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Springs in Rough Surfaces

Patrick Ford
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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