Physics Problems: Circular Motion

circuluar motion

02/28/2011

~ Gorilla

Very good. Explaination made understanding very easy.

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You can try to swing a full bucket of water over your head. You'll discover that you better swing it fast enough. If you swing too slowly when it's at the top, all the water dumps out on your head. It's real embarrassing. So what's the critical speed? It's an example of a circular motion problem. And we can apply the ideas of uniform circular motion, and Newton's second law, in order to calculate the required speed. And in the process, we'll learn a lot about the forces involved.
So let me imagine a bucket, which is traveling in a circular path like this. And here's the top. So let me draw a picture of the bucket. And the bucket, of course, is traveling around. So at various moments in time, it's in various places. And then there's something in the bucket. It could be water. It could be a lump of coal. It could be anything. What I'm asking is how fast do you have to be spinning this thing around so that this lump of coal does not fall down, or the water doesn't fall down on your head? It's a Newton's second law question. Let's use a force diagram. So let me pick this spot, because this is the one to be worried about, when the water is right above your head. And let us draw the force diagram right there, and see what's going on.
Here's the object. It might be a chunk of water. And no matter what is going on, there is always going to be mg, the weight, straight down. So is that the only physical force acting on the water? Well stop and think about it. You're swinging something over your head. It's in a bucket. Here's the water. Here's the bottom of the bucket. It's a contact force. The bucket pushes the water. It's not able to suck the water up, but it can keep the water down. So it's possible that there could be another force, a normal force or contact force, which is straight down in the same direction as the weight. Are there any other forces? Remember this thing is traveling around in a circle and it's at the top. There are no other forces in the vertical direction, none. They are the only two possible forces there are, so the sum of all forces in the vertical direction. Let's pick a coordinate system. Might as well pick up to be plus. It's completely up to me. So this is negative and this is negative. They're both in the negative direction. So negative N minus mg, that's the physical forces. And that's supposed to equal to , the centripetal acceleration times mass. R here is the radius of the motion of the bucket. V is the instantaneous velocity right up here at the top. And m is the mass of the little object. How about the sign, which way at this moment in time is the water accelerating? Remember it's traveling in a circle. And whenever you travel in a circle, you are always, at every moment on your path, accelerating toward the center. So there's a minus sign there. Everything in this equation has a minus sign, so I can cancel it through and solve for the normal force. Normal force is equal to, if I have plus signs everywhere, and then I bring this over to the right-hand side, this one gets a minus sign. So .
Now the only thing that I know about the normal force, remember when you have something touching something else solid, the normal force always adjusts itself. It picks the value that it needs to, to prevent the object from flying through. It can never be negative, so N always has to be greater than or equal to zero. If you go faster and faster, the normal force will get bigger and bigger, because this term is getting more and more positive, which makes sense. As you swing this thing faster and faster, the bucket has to push it down harder and harder to force it into a circle. If the bucket didn't push it down, nothing would be pushing it down. Gravity is not enough if you're moving it fast, and it would go in a straight line. The water would go flying out horizontally or follow a parabolic arc. So you can understand this equation. And in particular, since N has to be greater than zero, that tells you that v has to be greater than--let's see. I bring mg over here and notice that the mass cancels, . So there is my minimum speed. Take the acceleration of gravity, multiply it by the radius of the circle, take the square root, and you better swing it faster than that.
What if you don't? What if your velocity is really small? Well if your velocity is really small, this equation would give you N is negative something. It'll be a small number minus mg. And that's bad news, because N physically can't be negative. Normal force can't suck the water up. So what's going to happen is you won't have circular motion. You can't use the equations of circular motion anymore. And the water feels a force downward. It will accelerate downward. It will follow a parabolic arc, and will probably hit you. If you have v equals to equal to zero, it'll just accelerate straight down at you.
So this is the minimum speed. Let's work another example. And in this kind of situation, an example is worth a thousand words. Let me do the same story. Only instead of having a bucket, let's think of a ferris wheel. So here's a carnival ride. And in a ferris wheel, at the bottom, you're sitting in the bucket. So here's you. When you get to the top, you're still sitting, or standing. Here's the little seat. I don't know if I got the orientation right? You're sitting down in a ferris wheel, and it's designed so that the thing wobbles around. And when you're at the top you're sitting, and when you're at the bottom, you're sitting. So think about the force diagram when you're up at the top. Once again, you are executing uniform circular motion. So I know that at this moment in time, I am accelerating toward the center. And my force diagram looks like this. There's always gravity acting down on me. And now I'm sitting in a chair. It's not like the bucket situation where the bucket was pushing down. Now I'm sitting in a chair. The chair pushes me up. So I could call this the normal force. That's the contact force. I'm sitting on a chair. It's pushing me up. And those are the physical pushes and pulls on me in the vertical direction. And since I'm moving in a circle, . All of these minus signs are the same as in the last example. The only one that's flipped is N is now pointing in the positive direction. So look what happens. I now get a subtly different formula than I had before. Now I get .
So once again you stop and you think about this equation. Does it make sense? If you were just sitting still up at the top, N is equal to mg. V would be zero. That makes sense. If you're just sitting in a chair, it has to push up with your weight to hold you up. How about if you're moving very, very rapidly, well that looks like bad news in this formula. N is turning negative. And can N ever be negative? No, the chair doesn't hold you down. It can only push you up. So if this ever goes negative, we're in trouble. Our equations aren't satisfied. We won't have uniform circular motion any more. And think about. If you're in a Ferris wheel, and they crank that thing up too fast, everybody is going to go flying out at the top unless they got seatbelts, because seatbelts could apply a downward force. But with the situation as I drew it, you'd better have . It's the exact reverse situation now. Now my ms cancel and I get v. So this is telling me that the maximum speed of the Ferris wheel has got to be below square root of gr. If they crank the thing out too fast, everybody would go flying out at the top.
Uniform circular motion is a kind of a difficult problem to intuit sometimes, especially when there's more than one force in the problem. It's essential that you draw yourself a picture, and think hard about the physical pushes and pulls. You've got to understand whether they're pointing up or pointing down first. And then you can solve the equations and decide what's going on. Once you've drawn your force diagram properly, you're all set to go.
Let me make one final comment about this equation. If you are sitting in your car, and you go up over a bump, it's kind of like this situation. So the force diagram is the same for you, and the radius here is the radius of the curvature of the bump. Now what this is telling you is that the faster you go over the bump, the smaller the normal force is. What is the normal force? That's the force of the seat on you. It's what you feel. When you sit down, you feel the seat pushing you. So what you feel on the car is not gravity. What you feel is the seat pushing you. Gravity is always here. You don't really feel it. You feel the normal force. And the normal force gets smaller as you go up over a bump. And that's right. That's that feeling you get when you're going in the car over the bump and you feel all light. In fact, if you can get at high enough speeds, you can make N equals to zero, and you're kind of floating for a second, right, no normal force at all.
These equations really do make sense when you think about them physically. And the whole trick here is just draw your careful force diagram, and think about what the equations are telling you.^
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