[BLANK_AUDIO] Welcome to module 32 of two dimensional dynamics. Today we're going to derive the equations of motion for a body in 2D rigid planar rigid body motion and so you recall that we developed the angular momentum for 2D planar motion. This is the special case where P is either the mass center of the body or P has zero velocity, and we said that these were the definitions of the products of inertia and the mass moment of inertia. Now I want you to recall back we developed several modules ago, Euler's second law about the mass center. We said that the sum of the moments, about the mass center, for a body. This is a, body that's composed of an infinite number of particles, is equal to the time derivative of the angular momentum about the mass center. And so therefore, if we use that expression, with we derived earlier, we can put in the angular momentum and we get this. And so, normally let's now, look at this body from a fixed reference frame F, and normally we pick a body fixed reference frame that's welded into the body here. Little xyz, such that the moments of inertia and the products of inertia remain constant. And so they do not change with time. And by doing that. The, if we are in planar motion. This little body here with the little xyz axis is only going to rotate about the z axis. And the little xy axis become a time depend, dependent function of the big XY axis, with respect to the inertial frame. And so, and again I mention here that the little z axis and the little y axis are in the same direction, so the Z axis is not time dependent. Now we've done this differentiation several times before, at least a couple times before, I can take my big I and express it in terms of little i and j and my big J and express it in a little terms of little i and j. And then I can take the time derivative. And the result I get is, the derivative of i with respect to t, in the big F frame, is just omega j. And the derivative of j with respect to time, is just minus omega i. And so here's my results. And I'm going to use these expressions when I differentiate this equation. And so, when I differentiate it, I use my product rule, I have the product of inertia which is going to be constant because I welded my frame into the body. Omega dot times i, and then I'm also going to have omega times i dot because we know i dot is equal to omega j. And then I have I, yz, times omega dot j, plus I yz omega times the derivative of k, or excuse me, the derivative of j, which is minus omega i. And then finally we have I zz C times omega dot k and then the product rule would say I zz about C omega k dot, but since K, little k and big K are in the same direction that derivative is equal to zero and so this is the expression I come up with. And I can write it in terms of I collected the i and the j and the k components. And so I can write this now in scalar form and I can say that for my little body, the sum of the moments about the X axis through the point C is equal to. This portion of my equation, the sum of the moments through point C about the y axis is equal to this, and the sum of the moments about the Z axis is equal to the last part. Just by matching i components, matching gay, j components, matching k components. Ok, so, the moments about the x and y axis here must be balanced if we're going to maintain two dimensional planer motion only about the Z axis. We call these moments gyroscopic moments. And they result from non symmetrical distributions relative to the X and Y axis. So what happens is, we don't have symmetry and they are non zero products of inertia. To have planar motion, the sum of the forces also have to be equal to zero in the z direction so that we stay in the plane and don't come in or out of the z plane. So let's go ahead and look at an example of that. So, here's a body. Again, in 2D planar motion. It's going to, if it has perfect symmetry about the XZ plane, the products of inertia will equal zero and they'll be no reason for this thing to go out of plane. In fact, when you balanced, when wheels are balanced on automobiles you balance it so that the products' inertia vanish and so that you have pure planar motion. The other condition that we have to have is that the forces in the in the Z direction have to be equal to zero, because we don't want this wheel coming in and out of the plane in this direction. So, that's what planar motion is defined as. And so what we're left with is, if we have a balance of these moments in the X and Y axis, all we're left with for rotational motion, for two dimensional planar motion, is this third equation, the sum of the moments about C through the Z axis, is equal to I ZZ C alpha. And now this is, this, this equation is analogous to F equals MA. Instead of F now we're talking about moments, instead of mass we're talking about mass moment of inertia and instead of linear acceleration we're talking about angular acceleration. And so, with F equals MA... With F equals MA, if I have a body, let's say I have a, a small automobile. And I push it with a certain amount of force. Let's say I'm able to push it with 150 pound force. Okay, so for a small automobile, if I push that with 150 pound force, and then I compare it to pushing a large vehicle, with a lot of mass. With the, the same 150 pound force, think to yourself. Which vehicle will accelerate the fastest. And it should be obvious to you that the smaller vehicle, if you're applying the same force, will accelerate faster than the larger vehicle. So mass is sort of like a measure of resistance to linear acceleration. We can say the same thing for some of the moments equals I alpha. Mass moment of inertia is, a resistance to angular acceleration, given the same amount of moment. And so I want to do a little demonstration here for that. I am the body now. The Z axis is going to go through my head, down through the platform. I'm going to have to start out a whole lot of mass that's away from the axis. I'm going to keep my arms out. So I've got a lot of mass moment inertia. A lot of mass located far from my axis of rotation. And so then I'm going to bring my arms in. Which is going to reduce my mass moment of inertia. because I have less mass away from the axis of rotation. Which should allow me to increase my angular acceleration. And so we'll go ahead and do a demo here and so I'm going around and when I pull my arms in [LAUGH] I go faster. We'll do that one more time. Maybe not quite so fast. That's all right. All right, go ahead. There we go. And so I spin and I speed up when I pull my hands in. And now I'm dizzy. That's alright. [LAUGH] So, alright. So that's an explanation of, the sum of the moments equals I alpha, gives you a, a physical feel for that. And so, for planar 2D rigid motion. If I have my products of inertia equal to zero. And I don't have any forces in the Z direction. Then I'm going to end up with my two dimensional planar rigid body equation of the motion which are, sum of the forces of the X equals m x double dot some of the forces in the Y equals m y double dot and now I have this third equation, sum of the moments equals I alpha. So in, in addition to doing translation problems, which we did a couple modules back. I can now do problems that include both translation and rotation. And we'll start with those next time.