center of mass now i could start balancing things out further away that's fine but i could start taking some little things and some more mass over on this side and find a point that can get this to balance out so the force of gravity acting on this hook with the mass pulling it down on this side is balancing out the torque that's due to the force of gravity acting on the center of the mass of the object pointed down this way this one's producing one torque this one's producing another torque now they're different radio distances away and they're different masses but that's how torque works out right it's not just the mass it's not just the force it's where the force is being applied the radio distance away from the axis of rotation so again this is why it's important to be able to think about the center of mass of an object and be able to know what it is especially when we're talking about systems and balance where we've got equilibrium but we've also got the center of mass of the object maybe exerting a torque on on the object itself and i'm going to take it into account that's it so pretty cool little system here in this orientation some things to point out the force due to gravity is completely tangential here we got the radial vector pointing this way we've got the force due to gravity that way which is just the mass of the meter stick times g and then um the cross product between those two this gives us mgr and there's the torque that it produces and your reference frame you can say hey that would tend to produce a clockwise rotation so we could call this one a clockwise torque this one we've got this total mass here which is about 150 grams located at some distance away from the axis of rotation so we'd have another mg times a different r and that's producing a counterclockwise torque and if we went ahead and did some computations here we better find that the torque on this side is equal to the torque on this side they're producing opposite directions or they have oppositely directed torques that torque is equal to zero equilibrium equilibrium all right so that's kind of the idea with this not too bad and i'm just kind of look at that for now all right until next take care and be careful spinning hammers around