Star collapses

by Patrick Ford
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Hey, guys. So let's check out this example here. So ah, very common example of conservation of angular momentum questions is that of a star dying. And that's because the star spins around itself. And when the star dies or collapses, um, it will change not only its mass, it will become lighter, but it also reduced in size and radius and volume. Okay, so this is a good set up for a conservation of angular momentum because momentum will be concerned. So let's check it out. Um, when they start exhaust all of its stellar energy, it dies. That's why it's sad for think. Um at which point a gravitational collapse happens, causing its radius and mass to decrease substantially. So it's just tell me what happens. No actual information. There are 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 period. So period of the sun is around itself. 24.5 days if our somewhere to collapse and shrink 90% of mass in 90% in radius. In other words, our new mass. The new mass of the sun. I'm gonna call this M Prime. Um, is going to be It's shrinking 90% in mass. Meaning my new mass is 10% off the original mass. Okay, in the new radius is 10% of the original radius. I wanna know. How long would its new period of rotation take in days? In other words, if it's taken 24 on half days for the sent to spin around itself, how long would it take for the center spin around itself once these changes happen? In other words, what is my team final, right. Think of this as as T son initial. I wanna know what is my T final and what I want to do here is, um instead of writing l initial equals 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 gonna write actually like this, I'm gonna say l initial equals a constant. I'm gonna expand this. Okay. I'm gonna expand this the gonna be I initial Omega initial is a constant eye. So a sun the sun can be treated as a a zey solid sphere even though it's actually like a huge ball of gas. Eso so treating the solid sphere is kind of kind of bad, but it won't matter as I show in the second. So I have Let's just do that for now. Half m r squared, um, and then Omega, um I want not Omega, but I want period. And remember, Omega is two pi over t. So I'm gonna rewrite. This is two pi over t. And it's a constant because this is a proportional reasoning question. Um, this number doesn't matter in this number. Doesn't matter, Theo. Only thing that matters are the variables that are changing. So even though it was kind of crappy to, um to model the sun as a solid sphere, it doesn't matter because that fraction goes away anyway. So just just right m r Square and you're good. It's sort of what I did earlier. What I had like box m R Square, right, Omega So something like that, because the fraction doesn't matter. So here's what's happening. Um, this guy here is decreasing by by 90%. 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 art. Now the R is squared, so if you square 0.1, you get 0.1 So think of it as the left side of the equation here is being multiplied by a combination of these two numbers, which is 0.1 Basically, the left side of the equation becomes 1000 times smaller. Therefore, the right side of the equation has to become 1000 times greater. Okay, so the right side of the equation has to become 1000 times greater. So this side here grows by 1000. Now, the problem here is it's a little bit complicated because tea is in the bottom. So if the whole thing grows by 1000 that means that T, which is in the denominator, actually goes down by a factor of 1000. Okay, if your fraction goes up, your denominator went down, and that's how your fraction goes. up right. Imagine, for example, you have, ah, 100 divided by 10. That's 10. 100 divided by two. That's 50. Okay, your entire thing went up because your denominator went down. So if this goes down by a lot, right side right here goes up by a lot on the denominator. Goes down by a lot. So basically, your new period is 1000 times smaller than your old period, so I can write t final is, um, one over 1000 t initial. So it's basically 24.5 days divided by 1000. And then what you want to do is you want to convert this into hours, okay? Or some measure of that sort. So we're gonna do here, that one. They has 24 hours this cancer, this cancels and you multiply some stuff. Um, you multiply some stuff, and I actually have this in minutes. Okay, I actually have this in minutes, so let me change that in two minutes, so it's gonna be, um, 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, like, 0.55 hours and minutes makes more sense. Uh, it's easier to make sense out of it. Okay, so imagine this. This the sun takes 24 days to go around itself, and after it collapses and it shrinks significantly, it's gonna spend a 1000 times faster. So it's gonna 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.