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Yo-yo in vertical circle example

In this video David explains how to find the tension in a string that is whirling a yo-yo in a vertical circle. Created by David SantoPietro.

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Video transcript

- [Instructor] People find centripetal force problems much more challenging than regular force problems, so we should go over at least a few more examples, and while we're doing them, we'll point out some common misconceptions that people make along the way. So, let's start with this example, and it's a classic. Let's say you started with a yo-yo and you whirled it around vertically, and I think this is called the around the world, if you want to look it up on YouTube, looks pretty slick. They whirl it around, goes high, and then it goes low. So this is a vertical circle, not a horizontal circle. We're not rotating this ball around on a horizontal surface. This ball is actually getting higher in the air and then lower in the air. But for our purposes, we just need to know that it's a mass tied to a string. Let's say the mass of the yo-yo is about 0.25 kilograms, and let's say the length of the string is about 0.5 meters. And let's say this ball is going about four meters per second when it's at the top of its motion. And something you might want to know if you're a yo-yo manufacturer is how much tension should this rope be able to support, how strong does your string need to be. So let's figure out for this example, what is the tension in the string when this yo-yo is at its maximum height going four meters per second? And if it's a force you want to find, the first step always is to draw a quality force diagram. So let's do that here. Let's ask what forces are on this yo-yo. Well, if we're near Earth and we're assuming we're going to be near the surface of the Earth playing with our yo-yo, there's gonna be a force of gravity and that force of gravity is gonna point straight downward. So the magnitude of that force of gravity is gonna be m times g, where g is positive 9.8. g represents the magnitude of the acceleration due to gravity, and this expression here represents the magnitude of the force of gravity. But there's another force. The string is tied to the mass, so this string can pull on the mass. Strings pull, they exert a force of tension. Which way does that tension go? A lot of people want to draw that tension going upward, and that's not good. Ropes can't push. If you don't believe me, go get a rope, try to push on something. You'll realize, oh yeah, it can't push, but it can pull. So that's what this rope's gonna do. This rope's gonna pull. How much? I don't know. That's what we're gonna try to find out. This is gonna be the force of tension right here, and we'll label it with a capital T. We could have used F with the sub T. There's different ways to label the tension, but no matter how you label it, that tension points in towards the center of the circle 'cause this rope is pulling on the mass. So, after you draw a force diagram, if you want to find a force, typically, you're just gonna use Newton's second law. And we're gonna use this formula as always in one dimension at a time so vertically, horizontally, centripetally, one dimension at a time to make the calculations as simple as possible. And since we have a centripetal motion problem, we have an object going in a circle, and we want to find one of those forces that are directed into the circle, we're gonna use Newton's second law for the centripetal direction. So we'll use centripetal acceleration here and net force centripetally here. So in other words, we're gonna write down that the centripetal acceleration is gonna be equal to the net centripetal force exerted on the mass that's going around in that circle. So because we chose the centripetal direction, we're gonna be able to replace the centripetal acceleration with the formula for centripetal acceleration. The centripetal acceleration's always equivalent to v squared over r, the speed of the object squared divided by the radius of the circle that the object is traveling in. So we set that equal to the net centripetal force over the mass, and the trickiest part here, the part where the failure's probably gonna happen is trying to figure out what do you plug in for the centripetal force. And now we gotta decide what is acting as our centripetal force and plug those in here with the correct signs. So, let's just see what forces we have on our object. There's a force of tension and a force of gravity. So, when you go try to figure out what to plug in here, people start thinking, they start looking all over. No, you drew your force diagram. Look right there. Our force diagram holds all the information about all the forces that we've got as long as we drew it well. And we did draw it well. We included all the forces, so we'll just go one by one. Should we include, should we even include the force of gravity in this centripetal force calculation? We should because we're gonna include all forces that point centripetally and remember, the word centripetal is just a fancy word for pointing toward the center of the circle. And this force of gravity does point toward the center of the circle. So we're gonna include this force in our centripetal force. It's contributing, in other words, to the centripetal force. It's one of the forces that is causing this ball to go in a circle, so we include it in this formula. So mg is the magnitude of the force of gravity. We have to decide, do we include that as a positive or a negative? Many people want to include it as a negative 'cause it points down, but when we're dealing with the centripetal direction, it's inward that's gonna be positive, not necessarily up that's gonna be positive. If up happens to point in, then we'd consider it positive. So if we were down here, up is positive. But up here, down is positive 'cause it points in toward the center of the circle, and if we're over here, diagonally up and left would be positive because any force that would be pointing toward the center of the circle is gonna be included as a positive sign and there's a reason for that. The reason we're including toward the center of the circle as positive is because we chose to write our centripetal acceleration as positive. And since we know the centripetal acceleration points toward the center of the circle, if we make the centripetal acceleration positive, we've committed to toward the center of the circle as being positive as well. In other words, we could have decided that out of the circle's positive, but if we did that, this centripetal acceleration that points inward would have had to be included with a negative sign over here and that's just weird. Nobody does that. So we choose into the circle as positive. That makes our centripetal acceleration positive, but it also makes every force that points inward positive as well. And long story short, this force of gravity is gonna be counted as a positive centripetal force since it points inward toward the center of the circle. And we keep going. We've got another force here. We've got a force of tension. Do we include it in here? Yes we do because it points toward the center of the circle. And do we include it as a positive or a negative? Since it also points toward the center of the circle, we're gonna include it as a positive centripetal force. It is also one of the forces that causes this ball to move around in a circle. In other words, the combined force of both gravity and the force of tension are making up the net centripetal force in this case. So now, if we want to solve for the tension, we just do our algebra. We'll multiply both sides by the mass. Then, we'll subtract mg from both sides, and if we do that, we'll end up with the tension equals m v squared over r minus mg, which if we plug in numbers, we'd get that the tension in the rope is 5.55 Newtons. So this is to be expected. We subtracted the force of gravity, the magnitude of it from this net centripetal force, so this term here represents the total amount of centripetal force we need in order to cause this yo-yo to go in a circle. But the amount of tension we need is that amount minus the force of gravity, and the reason is, the force of gravity and tension together are both acting as the centripetal force. So, neither one of them have to add up to the total centripetal force. It's just both together that have to add up to the centripetal force, and because of that, the tension does not have to be as large as it might have been. However, if we consider the case where the yo-yo rotates down to the bottom of its path, down here, once the yo-yo rotates down to this point, our force diagram's gonna look different. The force of gravity still points downward, the force of gravity's always gonna be straight down and the magnitude is always gonna be given by m times g. But this time, the tension points up because the string is always pulling on the mass. Ropes can only pull. Ropes can never push, so this rope is still pulling the mass, the yo-yo toward the center of the circle. So now, when we plug in over here, one of these forces is gonna be negative. Before they were both positive and they were both positive because both forces pointed toward the center of the circle. Now, only one force is pointing toward the center of the circle, and we can see that that's tension. Tension's pointing toward the center of the circle. Gravity's pointing away from the center, radially away from the center. That means tension still remains a positive force, but the force of gravity now, for this case down here, would have to be considered a negative centripetal force since it's directed away from the center of the circle. So, if we were to calculate the tension at the bottom of the path, the left hand side would still be v squared over r 'cause that's still the centripetal acceleration. The mass on the bottom would still be m 'cause that's the mass of the yo-yo going in a circle. But instead of T plus mg, we'd have T minus mg since gravity's pointing radially out of the center of the circle. And if we solve this expression for the tension in the string, we'd get that the tension equals, we'd have to multiply both sides by m, and then add mg to both sides. And we'd get that the tension's gonna equal m v squared over r, plus m g. This time, we add m g to this m v squared over r term, whereas over here, we had to subtract it. And that should make sense conceptually since before, up here, both tension and gravity were working together to add up to the total centripetal force, so neither one had to be as big as they might have been otherwise. But down here, not only is gravity not helping the tension, gravity's hurting the centripetal cause by pulling this mass out of the center of the circle, so the poor tension in this case not only has to equal the net centripetal force, it has to add up to more than the net centripetal force just to cancel off this negative effect from the force of gravity. And if we plugged in numbers, we'd see that the tension would end up being bigger. We'd actually get the same exact term here, except that instead of subtracting gravity, we have to add gravity to this net centripetal force expression. And we'd get that the tension would be 10.45 Newtons. So recapping, when solving centripetal force problems, we typically write the v squared over r on the left hand side as a positive acceleration, and by doing that, we've selected in toward the center of the circle as positive since that's the direction that centripetal acceleration points, which means that all forces that are directed in toward the center of the circle also have to be positive. And you have to be careful because that means downward forces can count as a positive centripetal force as long as down corresponds to toward the center of the circle. And just because a force was positive during one portion of the trip, like gravity was at the top of this motion, that does not necessarily mean that that same force is gonna be positive at some other point during the motion.