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- [Instructor] Just for kicks, let's imagine someone spinning a flaming tennis ball attached to some type of a string or chain that they're spinning it above their head like this and let's say they're spinning it at a constant speed. We've already described situations like this, maybe not with as much drama as this one but we can visualize the velocity vectors at different points for the ball. So, at this point, let's say the velocity vector will look like this, the linear velocity vector just to be super clear. So, the linear velocity vector might look something like that and it's going to have magnitude V, the magnitude of the velocity vector you could also view as its linear speed. Now, a few moments later, what is the ball going to be doing? Well, a few moments later the ball might be let's say right over here, we don't wanna lose the drama, it's still flaming we're assuming, it's still attached to our chain right over here but what would its velocity vector be? Well, we're assuming it has a constant speed, a constant linear speed, so the magnitude is going to be the same but now the direction is going to be tangential to the circular path at that point. So, our direction has changed. Now, one way to think about this change in direction of velocity, it's a little counterintuitive at first because when we first think about acceleration, we tend to think in terms of changing the magnitude of velocity but keeping the magnitude the same, but changing the direction still involves an acceleration and at first, it's a little counterintuitive the direction of that acceleration but if I were to take this second velocity vector and if I were to shift it over here, and if I were to start it at the exact same point, it would look something like this, actually, let me do it in a slightly different color so it's a little bit more visible, so it would look something like this. This and this, they have the same length and they're parallel, so they are the same vector and so, in some amount of time, if you wanna go from this velocity vector and actually this should be a little bit longer, this should look like this, it should have the same magnitude as this one, so it should look like this. So, if in some amount of time, this and this should have the same magnitude. If in some amount of time you go from this velocity vector to this velocity vector, your net change in velocity is going radially inward. This right over here is your net change in velocity and so, in other videos we talk about this notion of centripetal acceleration. In order to keep something going in this uniform circular motion, in order to keep changing the direction of our velocity vector, you are accelerating it radially inward, centripetal acceleration, inward acceleration and so, at all points in time, you have an inward acceleration which we denote, the magnitude, we usually say is A with a C subscript for centripetal. Sometimes you'll see an A with an R subscript for radial but in this context we will use centripetal. Now, one question that you might have been wondering this whole time that we talked about centripetal acceleration is Newton's First Law might be nagging. Newton's First Law tells us that the velocity of an object both it's magnitude and its direction will not change unless there's some net force acting on the object and we clearly see here that the direction of our velocity vector is changing, so Newton's First Law tells us that there must be some net force acting on it and that net force is going to be acting in the same direction as our acceleration and so, what we're gonna do here is introduce an idea of centripetal force. So, centripetal force, if it's accelerating the object inwards in the inward direction, so we have a centripetal force that is causing our centripetal acceleration. F sub C right over here, you can view that as the magnitude of our centripetal force and the way that they would be connected comes straight out for Newton's Second Law. This isn't some type of new, different type of force, this is the same type of forces that we talk about throughout physics. We know that the magnitude of our centripetal force is going to be equal to the mass of our object times the magnitude of our centripetal acceleration. If you want, you could put vectors on top of this. You could say something like this but we know the direction of the centripetal force and the centripetal acceleration, it is inward. Now, what inward means, the exact arrow's going to be different at different points but for any position for the ball we know at least conceptually what inward is going to be. So, this is just to appreciate the idea. Centripetal acceleration in classical mechanics isn't just going to show up out of nowhere. Newton's First Law tells us that if something is being accelerated, there must be a net force acting on it and if it's being accelerated inward in the centripetal direction I guess you could say, then the force must also be acting inward and they would just be related by F equals MA which we learn from Newton's Second Law. And to appreciate the intuition for this, just remember the last time that you were spinning or rotating a flaming tennis ball attached to a chain above your head. In order to do that, in order to keep the ball spinning and not just going and veering off in a straight line, you can keep pulling inward on your chain so that the flaming tennis ball doesn't go hit a wall and set things onto fire. And so, what you are providing is that centripetal force to keep that flaming tennis ball in its uniform circular motion.
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