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Hey guys, let's look at this one out together here. So we have a block of unknown mass that's sliding along the floor with some initial velocity. I'm gonna call this V. I equals 30. But then what happens is it's going to hit this little rough patch along the road or the surface or whatever it's sliding on. And we're going to calculate the distance to the block travels before it stops. So later, once it passes through the rough patch, it's going to come to a stop right here and what that means that the velocity is going to be equal to zero. So, we want to calculate basically what is the distance that this thing travels across that rough patch? I'm gonna call that D here. All right. So what happens is we're gonna use conservation of energy for this. We've got our diagram. Let's go ahead and write out our energy conservation equation is really only two points. And initial and final here. So, this is my initial and this is my final. So, I've got K initial Plus you initial Plus. Any work done by non conservative forces equals K final plus you. Final. All right. So, we have some initial velocity here. So, we have some initial kinetic energy. There's no gravitational potential. You know, there's no spring energy or anything like that here. So, there's actually no potential in either case, initial or final. What about work done by non conservative forces? Remember work done by non conservative forces is work done by you and also friction. What happens is you're not pushing it? You're just watching this thing as it's sliding along. So there's nothing here but there is going to be work that's done by friction because as this block is sliding through this rough patch, there is a force of friction that's going to oppose that direction of motion here. So that's where our work is going to come into. What about K. Final here? Is there any kinetic energy final? We just said here that the final velocity is going to be zero once it comes to a stop. So there is no kinetic energy. So what happens here is you might be thinking, whoa, I thought mechanical energy has to be conserved. It's only conservative. You have only conservative forces, We have a non conservative forces acting here. So basically there's some work that is done by that non conservative force. So the point I want to make here is that whenever mechanical energy isn't going to be conserved, the work is always going to basically make up the difference. Let's check out how this works here. So, I've got my K. Initial which is going to be my one half M the initial squared and then the work that's done by friction. So the work done by friction, remember is going to be negative F. K. D. Right? So I've got negative. They're very important that you have that negative sign there because friction is going to remove energy from the system. All right, and this is going to equal zero on the right side. Right. Both of the terms cancel out here. So what happens is I can go ahead and rearrange um and actually sort of expand out this friction equation. So I'm gonna do this over here. So remember that friction is just equal to M. U. K. Times the normal force. And what happens is if you're just lighting along a normal a flat surface, then you have an MG downwards like this. And you also have a normal force. And these two things have to balance each other because there are only two forces in the vertical direction. So basically what happens is we're just gonna get Mieux Que times MG. So now what I can do here is I can say that one half M V initial square plus negative mu k M g D is equal to zero and I can actually just move the whole term over to the other side. So what I end up getting here um is I get one half M V initial squared equals mu K times M G D notice how the masses actually canceled, which is great because we didn't actually know what the initial mass was and now we just want to calculate the distance here. So basically what happens is when you calculate the distance, I'm just gonna go ahead and write an expression for this. You're gonna get the initial squared, you're gonna get one half of the initial squared and then you're gonna divide over them UK and also gravity some UK times G. So this is just gonna be um let's see here, this is gonna be the initial speed of 30 squared divided by two times um UK. Right? The coefficient here is 0.6 and then times 9.8 when you go ahead and work this out, what you're gonna get as you're gonna get 76. m and that's the answer. Right? So what happens here is that you've had all this, this initial kinetic energy, right? This one half mv initial squared but it gets removed by this non conservative force, basically friction removes that energy until the box is left with nothing. So what happens is is that this work done by non conservative forces always is going to make up the difference between the left and the right side of your equation here. And what happens is if you end up getting, you know, something that's non zero. Like we did hear that, that work done is always energy that's either added or removed from the system. Alright, so that's everything from guys, let me know if you any questions.

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