 Let's continue with motion very often in working with any kind of motion you are going to deal with constant circular motion I mean many things can be modeled a circular motion. So Not really any coding again. We're going to do. I just want to show you so just familiar with things that you might see when you are going to Encounter circular motion. So just an image here. We are dealing with constant circular motion. Here's a position vector Now imagine you can see the arrows here. Imagine there is something that you're modeling that is moving on this path Over time it changes at any time t. There's a position vector that points to that That'll be a certain angle away from wherever it started. So it started on the x-axis and a certain time later There'll be an angle change There'll be an angle change. Also note though that this position vector has a magnitude b Because that's just the radius of the circle. That's easy enough to see So just if we look at this image if we just do this the rate of change of that angle with respect to time We call that omega. That's this angular velocity. How many and what is the angular change per second? And that's what we define it as lowercase Greek omega, which is this curly w now I can Do a slight bit of algebra and what I'm getting is just a differential the the the change in theta equals The angular velocity times the change in time Now I can take the indefinite integral of both sides, which will just leave leave me with theta on this side And on this side it'll leave me This omega is a constant. I bring that out. So I'm just left with omega t plus some constant But we're dealing with angles here theta doesn't angle so this constant is actually also an angle and we just call it phi And that's just the initial it turns out just to be the initial angle. So I can start along here I could call this my zero Radians line and I can start on it then phi would be zero But I might start my t equals zero there and if I want to describe that according to this xy coordinate system It has an initial velocity, which will be this angle here, which we'll have to call phi So just putting phi as zero we have the following fact that you've seen this many times circular motion As far as a vector based equation is concerned will be the cosine of something comma the sign of something If you wrote cosine of something comma sign of something that is going to give you counterclockwise rotation And we put the b in front and the b in front there and I'll show you now why that works So constant circular motion That's the rate this this position vector is going to change over time if we model the following b times the cosine of omega t And b times the sign of omega t remembering there. I showed you omega equals Theta, I should say theta equals omega t. So it was the cosine of Theta the sign of theta, but I've just shown you now that it can be rewritten as omega t Let's see if that works. So if I do have a position vector as such And I take the first derivative of that with respect to time If you take the first derivative of that that's going to be negative b Times omega times the sign of omega t And the other side b times omega times the cosine of omega t That's just the first derivative of each of those separately as I said, I'm not using code here. I'm just using these Uh pictures to show you these images how to do it now. What is the magnitude of? The v of t so that means the speed. What is that? well That would just be The square root or the square of this plus the square of that and the square root of all of that If you were to do that You'd see very quickly that all you are left with is omega b because you're going to have a sign squared of omega t Plus a cosine squared of omega t by trigonometric identities those equals one And you're going to have the b squared the square root of b squared is just going to obviously be The square root of negative b squared and the square root of b squared. That's just going to end up with b there Now if If we take the first derivative of the v of t that's going to give us acceleration Or the second derivative of the position vector Then if I had to differentiate that again This is what i'm going to be in end up doing doing and do it on a piece of paper You'll see that's easy to see I can now clearly show that if uh if in here is Is is an The original position vector b cosine omega t. There is a b. There's a cosine omega t There is a b sine omega t. There's a b sine omega t So the only thing we left with is this common Constant omega squared and the negative. So if I bring that out It would be the same as negative omega squared times the original position vector and think about it. Why does Why does if if I were to distribute this negative omega squared into this position vector I'm going to end up with us and it tells us something it tells us now omega squared is a constant And it's the negative of that constant anything squared is going to be positive But there's a negative in front of it. So that's going to be a negative And that shows you if this is the position vector in this direction outwards Then acceleration to keep constant Constant circular motion is going to point in the direct opposite Direction it's going to always be in the direct opposite direction of the position vector And that clearly comes out just from this little bit of Of explanation. So obviously that would be some form of centripetal force Let's see if we can express that centripetal force remember force equals mass times acceleration We've just shown you what acceleration is it is negative omega squared r of t. So I'm just replacing a of t with that Quickly what would be the norm or the magnitude? Of that with in other words just just the size of that force not its direction Well, I just take the norm of this side Now a constant the norm of a constant is just the absolute value of that part So it'll be the absolute value of negative m omega squared And then the magnitude of the r of t Remembering though that means that upside down a little triangle of dots say that means because but because the r of t is that The norm of r of t will be this remember it's the square root of that part squared in that part squared that means this Position vector the norm of this position vector just as we had with velocity before I'm going to have this b squared And I'm going to have the cosine squared of an angle. Oh, there's a little plus missing there They should be a plus the sine squared that equals just one the square root of b squared equals this b Just to show you that the norm the magnitude of the position vector is the radius Ends up b if we write it like this If we write it like this b times the cosine of omega t b times the sine that b is always going to refer to the radius And Because of that remember we had the norm there the absolute value of negative omega Squared it's just m times the mega squared and I've just shown you the r of t Its magnitude is just b. So the magnitude of the centripetal force This is going to be mass times Angle of velocity squared times the radius So again, not very difficult All I wanted to show you is this for you to become familiar with some of these things You're going to see it all over the show and you can always fall back just to see how this was derived and how it all fits In together good