mechanical energy going to be the topic of this lesson in my brand new General Physics playlist which will cover a full year of University algebra based General Physics now mechanical energy simply put us just the sum of both kinetic and potential energies and to talk about these we'll talk about both conservative and non-conservative forces we'll also talk about the work energy theorem the work done by non-conservative forces and definitely the conservation of mechanical energy as well my name is Chad and welcome to Chad's prep where my goal is to take the stress out of learning science now if you're new to the channel we've got comprehensive playlists for General chemistry organic chemistry General Physics and high school chemistry and on chadsprep.com you'll find premium Master courses for the same that include study guides and a ton of practice you'll also find comprehensive prep courses for the DAT the MCAT and the oat all right so mechanical energy simply put is the sum of the kinetic energy and the potential energy and these are the abbreviations we're going to use here ke for kinetic energy p e for potential energy but you should know that these are not proper and not universally used by all textbooks a lot of freshmen physics textbooks will use them but the proper ones are actually K and it turns out uh my personal preference is definitely k e and p e it makes it much easier to remember what they stand for so but notice that your textbook may be a little bit different in that regard so but these are the ones we're going to use here and we'll start with kinetic energy this is the energy of motion so in this case it has a formula one half MV squared now for the next several chapters this is the only type of kinetic energy we will have presented it's called translational kinetic energy it turns out and it's the energy of something as it uh you know undergoes motion through space but we will learn another type later on several chapters down the road called rotational kinetic energy involving rotational motion instead so but for now when we say connect energy for at least the next few chapters will be specifically referring to translational kinetic energy and you should associate it with this theorem now if we take a look at that theorem there we can see a couple different relationships we can see that kinetic energy is proportional to mass so you double the mass of an object keep its velocity the same and its kinetic energy will have also doubled that's proportional so but we can also see that the kinetic energy is proportional to the velocity squared and so in this case if you double an object's velocity while keeping its mass the same you haven't well let's get an e in there as well so you haven't doubled the kinetic energy in that case if you double the velocity it's double squared you've actually quadrupled the kinetic energy in that case as well so a couple different relationships you should know there now we also want to talk about potential energy we'll find out it's kind of like the potential of being able to do work for an object so before we can talk about that there's a couple other terms we really want to introduce and there are conservative forces and non-conservative forces and all forces are generally going to fall into one of these two classes now conservative forces get their name because if the only force is acting on an object are these conservative forces then it's mechanical energy is going to be conserved that's going to be a big highlight towards the end of this lesson talking about the conservation of mechanical energy however if you've got non-conservative forces acting on an object then its mechanical energy will not be conserved all right so if we look at conservation I'm sorry if we just look at uh conservative forces gravity is kind of the textbook example we give to you at this point so it turns out force is conservative if you move it from one point in space to another and no matter which path you take it's going to require the same amount of work so and gravity is a great case if I want to move this marker from this spot to this spot so because I'm moving it upward it's going to take some work to accomplish that Against Gravity now it wouldn't matter if I went directly there in a straight line or if I went like on a zigzag pattern at all to get this marker there as long as I start in the same place and end in the same place the same amount of work would be done as far as gravity is concerned now we have what are called non-conservative forces that don't operate this way and we usually think of these as being somewhat of what we call drag forces friction and air resistance are the most common examples so and these always oppose the motion and so the longer your path typically the longer they get to act and the more they're going to decrease an object's mechanical energy and therefore that mechanical energy is not going to be conserved and along the way whether it be friction if you kind of look at you know rubbing your hands together you can find that your hands warm up and so these non-conservative forces take and convert some of the mechanical energy of an object either into things like heat or sound are the most common and so as a result an object is going to lose mechanical energy when these non-conservative forces are at work and so again going back to this path of the marker as the marker moves on a zigzag path or something like that or a straight line path as long as there's not you know any significant amount of say air resistance or friction so then we would say that only conservative forces are acting on it so and its mechanical energy is going to be conserved the sum of its kinetic and potential energies would be conserved and it wouldn't matter path as long as I start in one place and end up in a second place the path is independent if only conservative forces are acting on it so so those are those so it turns out this idea of conservative forces when these are possible and present that is when potential energy is present so it turns out gravity so as I move this object up it now has the potential to fall back down and if it you know hits something it has potential to do say work as it comes into contact with some other object like say pounding a nail in whether I pound an alien with a hammer or whether I drop the marker and it hits the head of the nail if it's moving fast enough it might pound it in a little bit too and do some work on that nail and so potential energy is kind of the potential to do work and potential energy is possible when there are these conservative forces like gravity or we'll see the electrostatic force in the next semester and things of that sort and so potential energy the only working version of potential energy we'll talk about right now is called gravitation additional potential energy and it turns out it's equal to mass times gravity times the height now some books will use an H right here and it's personally my preference is to use hate h as well but we'll use y so it's probably a little more common in most textbooks these days and so y refers to the height above some you know level uh and oftentimes the zero level is kind of uh often ground level but it might just be the lowest possible point on some object's trajectory so if I throw this marker so and it comes down and hits the ground we might call the ground level zero but if I'm on top of a skyscraper and I throw this marker and then it just Lands still on the roof of the skyscraper well then I might call this top of the skyscraper the zero level it's kind of arbitrary what we're going to find out is that the change in height is really the important thing so but typically the lowest point in the trajectory is going to be the zero point and that's quite often the ground all right so for potential energy a couple different relationships here as well we can see that potential energy is proportional to an object's Mass you double the mass you double its potential energy it's also proportional to the height so double the height and you double the potential energy there as well now technically it's proportional to gravity as well so but most examples you're going to be discussing are just going to be on the surface of the Earth and on the surface there's that gravity uh the acceleration due to gravity 9.8 meter per second squared is going to be constant now if you move to the Moon where there's only one sixth of the gravity well then we're going to find out yeah that would definitely affect the potential energy as well but for most the cases you're going to see we're going to be at the surface of the Earth and so gravity is going to be constant in those cases all right so we've got a couple other things to talk about the next one is what we call the work energy theorem and it talks about when there's a net amount of work done on an object and so it turns out when you do Network on an object it's going to change it's kinetic energy so it will either move faster or slower depending on if it's net positive work or net negative work so but if you don't do network if there's no network if the network is zero joules if you will or just simply zero then the kinetic energy is not going to change and that object's velocity is going to remain constant so it works both ways so for example let's say we take this lovely marker right here so and let's say I raise it up at constant velocity or I lower it down at constant velocity well because this is being done at constant velocity in either case there should be no change in kinetic energy velocity is constant right so the V stays constant the marker is not changing its mass well then kinetic energy is going to remain constant and this change in kinetic energy should be zero which means the network should be zero and that's exactly true so let's consider the case where I'm raising it up well as I raise it up at constant velocity because it's at constant velocity there's no net force no net acceleration and so what you find is that there's an upward Force so which is the pushing force on my hand or you might call it a normal force from my hand but then there's also gravity pointing down now because I'm moving upwards the force of my hand and the displacement both point up and that's going to be positive work so but it turns out the force I'm going to have to apply with is going to have to be perfectly balancing out the weight which points down and so as I move up the weight still points down but the displacement points up and that's going to be negative work and it turns out the positive work being done by my hand is exactly equal to the negative work being done by gravity and that's why the network is going to be zero and why the change in kinetic energy will also be zero now on the other hand let's say instead of having you know constant velocity here so let's say instead the force that I'm pushing up with my hand and the Force that gravity is pulling down with I.E its weight are not equal and let's say that you know the force of my hand is greater and so instead of moving up at constant velocity all of a sudden now because there's an overall net upward Force because again the force of my hand going up is bigger than gravity going down then all of a sudden this thing's going to accelerate Upward at least as long as it's in contact with my hand and while it's accelerating upward we can see then oh well now the force upward which is in line with the displacement is bigger than the force downward which opposes the displacement and so the work pointing up is going to be larger again force times displacement since it has a larger Force than the work going down the work going up was positive the work going down was negative and so all of a sudden now there's a net positive work and if there's a net positive work then we're going to see a positive change in kinetic energy an increase in this case in kinetic energy increase in velocity that's kind of how that works that's that work energy theorem so if you know all the forces acting on an object and you can kind of calculate all their contributions uh if you will uh that they would uh that they would contribute to the network then you could figure out the net kinetic energy or vice versa if you need if you can figure out what the change in kinetic energy is going to be then you could figure out what network must be being done or performed on an object all right next we'll move on to talking about these non-conservative forces now we're going to find out in a little bit here that conservation of mechanical energy that that's true again when there's only conservative forces acting on something that's how we kind of uh jumped into this in defining conservative forces so now if you have non-conservative forces though it's not going to be true and you're going to find out that the work done by non-conservative forces is always negative they always oppose the motion because they're always opposing the motion while force and displacement are in opposite directions that's negative work and negative and again non-conservative forces always are in the opposite direction to the motion and so the work they do is always negative and they always therefore decrease the total amount of mechanical energy well it turns out that the work done is going to be equal to the sum of the change in the kinetic energy and the change in the potential energy or simply put it's going to equal the change in the mechanical energy so whenever you have non-conservative forces going on there it will be a negative sum because there always going to be decreasing the total amount of mechanical energy so when we have only conservative forces we're going to find out that the uh the mechanical energy will be conserved but when we have non-conservative forces we're going to find out that the mechanical energy is going to decrease corresponding to a negative amount of work done by those non-conservative forces and then finally we'll talk about the conservation of mechanical energy here so and it just simply says that for an object only experiencing conservative forces no non-conservative forces that the kinetic energy initial plus potential energy initial will equal the kinetic energy final Plus potential energy final totally conserved now we could expand this out just a little bit so and we could say that one half MV initial squared plus m g y initial equals one half MV final squared plus mg y final and expanded out that way what you're going to find out though is that I usually prefer writing it out the first way first when doing certain calculations involving the conservation of mechanical energy and the reason being is you're going to find out that oftentimes so out of these four terms one or more of them are going to equal zero either in the initial or final conditions and so you're going to be able to just cross them out and so why write out the expanded version and write a bunch of stuff if you're just going to lose some of those terms because they are zero in such cases so but take your pick whether you want to represent it this way or this way it is up to you we'll work some problems you'll see kind of how it presents itself all right and now we're ready to present some problems so we've kind of set up a conceptual framework introduce some equations and now we're kind of ready to do a A random assortment of problems that are going to kind of give examples of all of these so we'll start with the first one a 1000.0 kilogram car has a velocity of 20.0 meters per second what is its kinetic energy and then how much Network must be done to bring the car to rest all right so we want the kinetic energy first you know in fact let's make a little room here we can see that kinetic energy is simply equal to one half MV squared and we can just plug in chug so we're told it's a 1000.0 kilogram car moving at 20.0 meters per second don't forget to square that that's one of the more common errors students will make in this section is just they write out the equation I just forget to include the square and then they refer to include it in the calculation so I was about to pull out the calculator but we can probably do this one in our heads as well so 20 squared is 400 400 times a thousand is four hundred thousand times a half would be two hundred thousand and so in this case our kinetic energy is going to equal 400 000 joules now it's not uncommon especially if you're getting like a multiple choice question it's not uncommon to see answers especially for very large numbers of joules expressed in kilojoules and so potentially you could also see this to convert to kilojoules we divide by a thousand and so you can also see it as 400 kilojoules now we've got to worry about sig figs here for a second so this guy's definitely got a a large number of sig figs five this guy's only got three we're gonna need three sig figs here well that's a pain in the butt with a bunch of zeros and I really prefer not drawing lines over zeros most of the time so I prefer to write in scientific notation so in this case I'm going to make this one 4.00 times 10 to the fifth Joule so one two three four five decimal places moved so times 10 to the fifth since the number bigger than one is a positive power of 10 or since I move the decimal to the left however you look at it so same thing over here we're going to make this one four point 0.00 times 10 to the second kilojoules so and again writing them in scientific notation my preferred way to get the proper number of sig figs rather than trying to put a line over some zeros to show that they're significant and things of that sort cool so that's the first half of this question second one though is how much Network must be done to bring the car to rest so it's got initial velocity of 20 meters per second we're trying to bring it to rest I.E where its velocity is zero and this is where that work energy uh work energy theorem comes into play and the key is if you notice it said Network right in the problem and that's what's in the work energy theorem so and we're doing a change of kinetic energy we're taking it from 20 meters per second velocity down to zero this is exactly what this theorem is made for and so in this case work net is equal to that change in kinetic energy which we might simply write again as kinetic energy final minus kinetic energy initial and so if you look here well we want to end up at rest which is the final amount of kinetic energy of zero if you plug a 0 in for velocity you'd get zero kinetic energy they're proportional or at least proportional to the square but still 0 is going to get you zero and so in this case that's a zilch minus the initial kinetic energy well we could go back and figure that out again but we already did right we already got the 400 kilojoules or 400 000 joules in there and so in this case I'll make it 4.00 times 10 to the fifth joules and 0 minus that gets us a negative number negative 4.00 times 10 to the fifth joules for our Network and does it make sense that it's negative so again when we do Network here that's negative this means the object is slowing down and indeed it is it had initial velocity of 20 meters per second it ends up at rest with no velocity that definitely corresponds to a loss of kinetic energy and therefore negative work being done so when work net is negative the object is slowing down when work net is positive that object is speeding up instead let's take a look at another okay so the next question here says a stone has dropped from a height of 7.35 meters what is the final velocity just before it reaches the ground and this problem should look somewhat familiar it's a kinematics question right and so you could totally solve this using kinematics so but a lot of students struggle with kinematics more than what we're about to do so notice it even says ignore air resistance at the end of the problem and if we're ignoring air resistance a non-conservative force this would mean we only have conservative forces actually on it and the conservation of mechanical energy G is going to apply what's nice is there's only one conservation of mechanical energy equation now it turns out you know later on we're going to find out that we can add some like rotational kinetic energy terms and we can add other types of potential energy into there as well so it can get a little more complicated truth be told but it's still in principle just one equation so that the initial amount of kinetic potential energies is equal to the final amount of kinetic and potential energy whereas with kinematics again we had at least four equations and students often struggle which one do I use and which situations and negative sort and if you did you might find that there's a number of kinematics problems that are now much easier to solve with the conservation of mechanical energy so let's see how this works so in this case again we've got kinetic energy initial plus potential energy initial get that right equals kinetic energy final plus potential energy final so and before we expand this out let's just take some things into account here so initially it's dropped from rest and if it's dropped from rest up here then it has no velocity and therefore no initial kinetic energy so and again here the lowest point in its trajectory is the ground level and so we'll probably define the zero height as ground level and so right as it's hitting the ground at the very end its height is zero and so it's potential energy with a height of zero is also going to be zero and that's the final potential energy and so here we're left with a situation where the initial amount of potential energy is going to equal the final amount of kinetic energy and so we can substitute our expressions in so mgy initial equals one half MV final squared and notice what falls out Mass falls out of this which shouldn't be too surprising the only conservative Force we have acting on this object and the only force in general is gravity and gravity doesn't care about your mass if you recall we said if you drop a feather and a bowling ball well we know the bowling ball is going to fall faster except that's due to air resistance if you did that inside a vacuum or on the moon where there's no atmosphere they would fall together all the way down and acceleration uh in a vacuum on Earth surface of this would be 9.8 meters per second squared on the moon it would be a little bit less or quite a bit less so but they would fall together and accelerate together would be the key and so we shouldn't be surprised that mass is going to fall out of this equation here with gravity being the only force acting and so if we go ahead and solve for velocity here we're going to get V final squared equals so multiply by 2 so 2 g y i and then we'll take the square root of both sides so 2 times 9.8 meters per second squared times in this case I've got our initial height was at 7.35 meters cool now it turns out this is fairly common result and it turns out if something is dropped from rust some people actually teach the equation that the final velocity will equal the square root of 2gy or 2 GH so I hate just memorizing equations that apply in certain situations I really like just setting up how to solve the problem from first principles like setting up conservation and mechanical energy things of A Sort so but if you've been taught to memorize that fair enough all right so here we've got the square roots of 2 times so let's put this all in parentheses square root parentheses 2 times 9.8 times 7.35 I'll close those parentheses and we're going to get 12.0025 meters per second which I'm going to round a let's see how many sig figs we're gonna round that to three sig figs that's going to be 12.0 meters per second okay so let's take a look at this just a little bit different way how might we have solved this using kinematics in this case in this case following a height of 7.35 meters we might have said well we don't know anything about how long this thing is falling you know time and so perhaps we would have used the final squared equals V initial squared plus 2 times a times Delta X and this is how we would have solved this back with kinematics and we would have said we want that final velocity squared the initial velocity though was zero and that falls out and notice what happens here you're going to take the square root of both sides you're going to have two times the acceleration in this case due to gravity 9.8 meters per second squared and then times the displacement which was 7.35 meters and ends up with the same exact calculation which obviously is going to lead you to the same result of 12.0 meters per second so but again a lot of students would struggle getting to the point of like which which of the kinematics equations do I use so whereas again in the context of conservation of mechanical energy there's only one equation life is good all right so our next question here says a baseball is thrown straight upward in the air with an initial velocity of 50.0 meters per second what is the maximum height reached by the ball ignore air resistance and again this is going to look a lot like a kinematics problem and again you can totally solve this with kinematics or we could take into account that we're told to ignore air resistance there's no non-conservative forces and so the conservation of mechanical energy should totally apply now if we take a look at the entire trajectory here so this thing's going to be slowing down slowing down slowing down until it reaches its maximum height here and at that point so its final velocity is going to be zero and that's going to be basis here so here we're going to be at a maximum of kinetic energy up here we're going to be at a maximum of potential energy in fact at its initial point it's all kinetic energy because it's at a height of zero and at the end it's all potential energy because its velocity is zero and therefore it's kinetic energy is zero and it's similar to the setup we just had in certain respects so but we set it up the same way so kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final and taking into account what we just said so initially it has no height so it's potential energy is zero and at the end of its at the top of its trajectory when it reaches the max height that's when its velocity and therefore kinetic energy is zero and so in this case our kinetic energy initial one half MV initial squared is going to equal a final potential energy mgy final cool and from here it is some plug-in and chugging now before we process this though let's see how much we've approached this with uh kinematics well again 50 meters per second initial velocity it's going to be slowing down by 9.8 meters per second every second due to gravity right so which is roughly 10 meters per second per second and so after one second it's going to be slowed down to like 40 meter second after a second second down to 30 meters per second approximately and so it's going to take roughly five seconds for its velocity to slow down to Zero from what we learned at kinematics okay so it's a five second Journey for the the way up if initial velocity is 50 meters per second we know its final velocity is zero it's really easy to see that the average velocity therefore is 25 meters per second 50 plus 0 all over two well an average velocity if you recall we had displacement equals average velocity times time was one way we did that and so an average velocity of 25 meters per second times five seconds would be roughly 125 meters and so that's what I'm expecting to get here when we do this calculation now it's not going to be exactly 125 because again we said well gravity is about 10 meters per second per second but it's really 9.8 but it should be in the ballpark here of roughly 125 meters as verified with our kinematics version of this calculation so let's do some plug-in and chugging here notice what conspicuously falls out again is the mass so and in here we actually want to figure out again that Max height so it's gonna be YF there so in this case we're going to have one half V initial squared all over G getting us that Max height so and that initial velocity was 50.0 meters per second don't forget to square it and again all over 9.8 meters per second squared gets us our final height which is the max height and once again we can ballpark this in our head so 50 squared well 5 squared is 25 and add two zeros onto that so 2500 2500 divided by roughly 10 would be 250 and 250 times a half would be 125 and so we can see this is going to come out to roughly 125 meters just like we thought it would but let's get an exact number on there and round it to three sig figs so we're going to have 0.5 times 50 squared don't forget to square it divided by 9.8 and we're going to get 127.55 so we'll round that up to 128 meters for three sig figs cool and again a lot of students really prefer using the conservation of mechanical energy than those old kinematics equations now they're going to be places where the kinematic equations are more convenient or where the conservation of mechanical energy doesn't apply or won't work so but for many of the kinematics problems this conservation mechanical energy uh might be a godsend for some of you all right so the next question says a 10.0 kilogram box slides from rest down a 5.0 meter long frictionless ramp inclined 30 degrees above the horizontal what is its final velocity at the bottom of the ramp okay now we could again use some kinematics for some Newton's Laws of Motion in the combination they're able to figure this out we could set up a free body diagram and figure out that the component of this object's weight down the incline is mg sine 30. so and then set mg sine 30 equal to ma according to Newton's Second Law solve for that acceleration and then figure out how long does it take to go five meters which would be a lot of work or we could realize that again there's no friction here because there's no friction so then there's no non-conservative forces and the conservation of mechanical energy is going to apply and you might be like but Chad this thing isn't falling straight down and conservative forces don't care they don't care about the path as long as we go from one height to another height it doesn't matter what path you take you take to get there and so the only thing that matters is the change in height in fact it wouldn't matter if we went straight down falling or falling down the ramp but ending up at this height compared to this height that's all that ultimately matters now if we take a look at this then conservation of mechanical energy kinetic energy initial plus potential energy initial equals kinetic energy final Plus potential energy final and again this thing is starting from rest if it's starting from rest and its initial velocity is zero which means it initially has no kinetic energy cool the lowest Partners trajectory is when it gets to the bottom of the ramp and so we'll call that the zero line and so at the end it has no height and so at the end it has no potential energy now one thing to note here we're taking the extremes here because it's really convenient here it's probably a relative relevant to this problem and stuff so but we're getting that the initial potential energy equals the final kinetic energy but again one thing you should realize anywhere along this trajectory is that the kinetic and potential sum will always be the same when we have the conservation of uh mechanical energy so even if we're in the middle of the trajectory here so in the middle it would have both some kinetic and some potential but the total between these two would equal the same total we're going to get here or the same total we're going to get here it's always the same when there's only conservative forces acting on the object okay so all the way down that ramp it's going to be getting faster and faster and faster so it's kinetic energy is going to be going up but all the way down the ramp its height is getting lower and lower and lower and so it's going to be losing potential but the sum of the two will remain constant all the way through all right so set this up then we've got m g y initial equals one half MV final squared and this should look a little bit familiar we'll see that the mass falls out yet again whether you're on a ramp or not gravity doesn't care about your mass as long as there's only conservative forces acting on it all right so in this case solving for that final velocity this should look familiar to g y I take the square root and there's our plugin and chugging but we got to know what that initial height was well we know the ramp is five meters long but we don't actually have the height above kind of the ground level zero level here in this case so we got to figure that out so you're gonna have to remember a little bit of your trigonometry here so if you notice this is the side that is opposite the 30 degrees and so sine of 30 would equal opposite over hypotenuse in fact let's write that out so sine of 30 degrees equals opposite over hypotenuse so it's the opposite side that we want and we can see that we can hypotenuse times the sine of 30 degrees which in this case is going to be 5 meters times sine of 30 degrees now I like using 30 degrees in practice problems quite a bit for lecture presentation because it's a nice convenience sine of 30 is one half and one-half times five is 2.5 meters so but obviously this could be any angle but for purposes of presenting a lecture it's just nice using numbers that come out nice and round like that all right so we'll get the square root of 2 times 9.8 meters per second squared times a height of 2.5 meters equaling our final velocity and we'll let our calculator go to work for us here so once again we'll take the square root parentheses 2 times 9.8 times 2.5 I'll close out those parentheses and it doesn't like my syntax let's try this again what did I do wrong I'll use the wrong bracket at the end all right that's going to get us oh I made the math nice here it comes out to exactly seven meters per second so in this case uh we're limited to two sig figs by the 5.0 meters and so we'll make this 7.0 meters per second okay so the next problem is a little bit related to this one so it starts off a 10.0 kilogram box slides down a 5.0 meter long ramp inclined 30.0 degrees above the horizontal if the velocity of the box at the base of the ramp is 6.0 meters per second not seven 6.0 meters per second how much work was performed by non-conservative forces on the box so here's the deal if it only has a velocity at the bottom of six meters per second again if conservation if mechanical energy is conserved in the conservation of mechanical energy applies we just figured out it would be 7.0 meters per second well now in this next example it's only six and so we can see that oh mechanical energy was not conserved when it's mechanical energy not conserved well when there are non-conservative forces at work and how do you calculate the work of those non-conservative forces if you recall we always said it's negative they're always going to lower the total mechanical energy and it's just equal to the change in mechanical energy all right so at the top of our trajectory here we have some amount of kinetic and potential energy so and then at the bottom here again we're going to have some amount of kinetic plus potential energy but because gravity is doing work in this case I'm sorry not gravity because friction is doing work here and opposing the motion we should expect this to be lower and again the difference between these two will be the work done by friction in this case by non-conservative forces and that work will be negative and so with our change we'll call this we can use initial and Final in this case so our work is therefore going to equal the change in mechanical energy which we could look at as change again in kinetic plus change potential which would be kinetic energy final well let's get that e in there kinetic energy final minus kinetic energy initial Plus potential energy final minus potential energy initial now again because we already have the difference in the final kinetic energies with and without this friction seven meters per second velocity would allow us to calculate kinetic energy in one case six meter second in the other case we'll get to take the difference in life and be good but had we not already performed that problem this is where we'd be going with this and this work done by the non-conservative force of friction and so if we take a look at this so kinetic energy initially is a big fat zilch it's dropped from rest so that term is zero so and then the potential energy finally is zero and so we can eliminate some of these terms here so in this case then our work done by non-conservative forces is going to equal one half MV final squared plus m g y initial oh there's a minus huh because plus minus plus a negative let's also track that off all right so let's do some plug-in and chug in so notice does mass fall out of this equation by the way like it's done before well mass is here mass is here can't we just take it out well no we can't in this case if we could Factor it out but we can't actually remove it because it's not equal to zero on this side so Mass totally matters it hasn't mattered in the past it totally matters now and so we'll need that 10.0 kilograms now final velocity in this case was 6.0 meters per second don't forget to square it minus again 10.0 kilograms times gravity 9.8 meters per second squared and that initial height again was that 2.5 meters that 5 sine 30 if you will cool and we can do some plug-in and chugging from here all right so 0.5 times 10 times 6 squared minus 10 times 9.8 times 2.5 and we're going to get Negative 65 here in this case foreign and that is joules and we need two sig figs based on the 6.0 meters per second and the 5.0 meters and that is two sig figs so there's your work done by friction it came out negative just like we expected it to for a non-conservative force uh and life is good now a couple other things to look at here so not the only way we could have approached this problem and a lot of students would have said well work work done by friction in this case will work as force times displacement and they would have looked at this and said oh okay well work at Force time displacement so in this case the force of friction is normal force times the coefficient of kinetic friction in this case and then we said oh the normal force is equal to mg cosine 30 great but the coefficient of kinetic friction is not given and at this point some students might be inclined if this was on an exam to go professionally like hey you didn't give us the coefficient of kinetic friction at which point your professor would say I didn't need to go back to the drawing board because that would be one approach to take at this there's more than one you know more than one equation now we've learned to have work in it and that one wasn't going to work force times displacement wasn't going to work because we had no way of actually calculating the force due to friction without the coefficient of kinetic friction being supplied so just one thing to keep in mind one other thing to take a look at here real quick so just a couple things so if we take a look at that initial total mechanical energy the kinetic and potential combined again that initial kinetic was zero but the potential here was the 10 times 9.8 times 2.5 let's work that out one more time so 10 times 9.8 times 2.5 and that's 245 joules so initially the total mechanical energy which was all potential was 245 joules now at the end it's all kinetic and none potential because we're at a height of zero and it was the one-half MV squared and so in this case uh 6 squared is 36 times 10 at 360 times a half is 180 and so at the end here this total comes out to 180 joules well the work done by the non-conservative forces just equals the loss of mechanical energy you might be like well how much mechanical energy would lose well from 245 joules down to 180 joules we lost 65 joules and the work done by the non-conservative force is that negative you got to make sure it's negative negative 65 joules so just another kind of conceptual way to look at it instead of just like I just plugged and chugged equations so you just understand that oh we had so much mechanical energy to begin with we didn't end up with the same amount mechanical energy was not conserved and so there must have been some work done by some non-conservative forces and it's the difference of the two and again if you do final minus initial that's how you'll get it to come out negative if you want a formula or you can simply remember that it's the difference between the two but it has to be negative for the work done by a non-conservative Force if you have found this lesson helpful consider giving it a like happy studying