Bowling ball in vertical loop

- [Narrator] Imagine that in an effort to make bowling more exciting, bowling alleys put a big loop-the-loop in the middle of the lane, so you had to bowl the ball really fast to get the ball up and around the loop and then only afterward, it would go hit the bowling pins kinda like mini golf bowling or something like that. Well if you were gonna build this, you'd have to know at the top of the loop, this structure's gonna have to withstand a certain minimum amount of force. You might wanna know how strong do you have to make this. You can't have this thing breaking because it can't withstand the force of the bowling ball. So let's ask ourselves that question. How much force is this loop structure gonna have to be able to exert while this bowling ball is going around in a circle and let's pick this point at the top to analyze. We'll put some numbers in here. Let's say the ball was going eight meters per second at the top of the loop. That's pretty darn fast so someone really hurled this thing through here. Now let's say the loop has a radius of two meters and the bowling ball has a mass of four kilograms, which is around eight or nine pounds. Now that we have these numbers, we can ask the question: How much normal force is there gonna be between the loop and the ball? So in other words, what is the size of that normal force, the force between the two surfaces? This is what we'd have to know in order to figure out if our structure is strong enough to contain this bowling ball as it goes around in a circle. And it's also a classic centripetal force problem, so let's do this. What do we do first? We should always draw a force diagram. If we're looking for a force, you draw a force diagram. So what are the forces on this ball? You're gonna have a force of gravity downward, and the magnitude of the force of gravity is always given by M times G, where G represents the magnitude of the acceleration due to gravity. And we're gonna have a normal force as well. Now which way does this normal force point? A common misconception, people wanna say that that normal force points up because in a lot of other situations, the normal force points up. If you're just standing on the ground over here, the normal force on you is upward because it keeps you from falling through the ground, but that's not what this loop structure's doing up here. The loop structure isn't keeping you up. The loop structure's keeping you from flying out of the loop and that means this normal force is gonna have to point downward. So this is weird for a lot of people to think about, but because the surface is above this ball, the surface pushes down. Surfaces can only push. If the surface is below you, the surface has to push up. If the surface was to the side of you, the surface would have to push right. And if the surface was to the right of you, the surface would have to push left. Normal forces in other words, always push. So the force on the ball from the track is gonna be downward but vice versa. The force on the track from the ball is gonna be upward. So if this ball were going a little too fast and this were made out of wood, you might see this thing splinter because there's too much force pushing on the track this way. But if we're analyzing the ball, the force on the ball from the track is downward. And after you draw a force diagram, the next step is usually, if you wanna find a force, to use Newton's Second Law. And to keep the calculation simple, we typically use Newton's Second Law for a single dimension at at time, i.e. vertical, horizontal, centripetal. And that's what we're gonna use in this case because the normal force is pointing toward the center of the circular path and the normal force is the force we wanna find, we're gonna use Newton's Second Law for the centripetal direction and remember centripetal is just a fancy word for pointing toward the center of the circle. So, let's do it. Let's write down that the centripetal acceleration should equal the net centripetal force divided by the mass that's going in the circle. So if we choose this, we know that the centripetal acceleration can always be re-written as the speed squared divided by the radius of the circular path that the object is taking, and this should equal the net centripetal force divided by the mass of the object that's going in the circle and you gotta remember how we deal with signs here because we put a positive sign over here because we have a positive sign for our centripetal acceleration and our centripetal acceleration points toward the center of the circle always. Then in toward the center of the circle is going to be our positive direction, and that means for these forces, we're gonna plug in forces toward the center of the circle as positive. So let's do that. This is the part where most of the problem is happening. You gotta be careful here. I'm just gonna plug in. What are the centripetal forces? To figure that out, we just look at our force diagram. What forces do we have in our diagram. We've got the normal force and the force of gravity. Let's start with gravity. Is the gravitational force going to be a centripetal force. First of all, that's the question you have to ask. Does it even get included in here at all? And to figure that out you ask: Does it point centripetally? I.e. does it point toward the center of the circle? And it does so we're gonna include the force of gravity moreover because it points toward the center of the circle as opposed to radially away from the center of the circle. We're gonna include this as a positive centripetal force. Similarly, for the normal force, it also points toward the center of the circle, so we include it in this calculation and it as well will be a positive centripetal force. And now we can solve for the normal force. If I solve algebraically, I can multiply both sides by the mass and then I'd subtract MG from both sides. And that would give me the mass times V squared over R minus the magnitude of the force of gravity, which if we plug in numbers, gives us four kilograms times eight meters per second squared, you can't forget the square, divided by a two meter radius minus the magnitude of the force of gravity which is four kilograms times G which if you multiply that out gives you 88.8 newtons. This is how much downward force is exerted on the ball from the track but from Newton's Third Law, we know that that is also how much force the ball exerts upward on the track. So whatever you make this loop out of, it better be able to withstand 88.8 newtons if people are gonna be rolling this ball around the loop with eight meters per second. Now let me ask you this. What if the ball makes it over to here, right? So the ball rolls around and now it's over at this point. Now how much normal force is there at this point? Is it gonna be greater than, less than, or equal to 88.8 newtons. Well to figure it out, we should draw a force diagram. So there's gonna be a force of gravity. Again, it's gonna point straight down, and again, it's gonna be equal to at least the magnitude of it will be equal to the mass times the magnitude of acceleration due to gravity. And then we also have a normal force, but this time, the normal force does not push down. Remember, surfaces push outward and if this surface is to the left of the ball, the surface pushes to the right. This time our normal force points to the right. And let's assume this a well oiled track so there's really no friction to worry about. In that case, these would again be the only two forces. So what about the answer to our question. Will this normal force now be bigger, less than, or equal to what the normal force was at the top. Well I'm gonna argue it's gotta be bigger, and I'm gonna argue it's gonna have to be much bigger because when you plug in over here, into the centripetal forces, you only plug in forces that point radially. That is to say centripetally, either into the circle, which would be positive, or radially out of the circle, which would be negative. If they neither point into nor out of the circle, you don't include them in this calculation at all because they aren't pointing in the direction of the centripetal acceleration. In other words, they're not causing the centripetal acceleration. So for this case over here, gravity is no longer a centripetal force because the force of gravity no longer points toward the center of the circle. This force of gravity is tangential to the circle. It's neither pointing into nor out of, which means it doesn't factor into the centripetal motion at all. It merely tries to speed the ball up at this point. It does not change the ball's direction, which means it doesn't contribute to making this ball go in a circle, so we don't include it in this calculation. So when we solved for the normal force, we'd multiply both sides by M, we would not have an MG anymore. So we wouldn't be subtracting this term and that's gonna make our normal force bigger. Moreover, the speed of this ball's gonna increase compared to what it was up here. So as the ball falls down, gravity's going to speed this ball up and now that it's speed is larger, and we're not subtracting anything from it, The normal force will be much greater at this point compared to what it was at the top of the loop. So recapping, when you wanna solve the centripetal force problem, always draw your force diagram first. If you choose to analyze the forces in the centripetal direction, in other words, for the direction in toward the center of the circle, make sure you only plug in forces that are into, radially into the circle or radially out of the circle. If they're radially into the circle, you make them positive. If they were radially out of the circle, you would make them negative. And if they neither point radially inward, toward the center of the circle or radially outward, away from the center of the circle, you just do not include those forces at all when using this centripetal direction.