Okay, so we're gonna be looking at problem number 20 from chapter eight, and this is a pendulum problem. So we have a pendulum here, and it is attached to a negligible it is. No, it is attached to a stone, and it is going to be swinging. Okay, so there's going to be gravity acting downwards, and it's going to be swinging this way, right? Swinging this way like this is gonna be an arch. Sorry. This arch is really weird. Okay, so this mass here weighs two kilograms, and we know that the string length l designated and l is equal to four meters. Okay? And we know that when the string passes through this point, it has a velocity of eight meters per second, and it's at the bottom here. So this is at a degree of zero degrees. This is at zero degrees. And so a A wants to know What is this speed when the string is at 60 degrees to the vertical? Okay, So less first, find out how much kinetic energy there is when the string passes through zero degrees. Right? And kinetic energy is found by this Formula 1/2. Um, be square. And so if you find the kinetic energy here, that's also that's equal to the mechanical energy off this of this system at 60 degrees. So whatever this value is, you know what? Let's actually just find that two kilograms and then the speed meters per second square. Okay, so it's 64. So 64 jewels, right? And let's use this equation after all, because energy conservation so mechanical energy, mechanical energy, mechanical energy is equal to 64 jewels and it's equal. So mechanical energy is equal to the sum of the potential and kinetic energies. So we have, um, basically mg Alan, this case and then whatever angle this is, they, uh, one minus the coast sign of that and then plus 1/2 em the square. And all you have to do now is basically substitute than the chemical energy, which is equal to 64 jewels into here. So you have 64 jewels there. 64 jewels is equal to the mass of the rock. Uh, which is to kill a grounds two kilograms gravity 9.8 years. Because I can square length. What is it? Four meters times one minus 60 co sign of 60 co sign 60 and then plus 1/2 em the square solve for Why did I want right? Um is to sow substitute in your values, right? And then all you have to do is basically solve for the sulfur. This V So what do you do? Right, you do subtract this to that sign. Whatever this quantity comes out to be subtracted to this site. Sulfur V and that will be your velocity at the bottom. All right, now, let's clear this out. Clear this out. Okay, Now, part B B is asking for what B is asking for. The greatest angle that the vertical Oh, the greatest angle with the vertical that the stream will reach. All right. What does this mean? Okay, well, let me explain. This means, like, basically what is? Well, let me just let me just do it. Four meters. Yeah, is over here. Right. So be We know the energy. Mechanical energy. Mechanical energy is equal to M g l. One minus co signed data. This is the equation we've been using. Right? And then we know that mechanical energy is still equal to 64. It's still equal to 64 jewels. So it's looking for basically what is the angle when it reaches the top right when it reaches the top and when it reaches the top here, we know that velocity is equal to zero because momentarily lost is there on the top. So we have 64 jewels, is equal to everything else, literally stays the same 9.8 meters per second square. Your length doesn't change, because why would it one minus co sign of data and then this is zero. So all you have to do is solve here for co sign. And how do you do that? Okay, you divide this entire thing over here. So the C two times for eight. Father, this is eight. So eight divided by 9.8 is equal to one minus co sign. Fade up so you can do minus 18 times 9.8 minus one is equal to negative coastline. They divide by negative one. Did my button I give one, and all you have to do is our coast on this side on that side, and then you'll get your angle of data. And now the last question is, if the potential energy of the pendulum Earth system has taken to use. There are stones. Lowest point was a total mechanical energy of the system. Well, we already where do you know that? You know, for this entire problem, this has been the lowest points. Been zero. So we know our mechanical energy is 64 jewels, and that's all for this question.