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welcome to our third example video considering spring potential energy In this video, we're going to look at a special scenario where you have multiple springs attached to the same object. The reason we're looking at this is because it tends to be very confusing when students come across it, and it's not an uncommon thing to be asked. Okay, so suppose we have two springs with different spring constants K one and K two, and we either hook them up in Siris or in parallel in Siris. Means there's one after the other in parallel means that they are parallel to each other and they're attached to this block em. So in order to figure out what happens here, what we want to do is we'd like to find a way to combine K one and K two so we can just find a total negative K total Times X. Because then the potential energy would simply be one half K total times X squared, which is however far we moved the mass. Okay, so let's see if we can figure out how we can maybe do that on the left hand side. You can see that when we compress here. Both springs must compress by the same amount. Uh, since that's true and we know withdrawing the free body diagram here if I compressed it an amount f that would be pushing back with an amount F one plus F two. That means F is equal to F one. Plus F two f is equal to negative K total times the displacement F one is equal to negative K one times the displacement and F two is equal to negative K two times the displacement the displacement termed cancels the negatives cancel and we find that K total is equal to K one plus K two. Okay, so this is the total spring constant of the system that we have here. And like I said, we can then take a total and plug it in and find the amount of energy we store in the system when we compress it in amount. And all sorts of interesting ideas here. Now, over here it's a little more complex because when we push it in spring one in spring 2 may not necessarily compress the same amount, which will make our potential energy a little more difficult to we'd have to write one half k one x one squared, plus one half que two x two squared. But how? How are we going to do this then? Well, we see, though, that if I compressed this a total amount Delta X, the amount compressed by X one and the amount compressed by X two would have to be adding up to that amount to the total displacement of the box. Knowing this and then also, let's look at our free body diagram here. If I were to pull out Mass this time with Force F, it would be pulled back in by the second spring. And then if I look at the point where the two are connected, I would see that the second spring is being pulled back in by the first spring. But if I'm just holding the mass m at a position Delta X from equilibrium, then I would find that the force has to be equal to have to, and the F two has to be equal to F one. So let's let's put that in here. You know that X for a spring is going to be equal to negative F over K. So if we do that, then we have negative F over K total is equal to negative F one over K one plus negative F two over K two. But if we're holding at equilibrium, all the F's cancel again. All the negatives cancel. And so we find that K total is equal to one over K one plus one over K two. So that's kind of handy. We can have these very simple techniques for solving for the total spring constant in systems composed of multiple multiple springs. And then we can use this in order to figure out things like, What's the force here? And once you know the force, then you can figure out Okay, well, I know that the force on this one you can calculate x one force on this one. You can calculate x two. And over here it's even simpler where you can simply plug it in as one half K Total times X squared

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