12. Rotational Kinematics

Converting Between Linear & Rotational

# Length of string rotating point mass

Patrick Ford

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All right. So here we have a small object that rotates at the end of a light string. So here's a string, and you've got a small objects. I'm gonna do this. Andi. It's spinning like this. Okay, imagine that the, um if you spin a string, it forms a circular path. Right. Theology. It reaches 120 rpm from rest. In just four seconds, I will give you a ton of information here. First, you start arrest. So, Megan, issue zero. Um, you reach the final rpm of 1. 20. And you did this in just four seconds. Okay? It also says you that the engine show acceleration. Tangential acceleration, remember, is a T after the four seconds is, um, 15 m per second square. And we want to know what is the length of the strength. Okay, we haven't talked about the length of string yet, but I hope you can figure out that the length of the string is the It's this distance here, right? It's the radius of the circle that forms the circular path you get. Or it is the radial distance from the center of rotation of which is your hand and the edge of rotation, which is where the object is. So little are the distance to the middle is your length. So essentially, what we're looking for is little are think of it as little are not as l because there's no l's in any of these equations, so you're not gonna find out. Okay, so this is a little bit of a mess, because we have we're gonna have to use a combination of equations here. Okay, So if you look through all the equations with you so far, you might first think about one of the three or four, um, motion equations, one of three or four emotion equations. And you might think of that because I gave you Omega initial zero I gave you Delta T. I gave you RPM, which you can convert to a make a final. And if you do that, you're gonna have three out of five variables once you convert. Okay, however, notice that if you look through all of those four equations, there are no ours in them. So you're not going to be able to solve for R by doing this. Okay, Now, if you look a little further, um I do have a nickel Asian that I give that I gave you recently that links up 80 with our 80 with our and that equation is 80 equals R Alfa and I know 18. So all I have to do is find Alfa r equals which is 15 right there, divided by Alfa. So we're gonna have to do is find Alfa and plug in here. Now, how do I find Alfa? Well, Alfa is one of my five variables of motion, so I'm going to be able to use thes three guys to find Alfa. Okay? And that's what we gonna do now. So first I'm gonna convert from rpm into frequency. I'm sorry into w final. So remember that w final is two pi f and F is our PM over 60 so 20 divided by 60. It's true. Therefore, Omega Final is two pi, And instead of that, I'm gonna put it to which is four pi. Okay, so I'm going to rewrite this year just to clean it up. Omega initial zero Omega Final is for pie. Delta T is four. We're looking for Alfa and they ignore variable is delta theta notice how I know three things. Um, ignore variables. Delta theta the Onley equation that doesn't have Delta. Fate is the first one out of the four Omega Final equals Omega initial plus 18. And if we're looking for Alfa, we just gotta move everything out of the way. Initial zero. So Alfa is Omega final, divided by time Or make a final. Is we found here? It's two pi divided by time. Time is four. I'm sorry for pie four cancels with four and Alpha's 3. 15 radiance per second square. Okay, let me disappear here so you can see. All right, Um, now all we gotta do is plug in this number here, and we're good, So 15 divided by 3.14. And if you divided to you get that are is 4.77 meters. Okay, 4.77 m. And that is the final answer. Cool. So whips. So that's it for this one. It's interesting question that combined sort of these two equations and the basic idea here is just that old school physics hustle. You got stuck in one and you're gonna have to go find the other and just kind of work your way through it. All right. There is not a very clear path. There's a few different ways you could have done this. But the most important thing is, you know, try to figure out what's an equation that has my variable and and then sort of look for all different ways to find all the letters, all the variables you have to solve for cool. Alright, that's it. Finished one. Let me know if you guys have any questions.

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