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10. Conservation of Energy

Springs & Elastic Potential Energy


Energy in Horizontal Springs

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Hey guys we've already seen one type of potential energy which is called gravitational and in this video, I want to introduce you to the other type which is called elastic or spring potential energy here. Alright, so let's take a look here. The idea is the same potential energy. Remember is just stored energy. So just like you store energy when you lift something. Springs store energy when you compress or stretch them. So this energy is called u elastic. So let's take a look here. Right? So if you're at the ground for gravitational potential, your gravitational potential energy zero. But if you raise it to some height of y then your Yugi just becomes MG. Y. Here. Well it's the same idea for elastics for springs basically the ground is like the equilibrium position when springs are relaxed, they have no stored energy. So there you elastic is zero. But what happens is that when you push up against them and you deform them by compressing or stretching them, then you have some applied forces and some spring forces. And the spring force here depends on your deformation which is this X. Here. Alright. So remember what we said, the relationship between work and gravitational potential energy was that W. G. Equals negative delta U. G. It's the same idea for springs. The work that's done by springs is going to be the change in spring or elastic potential energy. This negative delta U elastic. So what we saw here is that if this negative delta U. G. Equals negative MG delta Y. Then the equation for gravitational potential is just MG. Y. It's the same exact thing we can do for springs, we can basically cancel out these negative signs here. And we can say that the elastic potential energy is really equal to one half K. X squared. So this is the equation that we're gonna use our energy conservation equations. Now, how does this change our energy conservation equation? It actually really doesn't. We're still gonna write K. Plus you plus work done equals K plus you. The only thing is that up until now we've only been focused on gravitational potential energies, but now we're actually just gonna include elastic potential energy is because these things are the same type. They're both potential energy. So we can just combine them. So our potential energy is gonna be ug plus you elastic. So all you have to do is now just keep track or keep on the lookout for any springs in our problems. Let's go ahead and take a look here. We have a block that's attached to a horizontal surface, we have the kite K constant and we're gonna push the block with a force of 100 newtons. So I've got my applied force. The magnitude is 100 which means that the magnitude of the spring force that pushes back is also 100. So we want to calculate is the compression distance how far we've actually compressed the spring. That's actually this distance right here, this is X. So how do we solve this? Do we use energy? Do we use something else? Basically the idea here is that whenever you have spring problems in which objects are stationary like we have in this first part here we're still just holding the block up against the compressed spring. Then we're gonna solve this by using forces. And the idea here is that we want to solve the compression distance. Remember which is just X. We can solve this by using hooks law, which says that the compression sorry the absolute value of the spring force equals K. X. So we actually have the spring force and the applied force they're both 100 we have the K. Constant. So we can figure out what our X. Is by just rearranging for this. Let's go and do that. So X. Is really just gonna be equal to the magnitude of the applied force divided by K. Which is just 100 divided by 500. And you're gonna get compression distance of 0.2 m. Right? So you push the push this thing this block up against the spring that compresses by two m or 20.2 m which is like 20 centimeters. And that's the answer. Right? So then you release the spring. Now in part B so you're going to remove your hand. Now the spring gets now the spring releases and it shoots out the block to the rights we want to calculate this launch speed which is really just this velocity over here. So how do we solve this? Can we use forces? And the answer is no. Because remember that as this block travels to the equilibrium distance, the value of the force is constantly changing. The force is not constant. So we can't use any of our motion equations to solve this. Instead, we're gonna have to solve this by using energy. So whenever you have objects that are moving between two points on springs, you have to solve it using energy because the force is in constant. Alright, so we're gonna write out our energy equation. We've already got our diagram, we've got K Plus you plus any work done by non conservative equals K. Plus you final. Now we're just gonna go ahead and we're gonna eliminate and expand all the terms. So, let's take a look here. Well, in the initial right, so this is the initial And this is the final here, when you are pushing up against the block. Right? And you before you released it, its initial velocity is still zero. So it has no no kinetic energy. It still has some potential energy because now we know that potential energy is made up of gravitational potential and elastic potential. There's no gravitational potential because it's traveling horizontally, but we have a spring that's compressed. So what happens is there is going to be some elastic potential energy here. Now, what about work done by non conservative. That's either work done by you or friction. Work done by friction. And there's no forces applied by you or friction. So there's no work done by non conservative forces. Now, when you get to the end, right, once this block is released it has some V. Final. So therefore it does have some kinetic energy here. But the spring is basically fired this block already and it returns to the equilibrium position. So there's no more elastic potential energy. Okay, so really all that happens here is that all of our elastic potential energy becomes kinetic energy final. So we just expand those terms. This is one half K. X squared equals one half M. V final squared. So it usually happens that these problems is that the M. S. Won't cancel because you have K on one side and on the other, but the one half will usually cancel. So we can do is we can just go ahead and solve this V final over here. We have the final squared is gonna be this is gonna be K over M. And this is actually gonna be X initial times X initial squared. So what happens is we have the final is equal to the square roots. Now I'm just gonna start plugging in. My number's 500 divided by four times 0. squared. If you go ahead and work this out, you're gonna get 2.24 m per second as the final or the launch speed of the block. Alright, so that's it for this one, guys. Let me know if you have any questions and let's get let's go ahead and get some practice.

Springs in Rough Surfaces

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Hey guys hopefully got a chance to work this one out your own. Um Let's go ahead and check out together. So we have a four kg box that is moving to the right with an initial speed of 20 and it's going to collide with the spring. We have this force concept of the spring is 600. What happens is the spring is gonna compress. So this box is pushing up against the spring or collide with the spring and then it's gonna collapse like this. So what happens afterwards is you have this spring that is now compressed or coiled up a little bit more and it's coiled up some distance X. And that's what I want to find out here. Now. It's gonna be the maximum compression. What happens is the box um is going to have a final velocity of zero. It's given all of its energy to compressing the spring. So the maximum compression happens when the box comes to a stop. And so now we want to go ahead and write our energy conservation equation. We can't solve this using forces because remember that the force that happens throughout this motion here as you're compressing the spring isn't constant. It varies depending on how much you're pushing it right or how much you're compressing it. So we have to use energy conservation. So let's go ahead and check out our equation here. K. Initial you initial plus worked on non conservative equals K. Final plus you final. So we have some initial kinetic energy because the box is moving with some speed. What about potential energy? Remember potential energy could be either gravitational or elastic potential energy. What we said here was was sorry your initial gravitational potential energy is zero because you're moving along a flat surface there actually is no gravitational potential energy anywhere in the problem. What about elastic potential? What happens initial? Is that the spring isn't compressed yet? Right. The box hasn't hit it yet. So there is no elastic potential energy. There's no potential energy period. What about work done by non conservative forces? Well you're not doing anything right? You're just standing there watching and we're on a frictionless surface. So there is no work done by non conservative. What about the final? So Kay final is gonna be zero because we have the initial of the final speed of the block is gonna be zero. Right? So it's basically transferred all of its kinetic energy and it's become elastic potential energy. Right? So here we have some elastic potential energy. So let's go and write out our expressions. We have one half M. V. Initial squared equals and then the expression for you final is gonna be K. One half K. X. Final square. So this is actually we're looking for what is sort of like the compression distance. What is X final relative to the equilibrium position. Okay. So what you'll notice in these problems is that you actually can't cancel out masses on the left and right because you have M. S. On one side and case on the other. But you can't actually cancel out the one house. That makes things a little bit easier. So what happens is we're gonna write an expression, we're just gonna move K. To the other side and you're gonna get M. V. Initial squared divided by K. Is equal to X final squared. So you just take the square roots and X final is going to be the square roots. And we're ready to plug in. We have all of our numbers, we have the mass, which is just four kg. We have the initial speed is 20 m per second and we're gonna divide it by the K concept of 600. So, you go ahead and plug this into your calculator. Take the square roots and you're gonna get a compression distance of 1. m. So after the block has stopped, it's transferred all of its energy and its compressed the spring by 1.63 m here. So that's the answer. All right, so let me know if you guys have any questions and I'll see you guys in the next one

A 4-kg block moving on a flat surface strikes a massless, horizontal spring of force constant 600 N/m with a 20 m/s. The block-surface coefficient of friction is 0.5. Calculate the maximum compression that the spring will experience.