Welcome to a new problem this time were presented with the graph that shows the behavior of a rubber band. So we have one ex oxes, and then these are y axes will have force. So think about a rubber band that's being stretched on both directions, and we want to see a couple of things about the round man. If it obeys Hooks Law, the force applied on the rather barren happens to be that it three point five five x raised to zero point four eight seven one. So it's a variable force. You can see the ex works there, and it's it's exacted owner on both ends of the rubber band thie. Other thing that happens, the couple of values, for example, at when x zero point zero two. But this point happens to be around about five Newtons and when x zero point zero four at this point happens to be about seven mutants, so they're different. Values maximized at zero point one, and this is X in meters. So there's a relationship between the force Force is being applied on the rubber band and also the end, the displacement off the rubber band Ah Hook's losses. K F vehicles to K X WOR K is the constant force constant and excellent displacement. If this is the case, then we can solve who can end up having f over Ex becomes equivalent to the force. Constant. This is called the force Constant. There's a different, well studying the force constant where you used the time derivative. So we called it Kay uh, E f. F is derivative with respect to X off the force being applied because this is a a variable force. If the spring obeys hook slow, K f is going to be a constant. So events that the the ah Displacement X doesn't have any impact on the on the force. Constant k f f. So the first part of the problem you know, it's almost conceptual. Where the same does the rubber band obey Hoops law? If the rubber band obeyed hook slow, then the relationship would be a linear relationship ethic was too K X. Remember, Kay is like the slope. If you can recall from the graph ofthe algebra y costume MX plus B, it's similar to ethical Sze to K. It's where B zero in this case, but if you look a tte This graph. It's a nonlinear function. It's a nonlinear functions, so that means the gruff does not obey hooks law. So they thie rubber band. The rubber band violates some violets. Okay, hook slow. It doesn't the, you know, the relationship between the force in the rubber band, this force right here on the displacement X. So the forced air from the displacement X violets hook slaw with the violation because we don't have a linear relationship. If we had a linear relationship similar to Africa's Decay X, then that say that war the force is directly proportional to the displacement, and therefore it obeys hooks law. Ah, in the second part of the problem, we have a different case this time. And so, you know, we're required to check the stiffness off the spring. You know, we have the spring that obeys hook slow, and so they want us to determine K a. F f. Remember from the previous page, that's a different. We have seen the spring constant based on the derivatives. Okay, F f is a derivative with respect to the displacement off the force that's given Ah, so they want to find out. Is this value constant or is it increasing? Oh, decreasing a relative to X. So that's what they're asking. Is that value increasing or decreasing relative to X? So when we know from the previous page, that happens to be that we point five five. Okay, so F, which is the force being implied on the rubber band, equals two of thirty three point five five X rays to zero point four eight seven one. And so, if we want to get the derivative, we do it on both sides of the river to respected F of X equals to the gravity ofthe this expression X rays to zero point for eight seven one. And that means that when you get the derivative it if it's a simple problem like this, let's say X squared to get the derivative you bring the two down on. Then you subtract one, so you end up with two Xs and derivative. So in this case, I'm going to bring the zero point four seven four eight seven one down and then subtract one. So this becomes zero point four eight seven one times thirty three point five five x and then if we subtract one, we get negative. Zero point five one two night. Okay, that's what we have right here. We can simplify the relationship, so we have a k e i f f. The constant that were given is equivalent to the river to you of the respected ex off the force. This force right here that we're dealing with on DH, that's equivalent to If you multiply these two numbers out, you get sixteen point before this's raised to negative. If you ways are variable toe a negative exponents, that's the same as bringing thie exponents, bringing the variable and the exponents down and then changing the exponents toe a positive value. So these are the relationship between K F F and X is an inverse relationship, meaning that if one goes up there, the one goes down. So this is five one. Remember, this is one five one, the two night. So if, um, if exchanges, for example, if X increases this function, this constant is going to go down. So, um, it means that there's an inverse relationship if x his increasing X is increasing. Okay, FF is decreasing. Okay, that's what we have right there and We could also say that if X is decreasing, so it's the opposite. The opposite is going to happen. We get k e e F f is increasing, so it's an inverse relationship between the two. It's not constant. Its inverse on This is what we have in the second part of the problem, where they want us to find the relationship between those two and the third part of the problems to go back the next page, check the next page and that part of the problem, something extra happens. So they're saying, if you're stretching a rubber band, you have a rubber band and you're stretching it. Then in the first instant, gonna stretch the rabble ran from X equals to zero up until X equals to zero point zero fall zero zero meters and then you're going to extend it all the way. You know twice as much up until X equals two zero point zero eight zero zero meters. So you know what's what's the work done when you stretch it between those two points of the fast? Straight happens in between the zero and zero point zero four, and then the second stretch happens between zero point zero zero four and zero point zero eight meters. Is this his meter's? Remember that if you can recall if, as long as the forces variable wantto get the work, we have to do the derivative of that. But this time in the first instance, we have going from zero to zero point zero four meters. And so this is the integral of the force itself that we have, which is thirty three point five five eggs razed to zero point four eight seven one times three x. We want to get the derivative of that and then plug in the limits of integration because it's stretching from zero to stretching from zero to four. Remember, when you do the into go, you could move the constant out. That's the constants are we have thirty three point five five and then doing the integral me a hard one to this and then divide by the same value. So we divide by zero point for eight seven one plus one. We're doing the integral. So the outcome. His X rays, too. X rays to one point for eight seven one over one point four eight seven one. Don't forget the limits of integration. They go from zero to zero Ponzi off for meters like that zero point zero four meters thie into gold was with respect to the X. Now we have to plug in the numbers. Um, so x X is in is in meters and then the force eyes in Newton's. So right here. When you plug it in, you get three three point five five with plug in X, which is your point zero four zero zero meters. That's going to be raised to one point for eight seven, one all over and one for eight, seven one. Um, that's the first part of the Brockett. The second part. It's going to be zero. And the reason white zero is because off this limit right here, plug it in. It changes the whole thing to zero. So we end up with a solution off round about zero point one eight eight Jules. And that's the work required to move the robber bend from this location off until that location. This second part off the problem going to do that in the next change, it's still the same thing. So think about the rubber band and so initially went from zero ponzio for zero zero meters. You know it's stretched up until zero point four meters from X equals to zero meters up until there but then now was stretching it a stepfather. We want to take it all the way up until X equals to zero point zero. Eat zero meters like that. So initial stretches right there and then now once it gets that point, we stretch it again. Walk If the forces variable using into girls this time the limits of integration happened to be zero point zero four meters up until zero point zero eight zero zero meters. The force is given as previously three five. It's a viable force x zero points for eight seven one. That's DX during the interval from zero point zero for zero. Zero meters up until zero point zero eight zero zero meters. When you do the integral similar to what happened before we have three three five. Five x This becomes one point for eight seven one over one point for eight seven one limits of integration from zero point zero four zero zero meters up until zero find zero eight zero zero meters. So we have to plug in this one into the function in there, plug in that one and then subtract them from each other. So we'LL have a lovely step to show that. So the initial bracket is three three five five. Who x happens to the zero point zero eight zero zero raised one point for eight seven one. That's the first part of the bracket. Then remember, it's over one point for eight. Seven, one like that and then we do. The something was trucked. I thought we could have married easier. But you know, that's fine. We were who's trucked the same thing. So maybe we could of pulled out the constant. So that's what we're going to do. So this constant to make her life easier. Uh, this is going to be a constant. So we'LL pull it out because it's a constant. We just place it right here. Um, one point for eight, seven one and then, you know, we have a bracket there. Get rid of this other bracket. Oh, yeah. We need to get rid of this pocket right here. So, you know, these are just strategies to simplify the whole process. We just simplify simplifying the problem. Um and so it's algebra. So we have a minus here again. Plug ins. Your point zero four hundred meters. This is meeting us. Call. It's raised to one point four eight seven one. That's what we have now concludes the bracket. Um, if you simplify that, you get zero point three three nine jewels. That's the energy required to stretch the rubber band from opposition of zero point zero four top position of ze opens your aid. The last part of the problem. We can use thie energy or rather, the walk energy principle, which says the work done on the on the rubber band you know, the work required to stretch the rubber band from opposition off X equals to zero point zero four zero zero up until opposition of X equals to zero point zero eight zero zero. Uh, the work done as we can recall, is from the previous slide is zero point three three nine jewels point three three nine jewels. So from this point up until that point and using the walk and a vegetarian, that's the same was changing kinetic energy. I want to find the final blast e when you stretch it between those two points. Remember the initial block of lawsuit before you stretch it? One zero. So this is one half and the final squared minus one half and the initial squared. This is going to be zero because the initial velocity before you stretch it was zero meters per second. So zero point three three nine jewels equals to one half B Final squid. We divide both sides by one house. Him going one house him. So, you know, question is, what does the M stand for? The M stands for, like a block that when you release it, it's going to move back that way, you know, it's going toe accelerate that were So we're trying to find this velocity the final last year when you a kind of liken stretch it and that's your M o. That's the muscle. The block, these to cancel out. So the final equals two. Uh, you know, basing all steps on other break simplification sama. Zero point three three nine jewels over zero point five times the moss off the object. The muscle the object happens to be, You know, this object has a muscle zero point three hundred kilograms opened three zero zero kilograms and that's what we plug in here. Open three zero zero programs. That's the final spread. And so to get the the actual VI final, we take the square root off all all that zero point three three nine jewels off zero point five times zero point three hundred kilograms. Okay, and eventually that's going to give us one point five zero meters per second. So an elaborate problem where you know, initially we I wanted to check if the rubber band ho bass hook saw. And it doesn't because you can see that it's nonlinear if it was to obey hooks loyal millennia. The second part of the problem is we have a different force constant, which is derivative off the variable force. We get that reverted when we end up with an inverse relationship right here. And so because of us, if the displacement is increasing them, the thie constant K F F is decreasing life. The displacement is the quizzing. The opposite happens. It's increasing. Also, we want to check the energy involved in stretching from, say, zero to zero ponzu, for we use the integral because that's the area in the graph for variable forces, the energy being produced by variable forces. We do integral and end up with zero point one eight eight Jules. The second part we have from zero point zero four zero two zero opens your eight again. We do another integral because the forces variable, you know. So we end up getting this value right here. Zero point three three nine jewels. And finally, we have a block that's tied to the, um Rob. Abandoned once it stretched to zero points. You're eight meters. It's going to be released back that way, and we want to find out what's the velocity when it's released. So when it's passing through this point right here, the energy expanded his zero point three three nine jewels. So use that energy to find the end the walk energy trom remember the initial velocity zero and saw this whole initial Connecticut and Europa zero. We're only left with the final kinetic energy and the walk down in stretching, which is your point three three nine jewels. And so use that. And so for the the lost thanks for watching the video feel free to ask any questions you Khun post comments and hopefully you have a strong understanding ofthe hook slow based on this particular problem. Have a wonderful day and looking forward to the next.