10. Conservation of Energy
Energy with Non-Conservative Forces
Energy Conservation with Non-Conservative Forces
Was this helpful?
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.
Box Sliding on Rough Patch
Was this helpful?
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.
An 800kg car skids to a stop from 30m/s through a distance of 90m. Calculate the coefficient of friction between the car and the road.
Energy Conservation with Air Resistance
Was this helpful?
Hey guys, let's put this one out together here. So we have a two kg object, gets dropped from a height of 80 m and reaches the floor with 30. But in this case there's actually some air resistance here. We're going to calculate the work that's done. Let's check out this problem here. Let's go ahead and draw our diagram. So we've got the floor like this, this y equals zero. I've got a two kg object and it's gonna be dropped. Which means that the initial speed is zero, but once it falls to the floor, what happens is that it's going to reach the floor with some final speed. This is the final and this equals 30. Doesn't matter if it's positive or negative because their kinetic energy is always gonna be final, right or is always gonna be positive. Now, what happens is throughout the motion throughout the fall of this object, it being pulled down by gravity, there's MG. But there's also air resistance that I'm going to call this f air. Usually we kind of neglect air resistance but in this problem we want to calculate the work that is done by this force. So let's go ahead and check out our energy conservation equation. Right? We're gonna use conservation of energy. We've got our diagram and now we're gonna go ahead and do our conservation of energy. So this is gonna be K initial um Plus you. Initial plus work done by non conservative equals K final plus you final. So we got one last thing here, you're actually falling from an initial heights. This is why initial of 80. So this is the initial. Okay, so we have no initial kinetic energy because the initial speed is zero. We do have some gravitational potential because we're at 80 m and that's above are zero points. So we're gonna set the zero point of gravitational potential here. The risk that work done by non conservative forces because we do have a resistance and that's a non conservative force. Remember that worked? Non conservatives either work that's done by you which there's none of in this problem. Plus the work done by friction, basically friction and air resistance are kind of the same thing. Air resistance is really just friction through the air. So this is really just gonna be F air. The work that's done by this force here and this is going to be equal to K final place you final. So there is some kinetic energy final but there's no gravitational potential interview because you're at the ground. So what happens is our equation simplifies to this is gonna be M. G. Y initial plus the work that's done by F air and then equals the kinetic energy final is going to be one half mv final squared. So, let's see here, I know em I know g I know the initial heights. I also know em and I know the final squared. So all I have to do is just go ahead and move everything over. So this W. F. Air here, W. F. Air is just gonna be um this is gonna be uh let's see. You actually can't cancel out the masses because it doesn't exist in every single one of these terms, Right? This W. F. Air doesn't have an eminence. We can't cancel that out. So what ends up happening is you're gonna get let's see um you're gonna get one half. This is gonna be one half. Do I have this here? Yeah, there's one half of two Times 9.8 And then Times 80. So that's the first time. And then when you subtract it, you're gonna subtract it from two times 9.8. Uh and then this is gonna be I'm sorry. This is not A. T. This is the the final. So this is actually 30 square. There we go. Sorry about that. So you've got one half 9.8 and then we've got 30 squared minus two times 9.8 times 80. Alright, so that's initial heights. So then you go ahead and work this out. What you're gonna get is you're gonna get negative 670 jewels. So why don't we get a negative sign? It's because work is actually removing energy from the system. That's exactly which we should expect. So, we got a negative number here because we got we have energy that's being taken out by air assistance. All right. So, we can use energy conservation equations to solve problems with resistive forces. Like when we have air and water resistance basically because they just act like friction. So we can just sort of calculate them as a non conservative work. All right. So let me go ahead and quickly solve part B. Now. Barbie is now asking for the average force of air resistance. All right. So we want to figure out basically what is this F air? What's what is this F air? What happens is you can think of this F air as being a constant force that's acting on this object as it falls. So it's a constant force that's being act that's being exerted over some distance D. Which is really just my delta Y. So what happens is I can use the work energy or sorry, the work done by constant force equation. Work is really just gonna be F. Air time's D. Times the co sine of the angle between those two things. So you're displacements down your forces up. Therefore this is just going to simplify, this turns into a negative one. And basically what happens is you're going to get that the W. F. Air is equal to f. Air time's D. So now we want to figure out what's this force. We actually know what the distance is. It's just gonna be my delta Y. So I can go ahead and calculate this. So my F air is really just going to be the work that's done. Which I know is negative 670 divided by my delta wine. My delta Y. Is actually just 80. Or it's actually rather just negative 80 because technically um it's gonna go downwards like this. Also I just want the negative science to cancel so that I get the magnitude of the force and the force is equal to um what I get Is 8. Newtons. So that's the average force of your resistance. All right, so that's it for this one. Guys, let me know if you have any questions.