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this is Walter the penguin and Walter the penguin gets bored in Antarctica so he likes to run jump and then slide across the ice to a stop but Walter is a clever and curious penguin so while he's sliding he's thinking about energy conservation and he's confused because he knows that he starts off over here with some amount of kinetic energy but he knows that he ends over here with no kinetic energy since he slides to a stop so he wonders how can energy be conserved when he seems to be losing kinetic energy now if you would have asked this question when we dealt with forces you would have said oh well obviously this penguin is coming to a stop because there must be some amount of friction between the penguin and the ice maybe the ice is very slippery but it can't be frictionless or this penguin would probably keep sliding forever there might be some air resistance causing the penguin to slow down but it's probably mostly friction between the penguin and the ice but we're talking about conservation of energy here so how can we put this idea in terms of energy well the way we do it is we just say that this force of friction is doing negative work on the penguin and we know the work is negative because the force of friction is directed in the opposite direction to the Penguins motion the other way we can see this is that we could just use the formula for work done by any force that formulas F D cosine theta if we want to find the work done by the force of friction we would plug in the force of friction for our force the magnitude of it times the distance that the penguin slid to the right and then this theta in cosine theta is always the angle between the force and the direction of motion so this penguins sliding to the right the force is directed to the left you might think that's zero but that's not zero think about it the angle between leftward and rightward is not zero that's actually a hundred and eighty degrees so this angle would be a hundred and eighty right here or PI radians and cosine of 180 is going to give you a negative one so the work done by the force of friction on this penguin is going to be negative F kd- the force of friction times the distance the Penguins slid to the right but this still doesn't answer Walters question where did the kinetic energy go friction may have done negative work on this penguin but where did that energy end up and you probably have a good idea because when two surfaces rub together some of that energy of motion is going to get transformed into thermal energy in those two surfaces in other words this sheet of ice is going to have a little more thermal energy it's going to heat up just a little bit and Walters feathery coat is going to heat up just a little bit and they're going to have more thermal energy to end with than what they started with just like when you rub your hands together vigorously on a cold day to get warm you're turning some of that kinetic energy into thermal energy that warms up your hands and you might be like alright that's all well and good but how do we put this all together I mean I've got this idea of work over here I've got this idea of kinetic energy over here what unifying framework lets me think about this all in one package and that would be the statement of conservation of energy so we can say that the initial energy of any system plus any external work that's done on that system has to equal the final energy of that system in other words we could say that Walter the penguin started with kinetic energy so Walter starts with one-half MV squared and we don't have to worry about gravitational potential energy because Walter is not changing his height he's just sliding straight along the ice at a horizontal level and we can add the external work that was done which we figure that out we know the external work would be the work done by friction so we'd have a minus because it was negative work fkd and it's negative again because this force is taking energy out of the system we can set that equal to the final energy but Walter ends with no kinetic energy so we'd say that there's no final kinetic energy to end with there's no energy in our system to end with in which case we find out something that might be obvious to you is that one-half MV squared the initial kinetic energy that Walter started with has to equal if we add to both sides fkd the magnitude of the work done by friction but some people might object them I say wait a minute we just said there was thermal energy to end with how come we didn't include that in our final energy and the reason is that in this calculation here we assumed that Walter and only Walter was part of our energy system in other words Walter and only his emotional energy as kinetic energy was the only energy we keeping track of that's why we said that initially there was just Walters kinetic energy and this sheet of ice was external to our system not part of our system that's why it exerted a negative external work remove the energy from the system and Walter ended up with no kinetic energy but there's an alternate way to go about this calculation you could say all right instead of just considering Walter and Walter alone to be part of our system let's go ahead and include the ice as part of our system and all those places where the thermal energy can go like Walters feather e-coat if we include all the places energy can go then there won't be any external work so an alternate way to solve these problems is to use this same formula but now Walter and the ice are both part of our system our system would still start with the kinetic energy that Walter had at the beginning that doesn't change but now there would be no external work not because force of friction is enacting there's still a force of friction but that's an internal force between objects in our system because Walter and the ice are part of our system so there's no external work done now that might be a new or confusing idea to some people so let me just say if there's forces between objects within your system then those forces cannot exert external work and they cannot change the total energy of your system only forces exerted on objects within your system from outside of your system can change the total energy of your system so when this ice was not part of our system it was exerting an outside external force on Walter and the energy of our system changed we started with kinetic energy we ended with no energy but now that the ice and Walter are part of our system this force of friction is no longer external its internal exerted between objects within our system and so it does not exert any external work it's just going to transform energies between different objects within our system so that's why we write this zero here there'd be no external work done if we choose the ice and Walter as part of our system and this would have to equal the final energy and we know where this energy ends up is started with kinetic energy and it ends as this extra thermal energy in the snow and Walters feathery coat so I could write that as e thermal but I know how much thermal energy was generated this just has to equal the amount of work done by friction so even though this work done is not external it still transfers energy between objects within our system so when we write that that work was a negative fkd down here we mean that the force of friction took fkd from something and turned it into something else and that's all we need up here we need an expression for the thermal energy but if friction took fkd and turned it into something else the thing it turned it into was the thermal energy so that value of fkd that magnitude of the work done was how much energy ended up as thermal energy people might find that confusing they might be like wait a minute why why do we have this with a positive here and not a negative well in this work formula this negative is just saying that the force of friction is taking energy from something if you take energy from something you're doing negative work on it if I gave energy to something I'd be doing positive work so this negative sign and the work done just means that the force of friction took this much energy from something and turned it into thermal energy so when we run a write down how much thermal energy did we end with well we ended with the amount that we took so we took fkd the thermal energy ended with fkd and i can still set this equal to the kinetic energy that walters started with and i get the same formula I ended up with over here because I had to you because we're describing the same universe in the same situation so no matter what story you tell you should get the same physics in the end and we do but some people prefer one to the other some people like thinking of friction as a negative external work and not including the energy within the surfaces as part of their system and some people like including those surfaces as part of their energy system and just including that thermal energy on the e final side which is fine you can do either you just can't do both either the surfaces part of your system and you included in your final energy or the surfaces not part of your system and you include it as external work but you can't say it does external work and it gains some final energy over here because it's got to be either part of your system or not part of your system so long story short you can basically just think about the thermal energy generated by friction as fkd this is a formula the to let you solve for the amount of thermal energy generated when two surfaces rub against each other and we can take this idea a little further the force of kinetic friction is going to be equal to the coefficient of kinetic friction times the normal force between the two surfaces so we can rewrite this this thermal energy term can be re-written as mu K times FN times D and you might say well that's not all that remarkable it just looks even worse than it did before but if you keep going you realize that the normal force for something just sitting on a surface is just going to be M times G and we still multiply by the D but now that we've replaced the normal force with mg we notice that the mass cancels and this should blow your mind this means no matter what the mass of this penguin is if he starts with the same speed as some other penguin that's more or less massive he'll slide the exact same distance now some people will object they'll be like wait a really massive penguin is going to have a lot of inertia it really wants to keep moving it should slide farther but other people would say nunna the less massive penguin should slide farther because it has less frictional force but that's why it doesn't matter these two confounding effects exactly cancel in other words the more massive penguin does have more inertia and has more friction and the less massive penguin has less inertia but has less friction so the mass ends up canceling and all penguins no matter what their mass are slide the same amount if they start with the same speed and this also means that two cars are really tiny smart car and a huge SUV if they've got the same tires they'll have the same coefficient of friction and if they start with the same speed and slam on their brakes they'll both skid to a stop in the same distance again a lot of people would think that the really massive SUV has to slide farther but that massive SUV that has more inertia also has more friction so it stops in the same distance as the smaller smart car that has less inertia and less frictional force so even though the coefficients of friction would be the same for both cars the force of friction is going to be bigger for the larger car all right so to wrap this up let's do an example problem let's get rid of all this let's say Walter steps it up he's going to the extreme games of sliding and he's going to slide down starting at the top of this ramp that's completely icy this ramp has no friction but Walter has no fear so he goes to the top it's four meters tall he starts at rest Walter slides down speeds up but now he hits a patch that does have friction so Walter slides across this patch over some distance D and then comes to a stop and if we know Walter started four meters high and the coefficient of friction along this path is zero point two we could figure out what was the distance Walter slid on this horizontal surface before coming to a stop so let's do this we're going to use conservation of energy same formula we did before the initial energy of our system plus any external work done that is to say energy headed to or subtracted from our system has to equal the energy that we end up with in our system now what do we want to pick as our system personally I like choosing everything that could get energy is part of my system that way I never really have to worry about any external work because every force will just be internal so let's choose Walter the earth the ice the snow the incline everything's gonna be part of our system and let's consider our initial point to be when Walter started at rest at the top of this incline and our final point over here where Walter slid to a stop so what kind of energy is our system start with well Walter was up here he was at rest so he had no kinetic energy but he did have gravitational potential energy so that's MGH and again there was no external work done because even though there was friction down here I'm going to include this surface and Walters feather e-coat as part of our system so there was no external work done but there will be thermal energy generated and it will end up in our system so the thermal energy generated is going to be the force of friction times the distance that Walter slid across the surface that had friction so I'm not going to include the surface right here because there was no friction there I'm only including this distance right here and we know that the force of friction is going to be the coefficient of friction times the normal force and then we still multiply by the distance that Walter slid and because Walter was sliding over a horizontal surface the normal force on Walter is just going to be equal to the force of gravity on Walter so we can say MGH is going to equal the coefficient of kinetic friction and then FN we're going to replace with mg since the normal force is just equal to mg and I still multiply by D again something miraculous happens DMS end up canceling Walter could have been 100 kilograms or 2 kilograms would have slid the same amount finally I can solve this for D I'm going to say that D is going to equal gh Oh actually turns out the g's cancel to could have done this on the moon wouldn't it matter even if the G the gravitational acceleration was different Walter would still slide the same amount so I'm just going to get the D equals H and then I divide both sides by this coefficient of kinetic friction and there's my result it's really simple in other words H was 4 meters and then I divide by the coefficient of friction which was 0.2 and I get the Walter's going to slide 20 meters before coming to a stop so recapping when there's a force of friction acting on an object you can use conservation of energy by treating that frictional surface as part of your system in which case you would include it as a thermal energy on the final side or not including that surface as part of your system in which case you would include the same term with a negative sign as the external work done on your system regardless of what you do the thermal energy generated in such a situation is going to be the force of friction multiplied by the distance through which the object slides