Introduction to centripetal force | AP Physics 1 | Khan Academy

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. 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. 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 want to lose the drama; it's still flaming. We're assuming it's still attached, 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 is a little bit counterintuitive at first, because when we first think about acceleration, we tend to think in terms of change in 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 counter-intuitive, 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. It would look something like this. And 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 have the same length, and they're parallel, so they are the same vector. In some amount of time, if you want to 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. In other videos, we talk about this notion of centripetal acceleration in order to keep something going in this uniform circular motion.

To keep changing the direction of our velocity vector, you are accelerating it inward, radially inward: centripetal acceleration, inward acceleration. 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 you. Newton's first law tells us that the velocity of an object, both its magnitude and its direction, will not change unless there is some net force acting on the object. 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. What we're going to do here is introduce an idea of centripetal force. The centripetal force, if it's accelerating the object inwards— I guess you say in the inward direction— so we have a centripetal force that is causing our centripetal acceleration, f sub c right here.

We can view that as the magnitude of our centripetal force, and the way that they would be connected comes straight out of 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.

You could, if you want, put vectors on top of this; you could say something like this. We know the direction of the centripetal force and the centripetal acceleration is inward. Now, what inward means, the exact arrow is 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.

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. If it's being accelerated inward in the centripetal direction, then the force must also be acting inward.

They would just be related by f equals ma, which we learn from Newton's second law. 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 have to keep pulling inward on your chain so that the flaming tennis ball doesn't go hit a wall and set things on fire.

So, what you are providing is that centripetal force to keep that flaming tennis ball in its uniform circular motion.