Enough of blobs, let's get rigid. All right, so now we're going to deal with everything on, things are going to freeze. Instead of just general continuum, but I think this is a good classic thing. Earlier stuff, definitely things that I will put on the exam. Something like that's going to be on there guaranteed. But now we're going to freeze it. How do we do that? How does that change my definitions? How does it change momentum, energy, all this kind of stuff and we want to look at this. So now angular momentum about a point O, this is an inertial origin. I've picked that already. I can write that as my position vector or cross or with dots. That goes to each DM. I didn't draw the the DMs here anymore. But this is now a rigid boby. So if it's rigid, every point in the body has the same angular velocity. If I'm spinning, then the speed of my right hand is different than the speed of my left hand, the velocity, right? because one's moving forward, that's a different vector than a velocity vector moving backwards relative to me, opposite for you guys, all right. But the angular velocity, if we're spinning, everybody needs to be rotating at one degree per second. Otherwise if the panel number two is rotating at two degrees per second, panel number one is only rotating at one degree per second, those panels are going to catch up at some point, right? It's deforming, that's not a rigid body. So as before when we argued, RC was common with the body integrals here we would have tricks where we can say omega is constant on a rigid body or blog the jello. That's not true because everybody could be spinning in different ways right in that case. So that's one thing. Another one, this is the basic definition. You've seen now how we break this up. We write r as big R as Rc plus little r. And then same thing here, with thoughts, do the cross products on your own. Make sure you can derive this. And then you come there. There will be terms that vanish, but again the angular momentum about point O can be written as, this is the angular momentum of the center of mass plus the angular momentum about the center of mass. Kind of what we did with energy as well. So I'm leaving this up to you to practice and go through and make sure you can go from here to here. Now this part, that's your Orbits class. You've all taken some type of Orbits class probably. And you hang momentum on orbit and how it's conserved and from that you get all of the different chronic intersections and trajectories. And that comes from here. That's cool, but we don't care about that part. This class is all about attitude of spacecraft, right? The translational side, as you see through the super particle theorem, you can really handle separately and track what happens to the center of mass of a spacecraft. I care about what happens about the center of mass in this class. So I have wheels, I have gyroscopics, all the stuff, that's what I care about. That's what the attitude dynamics is. So instead of HO, I'm just going to keep this part which I'm going to write as HC. That's the angular momentum about the center of mass and that will simply be r cross r dots dm. Same definitions we had before right? So that's good. So now if this is rigid. People, if this is rigid, what happens to this derivative? >> r dot, it goes to zero doesn't it? >> Okay, who agrees with that? Who disagrees with that? And the rest of you are asleep, okay, or just confused, that's fair. I'm also confused sometimes. So, you agree that it would go to zero? Sorry, what was your name again? >> Ansel. >> Ansel, thank you. Why do you think it would go to zero? >> I thought that's how a rigid body was defined, that the points don't move relative to the center of mass. >> What does this dot mean? >> [INAUDIBLE] >> So this is your body. And to track this point, this little knob at the end of the pen, right? It's inertial, I mean it's an inertially fixed spin axis, it's a rigid body. Is the velocity as seen by you, an inertial observer, zero? Right, the position vector would be center of mass to this little knob, right? And that vector is clearly rotating as seen by you guys. So as seen by what frame would this time derivative go to zero if it is rigid? >> Body frame. >> The body frame. That's the key. So I'm glad you guys mentioned inertia because this an easy stumbling block. Everybody starts out with, it's rigid. All the dots go to zero. No. This is the inertial derivative of the velocity. If it's rigid, it just means as I'm rotating, my right hand is always to my right. It's not moving up, down, otherwise I'm not being a rigid body. So only as seen by the body are these time derivatives zero for a rigid body. That's the key thing that everybody always misses otherwise. So good, so now if we do that, we know how to do this transport theorem, right? Kinematics comes back and we take the body frame derivative, why? Because we know it's going to go to zero, we love zero. And what you're left with is omega cross r. Okay, good. We do that so r dot is nothing but omega crossed r and I can plug that in here. So I get an r cross omega cross r. This form is not very convenient. I said omega is constant as seen by the body itself. If it's rigid, every point in that body needs to be tumbling at the same rate or it's deforming. But in a cross product like this, I can't just pull the middle term out, all right. So I want to make it either at the left hand side, or the right hand side. Then I can take it outside of the body integral, and that's what I'm doing here. On A cross B is a same thing as minus B cross A. So I flipped it, and then I have r cross r cross omega with the minus sign. And in matrix form, this r cross r cross is nothing but r tilde r tilde. Again, that's where the minus sign comes from. And then now I can take the omega outside of that interval because it is rigid, right, that is the key assumption. Anybody recognize what this term is going to become? Angular momentum turns to be something times rates. >> Inertia. >> Inertia, yeah, moment of inertia, exactly. So these can become our inertia tensor that you've seen from your statics and mechanics classes. So we're just going to derive this now.
