Potential energy stored in a spring

Welcome back. So we have this green spring here, and let's see, there's a wall here. This connected to the wall. And let's say that this is where the spring is naturally. So if I were not to push on the spring, it would stretch all the way out here. But in this situation, I pushed on the spring, so it has a displacement of x to the left. And we'll just worry about magnitude, so we won't worry too much about direction. So what I want to do is think a little bit-- well, first I want to graph how much force I've applied at different points as I compress this spring. And then I want to use that graph to maybe figure out how much work we did in compressing the spring. So let's look at-- I know I'm compressing to the left. Maybe I should compress to the right, so that you can-- well, we're just worrying about the magnitude of the x-axis. Let's draw a little graph here. That's my y-axis, x-axis. So this axis is how much I've compressed it, x, and then this axis, the y-axis, is how much force I have to apply. So when the spring was initially all the way out here, to compress it a little bit, how much force do I have to apply? Well, this was its natural state, right? And we know from-- well, Hooke's Law told us that the restorative force-- I'll write a little r down here-- is equal to negative K, where K is the spring constant, times the displacement, right? That's the restorative force, so that's the force that the spring applies to whoever's pushing on it. The force to compress it is just the same thing, but it's going in the same direction as the x. If I'm moving the spring, if I'm compressing the spring to the left, then the force I'm applying is also to the left. So I'll call that the force of compression. The force of compression is going to be equal to K times x. And when the spring is compressed and not accelerating in either direction, the force of compression is going to be equal to the restorative force. So what I want to do here is plot the force of compression with respect to x. And I should have drawn it the other way, but I think you understand that x is increasing to the left in my example, right? This is where x is equal to 0 right here. And say, this might be x is equal to 10 because we've compressed it by 10 meters. So let's see how much force we've applied. So when x is 0, which is right here, how much force do we need to apply to compress the spring? Well, if we give zero force, the spring won't move, but if we just give a little, little bit of force, if we just give infinitesimal, super-small amount of force, we'll compress the spring just a little bit, right? Because at that point, the force of compression is going to be pretty much zero. So when the spring is barely compressed, we're going to apply a little, little bit of force, so almost at zero. To displace the spring zero, we apply zero force. To displace the spring a little bit, we have to apply a little bit more force. To displace soon. the spring 1 meter, so if this is say, 1 meter, how much force will we have to apply to keep it there? So let's say if this is 1 meter, the force of compression is going to be K times 1, so it's just going to be K. And realize, you didn't apply zero and then apply K force. You keep applying a little bit more force. Every time you compress the spring a little bit, it takes a little bit more force to compress it a little bit more. So to compress it 1 meters, you need to apply K. And to get it there, you have to keep increasing the amount of force you apply. At 2 meters, you would've been up to 2K, et cetera. I think you see a line is forming. Let me draw that line. The line looks something like that. And so this is how much force you need to apply as a function of the displacement of the spring from its natural rest state, right? And here I have positive x going to the right, but in this case, positive x is to the left. I'm just measuring its actual displacement. I'm not worried too much about direction right now. So I just want you to think a little bit about what's happening here. You just have to slowly keep on-- you could apply a very large force initially. If you apply a very large force initially, the spring will actually accelerate much faster, because you're applying a much larger force than its restorative force, and so it might accelerate and then it'll spring back, and actually, we'll do a little example of that. But really, just to displace the spring a certain distance, you have to just gradually increase the force, just so that you offset the restorative force. Hopefully, that makes sense, and you understand that the force just increases proportionally as a function of the distance, and that's just because this is a linear equation. And what's the slope of this? Well, slope is rise over run, right? So if I run 1, this is 1, what's my rise? It's K. So the slope of this graph is K. So using this graph, let's figure out how much work we need to do to compress this spring. I don't know, let's say this is x0. So x is where it's the general variable. X0 is a particular value for x. That could be 10 or whatever. Let's see how much work we need. So what's the definition of work? Work is equal to the force in the direction of your displacement times the displacement, right? So let's see how much we've displaced. So when we go from zero to here, we've displaced this much. And what was the force of the displacement? Well, the force was gradually increasing the entire time, so the force is going to be be roughly about that big. I'm approximating. And I'll show you that you actually have to approximate. So the force is kind of that square right there. And then to displace the next little distance-- that's not bright enough-- my force is going to increase a little bit, right? So this is the force, this is the distance. So if you you see, the work I'm doing is actually going to be the area under the curve, each of these rectangles, right? Because the height of the rectangle is the force I'm applying and the width is the distance, right? So the work is just going to be the sum of all of these rectangles. And the rectangles I drew are just kind of approximations, because they don't get right under the line. You have to keep making the rectangle smaller, smaller, smaller, and smaller, and just sum up more and more and more rectangles, right? And actually I'm touching on integral calculus right now. But if you don't know integral calculus, don't worry about it. But the bottom line is the work we're doing-- hopefully I showed you-- is just going to be the area under this line. So the work I'm doing to displace the spring x meters is the area from here to here. And what's that area? Well, this is a triangle, so we just need to know the base, the height, and multiply it times 1/2, right? That's just the area of a triangle. So what's the base? So this is just x0. What's the height? Well, we know the slope is K, so this height is going to be x0 times K. So this point right here is the point x0, and then x0 times K. And so what's the area under the curve, which is the total work I did to compress the spring x0 meters? Well, it's the base, x0, times the height, x0, times K. And then, of course, multiply by 1/2, because we're dealing with a triangle, right? So that equals 1/2K x0 squared. And for those of you who know calculus, that, of course, is the same thing as the integral of Kx dx. And that should make sense. Each of these are little dx's. But I don't want to go too much into calculus now. It'll confuse people. So that's the total work necessary to compress the spring by distance of x0. Or if we set a distance of x, you can just get rid of this 0 here. And why is that useful? Because the work necessary to compress the spring that much is also how much potential energy there is stored in the spring. So if I told you that I had a spring and its spring constant is 10, and I compressed it 5 meters, so x is equal to 5 meters, at the time that it's compressed, how much potential energy is in that spring? We can just say the potential energy is equal to 1/2K times x squared equals 1/2. K is 10 times 25, and that equals 125. And, of course, work and potential energy are measured in joules. So this is really what you just have to memorize. Or hopefully you don't memorize it. Hopefully, you understand where I got it, and that's why I spent 10 minutes doing it. But this is how much work is necessary to compress the spring to that point and how much potential energy is stored once it is compressed to that point, or actually stretched that much. We've been compressing, but you can also stretch the spring. If you know that, then we can start doing some problems with potential energy in springs, which I will do in the next video. See