 When you take an object and you look at what happens when you push on different parts of it, it turns out that's actually a very complicated process because if I push on one part of an object, the only reason the other parts move is that they're connected to it by 10 to the 23 tiny interactions between all the different parts. So this is actually a very complicated thing, but it looks kind of simple when you think of it as one big object. And the simplest thing about this object is the motion of its center of mass. If I look at a hammer, I know that this heavy end kind of counts more, even though a lot of the hammers over here, the center of mass is much closer to the big heavy end up here. And if I had something like a tube, then actually the center of mass is in the middle. There's no actual tube there at all, that's the hole, but the center is the exact center in there. And the center of mass is a very important thing when you're dealing with external forces because the center of mass moves very simply when you look at how it responds to external forces. Supposing I have more than one mass, supposing I have lots of masses. And these might not be completely independent masses, although they could be. These could be molecules of air, or they might be molecules in a hammer. So they might be strongly interacting or not. They might be part of one solid body and we've just broken up in little pieces. Supposing we have lots of different masses, we kind of know that the center of mass has to be closer to the heavier ones. So I'm going to label the position of each of these particles and I need to pick some origin and so I'll write the position of each of these particles as a vector in a particular direction and distance from that origin. And furthermore each particle is going to have a particular mass. So my center of mass is just going to be at a position. So it's going to be described by a position vector and it's going to be a sort of average position vector that's weighted by the mass of each particle. So in summation notation that would be, so the sum of all the masses added up, that's the total mass, and this is the mass weighted positions. So let's look at what this thing is. First of all it has units of position. So this is a position vector and this is a mass. So the top line is a mass times a position and the bottom line is a mass which leaves us with a position. We know that if we have just one particle so if we can cross out all these extra terms from the sum it's just the mass times the position divided by the mass so it's just an all position. So if I have a single particle its center of mass is where that particle is. So for example if I have two particles of equal mass then my center of mass vector is exactly halfway between the two masses. Now it's very easy to show that the motion of the center of mass of the system is very strongly related to the total momentum. So the total momentum of the system is just the sum of the momentum of each part of the system and non-relativistically that's just the mass times the velocity of each part. And the velocity remember is just the change of position divided by a change in time. And of course we can factorize out the time there. And this part here looks like a change in that part there. So if we take that which is the total mass and multiply that to both sides then what we find is that the total mass times the change in the center of mass is equal to this thing where we've got changes in positions. And that means that we can rewrite this equation here. The total momentum is just the total mass times the velocity of the center of mass. And that's interesting because all the forces between particles within our system so all the interactions between our pairs here we already know do not affect our total momentum at all. We already showed that because we get a momentum change to one particle and a momentum change to the other particle that's equal and opposite and so they all cancel. And so the only thing that causes the total momentum to change is external forces. So a force that has a pair that isn't considered in our system so there might be some other thing somewhere else that has an equal and opposite force indeed there must be, but if we're not considering it in our system then this is called an external force. And so those are the only things that can possibly change our total momentum. So what that tells us is that if we have a system no matter how complicated or unbalanced it is and we apply external forces to it then the motion of the center of mass goes just as though all those forces were located at a single point mass located at center of mass. So if I have this hammer and I throw it it's swirling around in a complicated fashion because gravity is acting on it but it's acting on all these different parts and they're interacting with each other. But the only external force is gravity and so the center of mass of this hammer is actually executing the lovely beautiful simple parabola that a ball bearing would or a point mass would. And so because the momentum is related to the center of mass motion the center of mass moves in a very simple way.