So in this problem, we have a diagram, and I'm gonna reproduce the diagram in pretty good detail because I think this problem is going to be kind of confusing. This is the X axis now on the Y axis. This is where is attached up at the top of the bottom. But it starts here, the particle and its unstrapped. And then it comes out to here. And we're saying that this distance that it is stretched is X No, it's unstrapped distance for the spring was l l. So when I draw the spring this way and the second spring this way, then the first thing I want to dio is I want to find, um this distance. Um, I'm gonna call this l two, so using the Pythagorean Theriault L two is the square root of l squared plus x squared. We also know that force equals okay, X, but this force is going to be in this direction or this direction. And I actually want to write del that X so we don't get confused between exes and whatever in here, this is just the generic formula. So the change in the length So in this problem. The force is k times the change of the length, which would be l two minus l. But L two is the square root of l squared plus X squared minus l. Now there is a component of each of these forces that is in the negative X direction. The force in the X direction equals that that force times the co sign of fate. But we know that the co sign of Fada is just the adjacent which is X over the high pot news, which is l two adjacent overhype oddness. And we also know that L two is the square root of l squared this X squared now in the X direction. This force is negative. So the force is in the negative direction and there are two of them equal to K. So I'm just putting this into here and multiplying by negative too. So that's going to be I still have the axe. Then I have the square root of l squared plus x squared minus l over square root of l squared minus X square. All right, negative two k x. I'm just going to distribute this to the first term and the second term, one minus L over the square root of l squared minus X squared. And that is in the X direction. So that is our answer. And now there's a part B You got to go to the next page to see Part B. We need to find the potential energy. All right, well, the potential energy is the opposite of the integral off the force D x. So the potential energy is I'm going to distribute the two K x again the integral of to Okay, X minus to K L X over the square root of l squared minus X squared DX. All right, So, um, let's see if we can do this. You of X is well, the integral of axe D X is X squared over two, so that first term becomes okay, X squared. But the second term is minus to okay, oh, times the integral of X over the square root of l squared minus x squared D x. I just want to make sure that I didn't make any mistakes already. Um, negative two k x. And it looks like I did No, I didn't que x two k x and minus No, I didn't make any mistakes yet. Um, minus two k l the integral of X over l squared minus X squared DX. All right, so now the difficulty is what is the integral of X Over. Whoops. The square root of L squared minus X squared DX. Well, I'm going to start by letting eggs equal l tangent of fada. Therefore, D x would be l driven. The tangent is the sequence squared of Fada di Fada. So if I substitute that in, I get l tangent of Fada over the square root of l squared minus l squared Tangent squared of Fada and I replaced d X with l seek and squared of Fada di Fada. All right, so that's gonna be the integral, Um, notice that the numerator, while numerator still just l tangent of Fada. But I have two l's there, so I'm gonna write l squared and then I've got seeking and squared of Fada de theater. But the denominator is now when a factor out l squared. So it's gonna be the square root of l squared times one minus tangent squared of Fada. All right, square root of l squared is just l So that's going to leave me with an l only one l Tangent of fade Seek and squared of fade Uh, over the square root of one minus the tangent squared of state. Well, that would be the square root. One minus two tangent squared is the sequence squared. Di Fada, which is gonna give me l times This integral seeking squared near the square root of the Seacon Square is just the seek. And so this is just going to be the tangent of fada times this seek it of Fada d theta. But now l tangent is sign over Cosa seeking to his one over co sign. So this is the same as L into Girl of the sign of Fada over the coast sine squared of data de Fade. So now I'm going to let you equal the co sign of data. So do you. Driven of the co sign is the opposite of the sign de theater. So now it becomes l in to grow in the numerator. I have sign of data d theta. You can see it right here. Sign of data death data. That's the opposite of de you so I can write books opposite of D. U. And then in the denominator co sign squared of data is use squared. So that's going to be negative. L times the integral of you to the negative to power do you, which is negative. L times you to the negative One power over negative one. We think about this again. This is negative. L you negative one power of her negative one. Okay, good negatives. Cancel out and it's just l over you, which is the same as l over you was the co sign of data way up above. We said X equals l tangent Fada. If I draw a rate trying with this being tangent I mean, with this being Fada tangent is opposite over adjacent using the Pythagorean theorem. This is the square root of l squared plus x squared. So the co sign of Fada would be the adjacent over the high pot. Innis who sign is adjacent overhype Otterness. So this now simplifies Teoh l over oh over square root of l squared plus x square. The coastline of Fada is the adjacent overhype Otterness l over the square root of l squared, plus x squared. I was just leaves me with word of l squared plus x square. So I went through all of that, and I got the square root of l squared plus x squared. So let's keep going here equals Okay. X squared minus to K l. And then the integral of X over the square root of elsewhere in minus X squared D X turned out to be squared of l squared plus x squared. Okay, I am somehow missing a term. All right. I think that what I'm missing is that when I take this into girl, um, of way down here that I have to add a constant because this is not a definite integral. So that's gonna be l over you. Plus a constant, which gives me the square root of l squared plus x squared, plus a constant. And so up here we took this in a girl regardless, there's gonna be a constant having trouble writing it in there. Okay, so let's just keep going. I want to say, um, you as a function of X is que x squared minus to K. L times, the square root of l squared plus x squared, plus a constant and according to what's written in the book. That constant must be two K l Square to K l squared. Now we do know the one boundary condition that the potential energy at zero would be zero. So that's gonna be K zero squared minus two K. What? What whoops. Que el square root of l squared plus zero squared plus c zero zero this negative two k backwards l squared plus c soc would be negative two k l squared. However, the square root of L square that we see right here could either be pot plus or minus l. And so it is possible that this could be, ah, positive. And so that could give us a positive, which is exactly what we want it Not exactly sure why we need to choose the negative here. Um, I believe we need to choose the negative so that the potential energy would generally be positive. So nevertheless, um, we do come up with the answer using the boundary condition that you as a function of X, is que x squared. Plus two que el times l minus the square root of X squared plus l squared. And I do believe that this were choosing the negative here so that this ends up with positive numbers. Okay, let's move on to see, we need a plot of you of X versus X, letting l equal 1.20 que equal 40. And there's even a part d after this. Well, first of all, I'm just going to go to a graphing calculator and I'm gonna put this in. So, um, 40 times X squared, plus two times 40 times l, which is 1.2 times 40. No, El is 1.2. AL is 1.2 minus square it of X squared plus l, which is 1.2 squared. And I am kind of interested. What if we had chose the negative up here, and then that would give us a negative? L yeah, that's going to give us potential energy that's on both sides of the That would be potentially negative. So that is the reason why that one has chosen. Okay, so I'm plotting potential energy vs X. I want to look at the question again, and so X is going to be positive where there's nothing stated about moving X in the negative direction. So we read it again, Maybe there is. And I think you can assume that it's it's gonna be pot pulled in the positive direction. Looking at my graph one. I guess that would be meter race. This is gonna be the potential energy in Jewell. Son is gonna erase this. I'm gonna write jewels here. I'm gonna try to So five, 43210 Okay, one comma. Five is there? Zoom in a little bit at 1/2. Its basically at point four. Do you not see if I noticed anything? Nope. Okay, so let's work on the rest of this question. I'm still recording. Yep. Looking for the question. Equilibrium points. Okay. All right. So stable equilibrium is when the potential energy is a minimal. So right there is stable equilibrium. Unstable equilibrium is where the potential energy is a maximum, but it never reaches a maximum. Um, goes off to infinity. Neutral equilibrium is when the potential energy is constant. And that does not occur either. All right, last question. So that's right. This is D. So we're saying that X now is 0.500 meters and I need to figure out the speed at X equals Zeer. All right, so all of the potential energy is going to be converted to kinetic energy. So potential energy of the spring is going to turn into kinetic energy potential energy this spring. I just need to put in X equals zero. No, no, no. I need to put in X equals 0.5. So let me go to my calculator. X is gonna be 0.5 five. That gives me a value of 0.4. So the potential energy in the spring is zero 0.4 jewels that's gonna be converted to kinetic energy, which is 1/2 M. What were we using for em? I don't seem to have written that down, so let's look in the question. Ah ha is a very beginning. M equals 1.18 kilograms. Connect energy is 1/2 times M V squared, so V is going to be the square root of to time zero point for Jules over 1.18 kilograms. Back to the calculator, where route two times 0.4. Divided by 1.18 eight to three meters per second. Um, now it would be going in the negative direction. So negative 0.82