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Energy Conservation with Non-Conservative Forces

Patrick Ford
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Hey guys. So up until now we've been using our conservation of energy equation, but we've been using with conservative forces. Remember the conservative forces are things like gravity and springs? We've been focusing more on gravity. So I want to do in this video, I want to show you how the conservation of energy equation works when you start throwing non conservative forces into the mix. Remember your non conservative forces or things like applied forces in friction. Let's check out our practice problem or example down here because it's exactly what's going on here. So I have a hockey puck, right? So hockey puck is traveling with some initial speed, so the initial speed is four. But then what I'm gonna do is I'm gonna push it with some hockey with my hockey stick. So my applied force is 200 I'm going to push it through some distance here. My D. Equals 0.3 m. And then what happens is this hockey puck? Because I've pushed it, we can expect that it's actually gonna be moving faster. So because I've pushed it it's going to have some final speed that's going to be greater than the initial four. And that's exactly what I want to calculate here. So how am I going to calculate this pucks? Final speed? I'm just going to use conservation of energy. Right? So I'm just gonna draw my diagram which have already done and I want to use my energy conservation equation here. So how do I do this? Well if I write my energy conservation, my mechanical energy going to write K plus you initial and this equals K plus you final. That's how we've seen this before. However, what happens is we're going to see that there is no potential energy because we're just traveling along this horizontal surface here. And what happens is we can tell without even calculating anything that are kinetic final is going to be greater than your Connecticut initial. Because again we said that the speed is going to be greater than the initial four because we've been pushing through some distance, We've been doing work on it. So what ends up happening here is that we're going to find out that the mechanical energy initial is not going to be equal to your mechanical energy final here, which is really for the kinetic final. And this should make perfect sense because the force that we have acting on this puck is an applied force and applied forces remember are not conservative. And remember the rule for conservation of mechanical energy. We said that non conservative forces do work, which means that the work done by these forces isn't zero. Then your mechanical energy is not going to be conserved. So here we have an applied force that's doing work. It's adding removing energy into the system. So your mechanical energy is not going to be equal on the left and right sides. So how do we solve these problems? Well, it turns out that we're actually still going to use conservation of energy and our equation to solve these problems. But now we just need one more term. So we're gonna have still capel issue on the left and K plus you on the right. But now the last term that we need is the work done by non conservative forces. So this is known as the full conservation of energy equation. We're going to write it like this from now on every single time. All right. So basically, instead of using this equation, now, we're actually going to write it using we're actually going to solve this problem by using the full conservation of energy. So this is K. Plus you initial. Plus the work done by non conservative forces equals K. Plus you final. Now we're still going to have no gravity potential energy, initial and final here. So now we're gonna go ahead and start eliminating and expanding out our terms. So our kinetic energy. So our kinetic energy remember is just going to be one half M. V. Initial squared. So what about work done by non conservative forces? How do we actually calculate that? Well, it turns out the work that's done by non conservative forces is just going to be the some of the work is done by these forces applied forces in friction. So the work that's done by non conservative forces is going to be any work that you do on the object. Plus any work that's done by friction are always just going to write that off to the side every single time. It's basically just any works that these forces actually will add or remove energy to the system. Right? So the work that's done by non conservative is going to be any work that's done by U. Plus any work that's done by friction. So do we have either of those? Well we're on smooth ice. There's no work done by friction. However, we know that the hockey puck is being pushed by you, you know with the hockey stick. So you're actually doing some work on this object. So how do we calculate the work that's done by you? Well really we're just gonna use our F. D. Cosign theta. The work that's done by any force. Is that force times deke assigned data. So really the work that you do is going to be F. A. Times D. Co signed data here. And that's how we sort of use this work done by non conservative forces. Now we just finally have the kinetic energy final which is one half mv final squared. Remember that we're actually looking for this V final here so we can go ahead and start plugging in all of our numbers. So this is going to be one half. We have the mass which is 05 and the speed initial which is four squared. So now we have F. A. D. Cosign Theta. Well you're Applied Force is 200. The distance that you're pushing it through 0.3. What about the coastline of the angle between them? Well remember what happens is that are Applied Force points to the rights and your distance also points to the rights. So the angle between these two things is zero degrees and you're gonna get a coastline of one. Or sorry, cosign equal to one. So it's gonna be 200 times 2000.3 and then this is going to equal one half of 0.5 times V final squared. So if you just plug in everything on the left side into your calculator, you're going to get 60 for jewels. So the 64 equals one half 0.5 V final squared. So what I'm gonna do is I'm gonna take the V final. All right, I'm gonna flip the equation and then I'm gonna have the square root of two times 64. That's what happens when you move the one half over to the other side. You're gonna divided by the mass, which is 0.5 here, you're gonna take the square root of the whole number and you're gonna get 16 m per second. So, as expected, the hockey puck is going to be moving faster in the final velocity equals m per second. And that's because you've actually done some work to it. So what I want to do finally is kind of like go back to this expression right here and show you what's going on. So I'm gonna write this out over here, this is my equation. So you're one half mv initial square, if you were to plug this into your calculator, which would get is you would get for jewels. And then if you plugged in the final uh the V final into your kinetic energy equation, you would get 64 jewels. So we can see that the mechanical energy on the left and right side isn't conserved. And that's really because the work that you have done onto the puck Is basically making up the difference. So the work that you're doing is equal to 60 jewels. So you have four plus 60. It's the work that you did, and then you have the final kinetic energy of 64. So that's how you do these kinds of problems, guys, let's move on.