Hey guys. So let's check out this example here. A very common example of conservation of angular momentum questions is that of a star dying. And that's because a star spins around itself. And when the star dies or collapses, it will change not only its mass, it will become lighter but also reduce in size, radius, and volume. Okay? So, this is a good setup for a conservation of angular momentum because angular momentum will be conserved. So let's check it out.
When a star exhausts all of its stellar energy, it dies. That's why it's sad. Poor thing. At which point a gravitational collapse happens causing its radius and mass to decrease substantially. So, it's just telling you what happens. No actual information there.
Our sun spins around itself at its equator at the middle point right there every 24.5 days. The time that it takes for you to go around yourself, for you to complete a full revolution of any sort is called the period. So, the period of the sun spinning around itself is 24.5 days. If our sun were to collapse and shrink 90% mass and 90% in radius, in other words, our new mass, the new mass of the sun, I'll call this m′, is going to be it's shrinking 90% in mass, meaning my new mass is 10% of the original mass. Okay? And the new radius is 10 percent of the original radius. I want to know how long would its new period of rotation take in days. In other words, if it has taken 24.5 days for the sun to spin around itself, how long would it take for the sun to spin around itself once these changes happen? In other words, what is my t_final? Right. Think of this as t_sun_initial. I want to know what is my t_final and what I want to do here is, instead of writing l_initial = l_final, because we don't have actual numbers here, we just have percentages in terms of drop. This is really a proportional reasoning question.
What I'm gonna do is I'm going to write it actually like this. I'm going to say l_initial = a constant. I'm going to expand this. Okay. I'm going to expand this. The gonna be I_initial Omega_initial is a constant I so a sun the sun can be treated as, as a solid sphere even though it's actually like a huge ball of gas. So, treating the solid sphere is kind of kind of bad, but it won't matter as I show you in a second. So I have, let's just do that for now, half m r², and then Ω. I want not Ω but I want period and remember Ω is 2π/T. So I'm going to rewrite this as ½mr2 and 2π/T.
So, this is a proportional reasoning question, this number doesn't matter, and this number doesn't matter. The only thing that matters are the variables that are changing. So even though it was kind of crappy to model the sun as a solid sphere, it doesn't matter because that fraction goes away anyway. So just write m r² and you're good. It's sort of what I did earlier, what I had like box m r². Right? Omega. So something like that because the fraction doesn't matter.
Here's what's happening. This guy here is decreasing by 90 percent. So basically, it's being multiplied by 0.1. Right? And then this guy here is being multiplied. Imagine that you're putting a 0.1 in front of the m and a 0.1 in front of the r. Now, the r is squared so if you square 0.1 you get 0.01. So, think of it as the left side of the equation here is being multiplied by a combination of these two numbers which is 0.001. Basically, the left side of the equation becomes a 1000 times smaller. Therefore, the right side of the equation has to become a 1000 times greater. Okay? So the right side of the equation has to become a 1000 times greater. So this side here grows by a 1,000. Now, the problem here is it's a little bit complicated because t is in the bottom. So if the whole thing grows by a 1,000 that means that t, which is in the denominator, actually goes down by a factor of a 1,000. Okay? If your fraction goes up, your denominator went down and that's how your fraction goes up. Right? Imagine, for example, you have a 100 divided by 10, that's 10. A 100 divided by 2, that's 50. Okay. Your entire thing went up because your denominator went down. So if this goes down by a lot, the right side right here goes up by a lot, and then the denominator goes down by a lot. So basically, your new period is a 1000 times smaller than your old period. So I can write t_final is one over a 1000 t_initial. So it's basically 24.5 days divided by a 1,000.
And then what you want to do is you want to convert this into hours. Okay? So we're going to do here is that, one day has 24 hours. This cancels this cancels and you multiply some stuff, and I actually have this in minutes. Okay? I actually have this in minutes. So let me turn that into minutes. So it's going to be 24 hours times 60 minutes. And when you do this, you get that the answer is about 35 minutes. I did minutes because otherwise you end up with about 0.55 hours and minutes makes more sense. It's easier to make sense of it. Okay? So imagine this, the sun takes 24 days to go around itself and after it collapses and it shrinks significantly, it's going to spin a 1000 times faster. So it's going to make a full revolution around itself in just 35 minutes. Okay? So that's it for this one. Let me know if you have any questions and let's keep going.
