Hey, guys. So let's check out this classic example of conservation of angular momentum. Okay? Remember, conservation of angular momentum involves objects that are spinning that will change either their m, and that will cause a change in omega, or they will change the radius of some sort, and that will cause a change in omega. Those are the two types that you have to look out for. So here we have an ice skater that has a moment of inertia of 6. So I equals 6 when she spins with her arms open. So her I is 6 when she spins with arms will open. And 4, if she closes her arm. It says here if she spins with 6 with a 120 RPM with her arms open. So RPM open is 120. What RPM will she have as a result of closing her arms? What will be RPM closed?
So you can think of open as initial because that's where we start, and you can think of close as final. K. So we're going to use the conservation of angular momentum equation, which is Li equals Lf. In this case, we have one person. So it's just going to be this is going to be just I omega for one person. Iinitial, omegainitial equals Ifinal, omegafinal.
The i's are given. So this is going to be initial is 6 and then I have omega. I'm gonna make a little space here and then this is 4 and then I'm gonna make a little space here. Now notice that this is omega and this is omega, but I gave you one RPM and I asked you for another RPM. So this whole question is in terms of RPM, but the equation is in terms of omega. Like usual, all of our equations are in terms of omega. We always have to convert RPM into omega. But what I want to show you is that you can actually rewrite this equation here in terms of RPM. So let's do that real quick.
Remember, omega is 2 pi f or 2 pi over T or 2 pi. I'm gonna plug in f here and it's gonna be RPM over 60. So what I want to do real quick is I want to show you that there are three variations of this question. Okay? So let's do that real quick. So this is like the official legit version number 1. Here's version number 2. Instead of omega, I'm going to write 2 pi f. And look what happens. Iinitial, 2 pi frequencyinitial equals Ifinal, 2 pi frequencyfinal. Notice that the 2 pis cancel, and you end up with Iinitialfinitial = Ifinalffinal.
If you do this with a period, this is version 2. Iinitial2 pi periodinitial equals Ifinal, 2 pi periodfinal. These guys will cancel, and you end up with Iinitialperiodinitial = Ifinalperiodfinal. And you can do the same thing for RPM. Okay. And this is the last one. That's the one we're going to use here. You can say Iinitial. Now instead of 2 pi f, we're going to use 2 pi RPM over 60. So 2 pi RPM over 60 equals Ifinal, 2 pi RPM over 60. Here, I can cancel the 2 pis and the RPM over 60, and you're left with Iinitial RPMinitial = Ifinal RPMfinal.
So this is the conservation equation, but you can think of it in these three alternative versions as well. This just makes it really easy for you to solve these questions, by basically briefly rewriting the equation. So one point that I want to make here is that a way to know how to make these exchanges very quickly is to look at omega. Omega is on the top. It's on the denominator, up numerator up here. F is on the numerator up here. They're both up top. That's why they both show up up top here. T is on the denominator. That's why T shows up at the bottom when you replace it. And RPM is at the top. That's why RPM shows up at the top here. K?
So in this question, we're actually we don't have to convert the RPMs into omega and then back into RPM. We can just actually plug in the RPM. So I'm gonna plug in RPM initial, RPM final. K? It would have been quick to just replace stuff, but I wanted to show you that we can do this. So RPM initial is 120, and then this is for RPM final. So RPM final will be 6 times 120 divided by 4. Okay. And the answer here is 180 RPM.
Now, the last point I want to make is notice that our I went from 6 to 4. It changed by a factor of 1.5. It went down by a factor of 1.5, and then the RPM went from 120 to 180. It went up by a factor of 1.5. And that's because conservation of angular momentum here, regular momentum, is a linear relationship. There's no squares or whatever. So if one goes down by 1.5, the other one has to go up by 1.5. Okay? Alright. So that's it for this one. This question is actually really easy. I just took a little longer because I wanted to do a little bit of analysis, and I wanted to introduce you to these three alternative versions of the conservation equation so you can solve some of these questions faster. Alright? So that's it for this one. Let me know if you need any help, if you have any questions and let's keep going.