A spring requires 12 J 12J of work to stretch the spring from 1.1 m 1.1m to 1.4 m 1.4m past its equilibrium. What is the spring constant?

this problem, we're told a spring requires 12 joules of work to stretch the spring from 1.1 to 1.4 meters past the equilibrium. What is the spring constant? Now recall that when dealing with these work problems where we had springs, there was an equation that we could use. We were told that work is gonna be the integral of our force function as a function with respect to distance. So we're integrating this function from the interval a to b. Now this is the general work formula that we can do for solving any of these types of work problems. But if we're specifically dealing with springs, our force function here is going to be Hooke's law, which is k, the spring constant, times x. Now what are we looking for in this problem? Well, in this problem, we're actually not trying to calculate work directly, because the work is already given to us as 12 joules. What we're instead trying to calculate is the spring constant. That's what it's asking for, which is this value right here, k. So to do that, well, I'm gonna set this up as though I'm solving a work problem and see if I can kinda isolate the spring constant. That's gonna be my strategy. So let's do that. And I'm gonna start by taking my force function k x and plugging it in for this function here. So So we're gonna have that work is equal to the integral from a to b of k x, this function, and all integrated with respect to x. Now k is the spring constant, so I can actually take any constant I have and pull it outside of the integral. So if that work is equal to k times the integral of x. Now where do I go from here? Well, what I can do is plug in the variables that I know about. So you can see right in the problem, we're given that the work done is 12 joules. So I can take my work here and replace it with 12. Now I can also see here that we're trying to find k, so this is going to be our unknown. And what about this integral? Well, the bands on my integral are going to correlate to the interval that we're stretching the spring past its equilibrium. Now notice that all of our units are in meters and joules, so we do have everything in the correct units here. So all I need to do is take my lower and my upper bound and put them in for the bounds of my integral. So we're going to go from 1.1 to 1.4, and that's going to be x integrated with respect to x. Now at this point, we can just go ahead and solve this integral. So doing this, we're gonna keep this 12 here, well, if 12 is equal to k times the antiderivative of x, which by using these rules, we know that the antiderivative here is gonna be x squared over two. That's the power rule used on this x. And this is all gonna be bounded from 1.1 to 1.4. Now next, what I need to do is plug in these bounds into everything I have in these brackets for x. So you're gonna have the 12 is equal to k, and then we're gonna plug things in. So I'm gonna first start with my 1.4 on top here, which is gonna give us 1.4 squared over two, and then we're going to have minus, and then I'm gonna plug in my lower bound of 1.1. So we're gonna have 1.1 squared over two, and then this is what we need to evaluate. Now this is something that you can evaluate on a calculator right here, everything within these brackets. If you do that, you should get that 12 is equal to k times everything evaluated here, which comes out to 0.375. And notice something. We're now at a point where we can solve for k. All we need to do is take the 0.375 and divide it on both sides of the equation. Now doing this, I can cancel the 0.375 on the right side here, so all I'm going to be left with is the spring constant k. And 12 divided by 0.375 turns out to be 32. So the spring constant is going to be 32, and we're going to be newtons per meter. Because suddenly we were dealing with meters for the bound, and we were dealing with joules for work. So that means that the spring constant units would be in newtons per meter, and that is the solution to this problem. So this is how we can calculate the spring constant when it comes to these situations where you need to find it. So I hope you found this video helpful, and let's move on.