Hey, guys. Let's work this one out together here. So we have a block of unknown mass that's sliding along a floor with some initial velocity. I'm going to call this \( v_{\text{initial}} = 30 \). Well, 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 that 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 with that means that the velocity is going to be equal to 0. So we want to calculate basically what is the distance that thing travels across that rough patch. I'm going to call that \( d \) here. Alright. So what happens is we're going to use conservation of energy for this. We've got our diagram. Let's go ahead and write out our energy conservation equation. There's really only 2 points, an initial and a final here. So this is my initial and this is my final. So I've got \( K_{\text{initial}} + U_{\text{initial}} + \text{any work done by non-conservative forces} = K_{\text{final}} + U_{\text{final}} \). Alright. So we have some initial velocity here. So we have some initial kinetic energy. There's no, gravitational potential or, 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 \( K_{\text{final}} \) here? Is there any kinetic energy final? We just said here that the final velocity is going to be 0 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 conserved if you have only conservative forces. We have a non-conservative force that's 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 \( K_{\text{initial}} \) which is going to be \( \frac{1}{2} m v_{\text{initial}}^2 \), and then the work that's done by friction. So the work done by friction, remember, is going to be \(- f_k d\), right? So got negative there very important that you add that negative sign there because friction is going to remove energy from the system. Alright, and this is going to equal 0 on the right side, right? Both of the terms cancel out here. So what happens is I can go ahead and rearrange, and actually sort of expand out this friction equation, so I'm going to do this over here. So remember that friction is just equal to \( k \) times the normal force, and what happens is if you're just sliding 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 2 forces in the vertical direction. So basically what happens is we're just going to get \( \times mg \). So now what I can do here is I can say that \( \frac{1}{2} m v_{\text{initial}}^2 + \text{negative}\mu g d = 0 \), and I can actually just move the whole term over to the other side. So what I end up getting here, is I get \( \frac{1}{2} m v_{\text{initial}}^2 = \mu k \times mgd \). Notice how the masses actually cancel, 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 going to go ahead and write an expression for this. You're going to get \( v_{\text{initial}}^2 \), you're going to get \( \frac{1}{2} v_{\text{initial}}^2 \), and then you're going to divide over the \( \mu k \) and also gravity. So \( \mu k \times g \). So this is just going to be, let's see here. This is going to be the initial speed of 30 squared divided by 2 times \( \mu k \). Right? The coefficient here is 0.6 and then times 9.8. When you go ahead and work this out, what you're going to get is you're going to get 76.5 meters, and that's the answer. Right? So what happens here is that you've had all this initial kinetic energy, right, this \( \frac{1}{2} mv_{\text{initial}}^2 \), 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 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 here, then that work done is always energy that's either added or removed from the system. Alright? So that's it for this one, guys. Let me know if you have any questions.
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10. Conservation of Energy
Energy with Non-Conservative Forces
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