Continuous system, basically jello, blobs of jello, right? This is kind of what we're starting out with. So we have a dynamical system, that's this and our every self-respecting dynamics class has to have these amorphous flying potatoes right. It's just a general thing and we want to now compute certain properties of it. Newton's Law is simply F = MA, written in this form. You can see. And now, because I have a continuum, it's kind of hard to do it for the full thing, a blob of Jello, right? Different things will have different forces acting on it. So what we want to do is track infinitesimal little elements here, and that's my little box that's DM infinitesimal object. Just like classic calculus that you've seen many many times before hopefully. So df is the force that's acting on the differential element, dm. R is my inertial position vector, so big R is the inertial position vector. Little r is my position vector relative to the system's center of mass. So whatever this bulb is you have an infinity of little DM's that all add up to this thing. They have in common at this instant that's where the center of mass is so that's a classic location. We'll be writing all our equations about that point so that's it. That's basically F = MA times that. So but for F = MA it was important that we have an inertia derivative so these dots are derivatives as seen by the inertial frame, and you have to write your position relative to the inertial origin. That's why you wouldn't do little r double dot, you have to do big R double dot, relative to the inertial frame then dF. Now these forces, we can categorize them generally. There's external stuff and internal stuff. Who can give me an example of an external force that might be acting on a spacecraft? >> Drag. >> Drag, right, atmospheric drag is one of them definitely. Gravity could be an external force acting on it. You could have solar radiation pressure. You may have your thrusters. This is a man made object, maybe you have some way to control it. Now you have to do some stuff. That's all the external things, so what are examples of internal forces acting on this blob of jello? >> Torquers? >> Actually, good question. So magnet torquers are actually objects that push off the external magnetic field. I would see them as external [INAUDIBLE] influences acting on these elements. The element might be inside the craft, but it's still pushing off an external field. So what would be an internal force then that you might have? >> Reactions. >> Reactions is a good example. because you have these fly disks that we put in, sometimes called flywheels or reaction wheels or control movement gyros that are all spinning objects inside our spacecraft, and then they have motors attached that will spin them up or spin them down. That motor is taking the wheel and it's pushing off the space craft. So as a dynamical system it basically means some part of jello is pushing off the other part of the jello and that's what cause the spinning. This would be an internal force for the system all right? If you do a free body diagram of just the reaction wheel then maybe we can treat it as an external one on that system. But here we have space craft jello could be fuel and all this stuff. Fuel's another one. Fuel slash backing back and forth. Those are all internal forces. And that will be key because we'll be talking about internal forces, external forces, and how do things impact momentum and energy. And there are some good high level things that are really useful as a spacecraft analysts. So we're going to break it off in two ways. Now if I need a total force acting on jello, I have it gazillion dm's, I have to integrate over them. So if you look at chapter 2 we do a summation because we have a gazillion of them. And if you have a continuum you simply replace the summation with an integral sign. That's really, if you go back and look at calculus and how you did integrals, you kind of broke it up into little bitty chunks and then made those chunks infinitesimally small. It's basically this big summation. So if you're summing up over this, the claim here is the total force acting on this blob is the sum of all the forces acting on the individual DMs. Which is simply going to be the sum of the external forces. What happens to the internal forces? >> It blots it, it cancels out. >> Yeah, which law is that? >> Newton's third law. >> Yeah, it's one of Newton's laws, right? So if you take a wall, and push against the wall, the wall is actually going to push back with you exactly the same amount but in opposite directions. So as I'm torquing on a spacecraft, applying forces to the wheel, you're getting equal and opposite forces applied to the spacecraft. When you sum them up all those things have to mutually cancel. So internal forces never contribute to a net force on the system, only the external ones do that. So I'm sure you've seen that before. Now some basic properties of jello is any continuum, any dynamical system is whatever the mass distribution is, it will have a center of mass location. And there's many ways to define it. I'm going to show you it's at least two here. One of them is the center of mass is essentially your mass averaged location. So Rc is the center of mass of this blob. If you look at R times dm. You really, you think of it you got one kilogram, half a meter out. Two kilograms, a full meter out. And five kilograms, a half a meter to the left, right? You would add them up with a distances divide by a total mass. That's going to be the center of mass location, if you have this mass distribution on a system. If it's continuum, instead of summing you do integrals of mass, you know, the mass of the object times its location. And then we divide by the total mass to get Rc. So that's a classic definition of center of mass, this works for any continuum dynamical system. Now we can break this up because these are vectors, the inertial position vector. So we can say R here has to be RC pass the r. Again, this is done in a purely vectorial way. I'm not saying r has to be expressed in the body frame or in the inertial frame. That's a book keeping element we keep to later. Right now we're just adding vectors. Vectors plus vectors and doing with that. So if you do this you can see this will actually break up and let's do that one quickly here. We had that. Rc + r. Dm that's the sum of two vectors. So I'm going to do this, and say okay. That's going to be Rc dm + rdm. And my little subscript b that notation is actually a triple integral. I have to integrate over the entire body. But instead of writing triple integrals with all the limits, this is a short kind. I just do one integral with b that mean it understood to be integrating over an entire, whenever this body is, all right? So we can break these things up. That's nice. That should be a vector. Then there's some simplifications that are going to happen. We knew earlier, by definition, this was M times Rc right? This term can somebody say how this is going to be refined? All right? Kevin. >> You'll get RC out of the interval >> Okay. >> We get big M times R. >> Yeah this is nothing but the sum of all the mass elements which is the total mass. You can get there. So now Kevin can you explain to me why it's okay to take Rc varies with time and everything. Why is it okay to take it outside the integral? >> because we're only intergrating the phase. >> Right. >> We're integrating over the body. So when you see these B integrals, think of that, over this body, let's say my center of mass was right here by the microphone just easy right? Stop I'm looking at this and going at this instant, whatever the mass distribution is, my left hand is part of the system the center of mass for the left hand is still down here, right, for the system. That Rc location for the right hand is also going to be here. Every element of my body, because it's a continuum, it shares the same center of mass location. So as far as the body integral goes, it's just a constant. Just like integral of 3 times x. You pick the 3 outside the integral and just do integral of x right? And continue, that's how we can treat it. So this is just a spatial integral. Yes rt varies with time but we're doing a spatial integration over the body volume right? And that location is a fixed it doesn't change depending on if you're looking at the nose of the spacecraft or the tail of the spacecraft. It's one system and that system has one center of mass location. So yes that's the reason, and we'll use this trick several times today. So if it didn't quite make sense now hopefully it will sink in more and more. So you can do all of this. That gives me MRc Plus little rdm = MRc. These cancel and you will have little r times dm = 0. This is the other way. If you do your classic sophomore level center of mass stuff, you have 5 kilograms here, 2 kilograms there, you add them up. If all your positions are taken relative to the center of mass. All these mass times positions have to sum up to be 0. That means you've balanced it perfectly. And that's what it looks like for generally speaking. We have the little r dm's body integrals have to be equal to 0. That's just a complete 3D version that works for any jello. So good. So that, from here, a few steps you can prove this. These are the kind of things I would definitely expect you to be able to do easily and quickly in an exam. So if you haven't done much of this, practice. Now, what we're going to do next is we're going to differentiate this. So we're going to take that definition that we just had, the center of mass location, all this is good, and I'm going to take two inertial time derivatives. So that means on the left hand side, mass is constant. I'm not losing or gaining mass in this dynamical system. And RC, I just have to put two dots over it and I'm done. This is kind of like homework 3.6 that you guys did that had inertial derivatives and stuff. In vectorial form this is dead simple, two dots and you're done. The right hand side dm I guess doesn't change, we're not losing or gaining mass with the system. This is a body integral, as Kevin was saying. And so the only thing that varies is r. So we get big R and it has two dots over it. But then you go back and look at Newton's equation that mass times inertial acceleration is the force acting on that one. And so this can be replaced with also the force. The integral over the DF force and we show that the integral of the DF force is nothing but the net external force. You can ignore all the internal forces. So you end up with an equation that should look pretty darn familiar. Mass acceleration equal to force. But you've used it primarily I'm assuming on particles. This is the particle, here's the mass, here's the force this is where it's going to go. This actually also works on a continuum. Blobs in space. Anybody seen the Muppet Show? There was pigs in space, you know, a lot of jiggling going on. That's that kind of stuff. This works for anything, even muppets. Okay, so this is the key thing. We call this the superparticle theorem. What this basically says, no matter what the dynamical system is, this complete closed system, you can treat the center of mass of that system will act just like a particle. If I know there is two forces acting on it from a megatometer and maybe from the atmospheric drag force that Daniel was talking about, that's it. Then those are the only two forces I have to consider acting on the center of mass and I will predict the translational motion perfectly. Now that blob may pull apart. It may do all kinds of weird stuff. So if your external force depends on the shape, then it gets more complicated. So if this blob would be pulling apart, then with gravity gradients the part's closer to the planets would have different force than the parts top. The net forces change because of the shape. That makes it more complicated. But we didn't have gravity. We have just net thrusters you've got blobs in space. Somebody fires a thruster. And this thing is going to take off and go. Just by looking at the force and knowing the mass, I can predict what's going to happen to the center of mass. I don't know what happens to the shape yet, but I do know what happens to the center of mass of that system. Which is a very powerful argument. So that's what's called the super particle theorem. Any blob in space. If you just are concerned with the center of mass, the center of mass will act like a particular. F = Ma holds but F has to be the net external force. M is the total mass and Rc double dots has to be the inertial second time derivative.