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>> Well, let's think about this for a second, all right. I've got a ball on a string, okay, and here's my string, and that ball is falling, and we want to know when it's at the bottom, what are the forces that are acting on it? Okay. So, let's pretend, for a second, that it's just hanging. If it's at rest, right, and this is the string that's hanging from the roof, if it's at rest, what does it look like? Well, it looks like exactly what we just drew. There it is. What does the free body diagram look like for that? What force should I put on there for that? What's the first one we always draw? >> Gravity. >> Gravity. Gravity is down, okay. What else? Tension, T, going up. Should tension T look like that? People like that? >> Up and down [inaudible]. >> Okay. We don't like that. You said it's got to be equal in length to mg. Why does it have to be equal in length to mg? Because the ball is at rest. If it's at rest, there's 0 acceleration, and so the forces have to exactly balance, right. Up has to be balanced with down. We can't have any net force because if we have a net force, we will have acceleration. So, this is correct. Is there any other force here in this picture? No. That's it. So, let's go back over here for a second. When I think about the ball at the lowest point, what forces are acting on the ball? Gravity. Gravity is not going to change, right, has to be exactly the same whether it's hanging at rest or it's in motion. If we're near the surface of the earth, it's always mg. What about tension T? Tension T is still going to be along the string, but how big should I make it? Should I make it equal to gravity? Let's start with that, and then let's figure out what other forces we need to add. Everybody said e, right; not everybody, but a bunch of people said e on this. So, what is this force? What is that force? Is that the force of motion? There's no force of motion. That's not a real thing, okay. There is, in fact, no force in the horizontal direction when it's at its lowest point, okay. There's no air resistance, so we're not worrying about any of that. There's no force to the right anymore. These are the only two forces. So, the only question is do we have this diagram correct, or do we have it incorrect? And to figure that out, we got to go to Newton's second law. The sum of the forces equals the mass times acceleration. If these are equal and opposite, I'm saying the acceleration is 0. Is the acceleration 0 for this ball? What do you think? Remember, this thing is swinging in a circle. So, is that ball accelerating? If it's going to change its direction, it is accelerating, and so a is not equal to 0 because we have centripetal acceleration. And so this is not quite right. I need to draw it like this. Tension T has to be bigger than mg, all right. How do we see that mathematically? It's very simple, right. We know that the sum of the forces in the radial direction have to be mv squared over r. We have T minus mg equals mv squared over r. T has to be bigger than mg to get this positive quantity over here on the right. All right. So, the correct answer on that one was, in fact, c. Anybody in here put c? No? Okay. Good. So, we learned something today.

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