Hi and welcome to module 2 of 2 dimensional dynamics. Here are the learning outcomes for today's module. We're going to explain the concept of reference frame as applied to vector derivatives. We're going to describe the particle kinematic relationships of position, velocity, and acceleration and we're going to define Rectilinear motion. So here is a a, it could be a vynal record from my days, or a CD, and i've drawn a vector on that, that rotating disc, and the vector goes from point P to point Q, and we call it R. And since now we have motion, as opposed to the study did we on statics, I'm going to need to take time derivs. How the vectors change with time. And so here I have the derivative of that vector R with respect to time, or we can write as, as R vector dot. So first of all, as a review I'd like you to recall in your own words what we mean by a vector. So a vector, if you recall has 2 components. It has a magnitude and a direction. And so when we take the time and [?], that vector can change in both magnitude and direction. And so it depends on what reference frame you're looking at With regards to how you take the derivative. So if I have a vector of constant magnitude, so the magnitude's not changing, it's the same length in the vinyl reference frame, would the derivative of r with respect to time be equal to zero, or be equal to something other than zero. So in that case, the derivative of R with respect to time from the reference frame of the, of, of the vinyl record, or it could be the CD, is 0. Because if I shrink, let's say I shrink myself down and I stand on this disk and I look at that vector, from the, from the reference frame of the disk itself, if I look at the vector, I know that its magnitude doesn't change, because we said that was constant, but as far as I'm concerned, it's not changing in direction either, because I'm moving around with the disk, and so it looks like that vector's just standing still, and so the derivitive of R with respect to time is zero. Now let's look at it from the reference frame of the Earth. Standing and, and, and looking from the side at the disc, or the vinyl moving. In this case, is the derivative equal to zero, or is it equal to something other than zero? So in this case, you should've said that the derivative of R would respect the time Does not equal to zero, because as far as the change in magnitude is concerned yes that's equal to zero because we said it had a constant magnitude. But as far as the direction is concerned, if I'm standing here off the disk and the, and the disk is rotating around then the vector is changing direction as well. And so that time derivative cannot be equal to zero. So we find that when we're taking, vector derivatives, we need to specify the appropriate reference frame to make sure that, we get our answers correct. So let's say that for a Vector R which we describe with a unit vector's E1, E2 and E3, having components R1, R2, and R3, respectively. These could be, and we'll talk about coordinate systems later, but e1, e2 and e3 might be your typical rectangular Cartesian coordinate system of i, j, and k. So when I take the derivative, you see now each component has a magnitude, direction, magnitude, direction, magnitude and direction. And so, I've got to use the product rule here and I've got to specify which product frame I'm, I'm taking the derivative in. So let's say I'm taking the derivative from the Earths reference frame, or I'll call it The fixed frame out here. Then the overall derivative of R with respect to t with that F frame, is how R1 is changing with respect to time, how the component of magnitude, the, e 1 direction is changing. Time E1 plus that magnitude times the change of the direction itself with respect to the F frame. Similarly for R2 it's the change of magnitude for R2 times E2 plus the R2 times the change of the direction of E2. because E2 is going to change as it rotates. Same thing for, for R3. And so that's a complete [no period] Description of the derivative of r with respect to the f frame. So now let's look at the particle kinematic relationships. Let's first look at position. So for position here's my fixed frame f, might be the earth and here's a particle moving along some path p, and so the position is from 0.0 to 0.p. So I write. r is a vector from O to P. Next we want to find velocity, and so, my question to you is, how do you find velocity given position? And what you should say is you have to take the time derivative, so the velocity is the velocity of point P, now And so we've got the velocity of P is equal to the time derivative of the position, or R, O to P dot. Then we go onto acceleration, and again, how would you go from velocity to acceleration? And your answer should be, take the derivative again so we have the acceleration we'll call it a sub p is now equal to the second derivative of the position vector with respect of time. Or r o p double dot. which is the same as the derivative of the velocity or v p. dot. So my next question would be how do you go from acceleration back to velocity? And the answer should be instead of the derivative now. What we want to do is integrate. And to go from, velocity to position, again, we integrate. And so those are the particle kinematic relationships. Now let's look at a special case of particle motion, called rectilinear motion. So, we have a body, a reference frame, f, and instead of moving in, in a 3-dimensional manner This particle P moves along a straight line. So it's only got one component of motion, in what we'll call the eye direction. So it's straight line motion, there's only one component of acceleration and our mode of P is the distance X is a function of time. As far as we go in the i direction. Velocity is just the derivative of the position. In this case i doesn't change because there's only one direction of motion. So the only thing that changes is x and so we have x dot for the magnitude of the velocity and for acceleration we have x double dot for the magnitude of the acceleration and so there's no change in the direction. All of those kinematic, relationships are in the i direction. Therefore, we're able to work with scalars, and we'll use that in the next module to solve an actual problem.
