**Concept:** Conservation of Total Energy

Hey guys, we're now going to start talking about conservation of energy which is one of the most important topics in physics and you have to be really good at it, I'm going to start by giving you some conceptional information and then we're going to jump into some problems let's check it out, so in this video I want to talk about the 2 rules for conservation of energy and you should understand these are conceptual level. Both rules have to do with the idea of a system so conservation of energy often refers to a system of objects, now a system of objects is just an arbitrary group of objects, group of objects arbitrary means that you make it up you determine what the system is so for example if I have 2 boxes going against each other I could say let's consider the system to be just this box and if this box has an energy of 10 joules the system has an energy of 10 joules but I can also say well I want the system to be these two boxes here and if they each have energy of 10 Joules then the system has an energy of 20 Joules, OK? The energy of the system is just the summation of all the energies of the individual objects and it's up to you to determine what the system looks like, some problems will tell you the system is made up of these 2 things and then we have to go with it, cool? So, you have to understand what systems are it's just a group of objects so you can think of this as sort of an imaginary container of objects, alright? System collection of objects.

So, the first rule says that the total energy of a system, system, right? groups of objects is conserved if the system is isolated, OK? The two key words here are total and isolated, total energy well remember total energy let's call that E is a combination of mechanical energy plus non mechanical energy those are the big groups of energies you could have, the idea here is that if the total energy of the system is conserved this is always the same number but you could lose some mechanical into non-mechanical and vice versa as long as the total number is the same, one analogy is if you have sort of a Styrofoam cup and there's some coffee here and some air here and it sealed if you close Styrofoam cup the heat will stay there for much much longer than if you open it and that's because this is an isolated container so some of the heat will go from here to here but it still stays trapped inside of the Cup, so that's what kind of an idea for the isolated system though the isolated system doesn't have to be physically closed in our bounds, OK? A system is isolated lets the find what it is an isolated system so the energy is conserved if the system is isolated and the system is isolated if there are no external forces, OK? But obviously you have to also know what is an external force? So, no external forces doing work on the object so you could have external forces but if you have one there has to be another one to cancel it, there is no external work on the system so the work done by external forces that's what that means equals 0, OK? No work done by external forces, forces can be either internal or external depending on how you determine the system for example if I have the system made up of just this one object here then this guy the second block is considered external to the system it's not part of the system, OK? The best way to look at forces internal vs external is to do an example, alright? But the basic idea if I could summarize all of this everything in this box in one line it would be that if you have if there is no external work if the work done by external forces is 0 then you have that the system that the total energy is conserved, OK? That is sort of the one line idea there, now how to determine force are internal or external? So, let's see here we have an example box slides with a force floor lets the box goes to the right of the velocity V, I have kinetic friction kinetic is the rubbing friction so it's opposite to motion meaning to the left over here and this friction is between the floor and the box, it's the only force doing work on it, OK? We have MG and normal but they're perpendicular to motion so they don't do any work for each one of these choices of the system below for each of these choices of the system below we're going to ask a bunch of yes or no questions here, so first we're going to say that we want to call the system remember we pick we're going to say the system is just the box and then we're going to say the system is the box with the floor, it's a little bit more inclusive of a system there so here is friction internal? No friction is not internal, friction is external to the system because we determine that the system is just the box and the friction is sort of outside of the system because it's between the box and the floor and the floor is not part of the system, here the floor is external to the system, OK? Is this system isolated? Well no because there is a force that's coming from outside of the system, OK? Think about if you now open this cup and you poured some stuff in there, right? Something like that so this is no longer isolated because there's an external force that's doing work, the work done by external force is not 0, OK? Is the energy conserved? No, the energy the total energy is not conserved because now that there's work and remember work is energy there is work being pumped into the system or out of the system the total energy is going to be changing, OK? No and that's because the system is not isolated, OK? Here it's a little bit different because we now include the floor so now friction is an internal force because it's part of this, the system is isolated because the only force doing work on the system is internal, OK? So, it is because friction is internal and energy is conserved because the system is isolated, OK? These things are all dependent on each other, so if the force is not internal the energy not conserved if the forces are all internal then the energy is conserved That's it for this one.

**Concept:** Conservation of Mechanical Energy

Alright so the first rule of conservation of energy had to do with the total energy of the system, the second one has to do with just the mechanical energy of the system, remember the total of energy is made up of mechanical energy and non-mechanical energy and we're now going to talk about this one and this rule is actually the one that's the most important at this point in physics, OK? So the mechanical energy of the system is conserved if the system is conservative, now this is kind of a crappy rule because it doesn't really tell you much and just like how the other rule worked it will depend heavily on the definition of conservative so a system is conservative and this is really crappy too if no non-conservative forces do work on it, so if there is a non-conservative force and I'll tell you what it is in a second if there's a non-conservative force acting on the system then the system is not conservative and if the system is not conservative It means that the mechanical energy of the system is not conserved so conservative system has to do exclusively with mechanical energy, total energy has to do with isolated system, mechanical energy has to do with conservative system, OK? So, this means that the work done by Non-conservative forces equals 0, so remember we have internal force and external forces we also have conservative forces and non-conservative forces so there's two ways of categorizing forces, now conservative vs non-conservatives is pretty easy for the most part is there's really only 4 forces you have to worry about and I'll tell you which one is which, conservative forces are going to be gravity or more technically the weight force, right? Gravitational attraction and any kind of spring force these are conservatives so you just need to remember that weight and spring are the only two conservative forces, non-conservative forces are any kind of applied forces for example if you push on something the force of you, OK? Or any kind of friction is non-conservative, OK? Now there are some properties here that you should probably remember especially if your professor likes all conceptual type of questions, right? Conservative forces, the idea of conservative forces is that there will be some transfer of energy but that the total mechanical energy of the system is conserved, remember mechanical energy is kinetic + potential so for example if an object is falling at the absence of air resistance it's gaining kinetic energy while losing potential energy, kinetic energy has to do with velocity so it's getting faster, potential energy so it's gaining kinetic energy potential has to do with height gravitational potential has to do with height for example and if something's falling it's losing height so you're losing one while gaining the other but the total amount of energy the total amount of mechanical energy stays the same, OK? The total energy there stays the same so the energy is transferred from one to the other but the mechanical energy is conserved if only have conservative forces, if you have non-conservative forces they were add or remove the total amount of mechanical energy from the system so the total amount of mechanical energy is not conserved because the numbers changed, right? and we'll see a lot of this, this rule is more important than the other but you should probably know both again especially if your professor likes conceptual types of questions something like this can show up in multiple choice and the distinction between the two subtle but important, right? So, I have an example here to help illustrate the difference between internal, external and conservative and not conservative, OK? So it says here each arrow below represents how energy in the system changes due to a different force indicate whether the force is internal or external or conservative non-conservative so this bigger box here the red box is the total system the system is made up of objects A and B so the total energy of the system is just a combination of A and B and here A I show all the different types of energies that A can have all the different types of energies that B can have, remember total energy is mechanical plus non-mechanical, OK? And mechanical breaks down to potential and kinetic so this is a diagram showing all the possible energies or all the possible categories of energy that these two objects can have, this first arrow here represents something coming from outside of the system so this is the work some force coming from outside the system is doing work on the system because it's coming from outside it is external so number 1 is external, number 2, 3 and 4 all the other 2, 3 and 4 here these are all forces there are transferring energy within the system, the system is a big box so they're all internal, OK? So, if it's trapped within what we defined the system to be that it's internal, if it's a force coming from outside it's external, OK? For example, if you have a box and I say the system is just the box and then you come and you push against the box you're not part of the system so your force your push is an external force therefore the system is not isolated and the total energy is not going to be conserved because you are adding energy to the system, OK? Now is it conservative or non-conservative? Conservative and non-conservative has to do with whether the mechanical energy changes, well let's see here your bringing some energy from the outside so the mechanical energy must change you're adding potential energy which is part of the mechanical energy or removing, right? So, the mechanical energy here is changing so this force is non-conservative, OK? Now force number 2 you're going or you're transferring kinetic energy into potential energy in this case both of these kinetic and potential are mechanical energies so the total amount of mechanical energy doesn't change because the transfer of energy state within mechanical energies this is a conservative force, here we're going from kinetic of this object to potential of this object now you might think those are two completely different objects, well but the mechanical energy of the system is the same this guy lost some mechanical and this guy gained some mechanical but the total mechanical energy of the system is the same even though the transfer of energy is happening between objects, OK? Conservative, here just within B kinetic energy is being transferred into other types of energies, non-mechanical so you're losing some mechanical gaining some non-mechanical so because we're losing mechanical energy this is non-conservative, alright? So that's the basic idea just so you understand the distinction between internal, external, conservative, non-conservative and also the rules that the determine whether the mechanical energy of is conserved and the total energy of a system is conserved in terms of practical applications not much of this will be used in problem solving, I'll mention these when we get to that point but again this is especially useful if your professor does conceptual questions, alright? That' it.

**Example:** Conservation of Energys

Hey guys, So in this video I'm going to quickly summarize the two rules for conservation of energy, show you some examples and then show you the conservation of energy equation which is one of the most important questions in physics, let's check it out so remember that the first rule is that the total energy mechanical and non-mechanical the total energy the system is conserved if the system is isolated and isolated system is a system where the work done by external forces is 0, isolated means that you have a closed system that there is no forces that come in or no work done no energy coming from the outside, mechanical energy is a part of the total energy mechanical energy is conserved if the system is conservative the system being conserving means that the work done by non-conservative forces is 0, conservative forces are these 2 weight and spring, non-conservative force are these two friction and applied force, now one of the problems with this definition is that it depends on the definition of conservative and knowing which forces are conservative a more straightforward way of thinking about this is right here, mechanical energy is conserved as long as there are none of these forces doing work on the system, OK? As long as there are no applied forces or frictional forces doing work on the system so if you look at a situation and we'll do this a whole lot of times as long as there is no friction and no applied force we can say that mechanical energy is conserved, OK?

Let's do an example here or there are situations, I want to know is this system isolated? Is it conservative? And I want to describe the energy transfer so I have a block the falls a block that is in freefall so there's no air resistance and we're going to the define the system to be made up of the block, the air and the earth so I'm going to draw that really quick, here's a block and then here's the earth and then that's the system, OK? It's in freefall which means the only force acting on the system is the weight force, weight is a conservative force, OK? So, the only force is the weight force which is an attraction between the earth and the block, it's the earth pulling a block because our system comprises the earth and the block and the forces between those two then we say that the system is an isolated system because here weight is internal, it's an internal force to the system, is the system conservative? Well the only force is weight and weight is a conservative force so yes, the system is conservative, Ok? Again, because weight is a conservative force, weight is internal because of the way that we define the system and weight is conservative because weight is always conservative, OK? So, this one depends on how we define the system this one is always going to be conservative, what kind of energy transfers are going on? Well as you're falling you are losing gravitational potential energy which if you remember has to do with your heights so you're losing this and you're gaining kinetic energy which has to do with your speed, OK? So, your potential energy is going into kinetic energy, OK? You're losing potential but that goes into kinetic, alright? Now I have a block that is falling fells the resistance force or air drag and we define the system to be the block, the air and the earth so the system is just as here the only difference is that now this block has air drag so I'm going to call this friction of air resistance and then there's obviously an MG pulling down the block, is the system isolated? Well the friction of air is a force between the block and the air and those are both part of the system and the MG is a force between the Earth and the block and those two are part of the system as well so this system is internal because both forces are internal, weight and friction are both internal. Is this system conservative? The system is not conservative no that's because friction is a non-conservative force, every time there's friction the system is not conservative, what kind of energy transfers are going on here? Well you were losing potential energy, you're losing height you're losing gravitational potential energy but it's not just going into kinetic energy, some of it is being dissipated into heat some of this energy is going into heat so you are losing some mechanical energy into heat, remember these two are mechanical forms of energy this is a non-mechanical form of energy so you are losing mechanical energy and that's why we say that the system is non-conservative, OK? Here assumes a Styrofoam cup is sealed so no energy enters or leaves the cup so it's perfectly sealed no energy goes in and out, the cup is at rest but the coffee is heating the air inside of the cup So there's coffee here, there's air here and the system is defined to be the cup and the contents so the cup with everything inside is the system, is the system isolated? Yes, because the cup is sealed and no energy leaves or enters the cup which is the system so this is Yes, there is some transfer of energy going on here some of the heat from here is going into here we can think about it that way but that transfer of energy is internal so the heat transfer is internal, OK? Is this system conservative? This is a little more tricky the system is conservative because the mechanical energy is not changing, OK? Mechanical energy if you remember has to do with kinetic and potential, Mechanical energy has to do with kinetic and potential the system doesn't move so the kinetic energy of the cup is the same and the system isn't gaining or losing height so the potential of the system is the same as well so because there's no change in mechanical energy the system is actually conservative and what kind of energy transfer are going on? Well heats from the coffee is going into the air so I can say Q is the unit for heat you don't have to worry about that or at least not for now and then Q goes into air, OK? Again, you don't really have to worry about this exactly but I just want to make the point that in a system like that it would be isolated because everything is confined to the cup and it is conservative because mechanical energy is not being lost, OK?

I want to make a couple of quick notes here that are important, the universe is considered an isolated system, OK? And if you think about it if this is the universe there's nothing outside, right? Because if there were things outside of a determined area we just say well that's part of the universe too everything is part of the universe so the universe can be thought of as an isolated system therefore the total energy of the universe is always conserved, right? If you're isolated your total energy is conserved and think about this if the energy of the universe were to change what is that energy coming from? There's nothing outside of the universe so the energy of the universe is trapped in there, OK? That's the first point, the second point is whenever gravity does work on the system and remember gravity technically this is weight but we can refer to it as gravity as well that's cool whenever gravity does work on the system and the work done by gravity is MG the change in height so in other words whenever an object is moving up and down gravity is always doing work, the earth must be part of the system so here when we do defined that this block was falling and then the earth is down here and we said that the system comprises all of this whenever there's a block falling we always have to say that the earth is part of the system, OK? The Earth is part of the system therefore because weight is between the objects and the earth then weight is always an internal force, OK? Weight is always an internal force and when weight is also always conservative and the reason I'm telling you this is not so that you have more stuff to remember it's just that this simplifies a lot, you don't have to worry about weight and think about whether it's internal or external conservative or non-conservative it's always internal and always conservative, So that's it for now.

**Concept:** The Conservation of Energy Equation

Alright so now that you have a good understanding of the different rules for a conservation of energy both total energy and mechanical energy we're going to focus on a mechanical energy the most in problem solving, I'm going to combine a bunch of equations and we're going to write this new equation that's super awesome It says here new sexy and ridiculously useful conservation of energy, OK? Conservation of energy equation which is technically an equation that shows the conservation of mechanical energy and it's the one we're going to be using over and over again to solve problems, OK? So, I'm going to derive this really quick because there's a lot of components to this that you need to know I'm going to summarize it at the end, it's a little painful but just work with me here, alright? So, remember the work done by weight work done by G or MG both notations are fine is the negative of the change in potential energy, we talked about this earlier the work done by a spring or by the spring force is a negative of the change in the elastic energy so weight has to do gravity so it's the change of gravitational energy and spring is the change in elastic energy make sense, right? And they're both negative, now if I combine these two I want to remind you that gravity or the weight force and the spring force are the only two types of conservative forces that we discussed so if we add these 2 this is the work done by all conservative forces, right? Those are the only 2, the work done by conservative forces is the work of these two combined, these two are the two types of potential energy we have discussed, right? The two types of mechanical potential energies and so if I add this I can combine it and say that this is simply the change in total potential energy because if you write just U remember in means Ug + Uel, OK? So that's the first equation that we need to know, alright make this a different color.

So the second one I want to show you already know this one this is called the work energy theorem and it says that the net work the work done by all forces is the change in kinetic energy now what I want to do is I want to change this up a little bit, well the net work is the work done by all forces conservative and non-conservative so the work done by net force or the net work rather is the work done by all conservative forces plus the work done by all non-conservative forces and it just equals the change in kinetic energy, now I have this here that I can replace and look what's going to happen this is negative delta U + work non-conservative = Delta K and if I move the delta U to the other side I get that the work done by non-conservative forces is Delta K + Delta U, I hope you recognize that Delta K + Delta U are the two types of mechanical energy, so this is the change in the total mechanical energy and that is the second equation you need to know, alright? Actually, that's the third equation you need to know the second one was over here that you already knew but it's the new equation here that I'm telling you that I'm giving you is that the work non-conservative equals the change in mechanical energy, OK? And the change in mechanical is by the way every time you have the delta it's just mechanical final minus mechanical initial, we're almost done so there are three equations you need to know the work each type of work equals to a change in some type of energy, the work done by conservative forces is the negative of the potential the work done by non-conservative forces is the total mechanical energy and the work by the net work or the total work rather is a change in kinetic energy, OK? So these are the three equations that you need to know but the more important one is what happens when you combine it all and I'm going to rewrite this here I'm going to combine this guy with these two guys here in I'm going to rewrite this equation, I'm going to say that the final mechanical energy equals the....I'm sorry I'm going to say that the initial mechanical energy I'm going to move this over to the left becomes positive plus the work done by non-conservative forces equals the final mechanical energy and when we read this is that the final energy is the initial plus work, work is an addition of energy whatever energy I had plus whatever was added is how much I have now, OK? So, the last thing to do here is I'm going to expand this, mechanical energy is made up of kinetic and potential, so kinetic initial + potential initial + the work done by non-conservative forces equals I'm going to expand this as well kinetic and potential. And this is the equation we end up with, this is the conservation of energy equation, OK? I'm just going to pull an arrow from here since it says conservation of energy this is the conservation of energy equation we're going to start most of the problems that we're going to solve or start solving now are we're going to be begin with that equation, OK? So this is massively important the last thing I want to tell you here is that this guy work non-conservative is remember there are two forces that are non-conservative friction and the applied force by you so or by some sort of object or animal rather so the work done by friction plus what I call the work done by you so I just showed you 4 equations, this is by far the most important one and we're going to be using this over and over again to solve problems, Let me do a quick example so we can see how this works, OK? It says here you release it to 2 kilogram object from 100 meters so 2 kilogram object you release it from 100 meters and I want to find the speed just as it hits the ground, so the speed right here right before it hits the ground and it drops a height of 100 meters the initial velocity is obviously 0 because we're releasing and I want to know what is the final velocity just before it hits the ground and we're going to use conservation of energy which means I'm going to write that equation, so we always start here and now what we're going do is try to figure out which types of energies exist, OK? This is a quick preview. The kinetic energy in the beginning, well kinetic energy has to do with velocity or speed, right? The object is not moving so there is no kinetic energy in the beginning, potential energy has to do with height the object does have a height so I do have potential energy, work non-conservative has to do with the work done by you + the work done by friction, we're going to ignore air resistance we always ignore your resistance unless told otherwise so there is no friction and you're not doing anything you just release this thing you're just watching therefore there is no work done by conservative forces. Is a kinetic energy at the end? Well kinetic energy has to do with speed and just before I hit the ground I have some speed therefore there is kinetic energy, potential energy has to do with height and just before you hit the ground your height is 0.00000 whatever very tiny number so we're going to say that there is effectively no height, OK? So, height final is 0, haven't touched the floor yet but it's basically 0 and then I end up with simply these two terms here, OK? I end up with the whole thing simplifies into U initial (potential initial) equals kinetic final now you can think of this as an equation, left equals right but you can also think of this is a transfer of energy, right? My initial potential energy became kinetic energy, alright? The next step so you write the equation clear everything out the next step is once you have this short form here you're going to expand this and what I mean by that is you're going to replace U with what it stands for and MGH this is the initial height because it's the initial U and you're going to replace, OK? 1/2MV squared and this is final because it's kinetic final, OK? And now we can go on and try to solve for velocity, alright? You notice in solving for velocity that the masses will cancel and this is going to happen a lot in motion or energy questions the mass is going to cancel most of the time actually so you should be looking out for that and if I solve for the final speed here, final speed will be I just have to move the 2 to the other side so it becomes 2GH initial and then I take square root of both sides and I get this, OK? Now before I plug in numbers and we get the answer I just want to kind of highlight this because you're going to be seeing this answer a lot and I want you to become familiar with this so that when you see the square root of 2GH later on It serves as a sort of a checkpoint you now know that you're probably right this looks familiar, right? So, I want you to recognize this later because we're going to be seeing this a lot now it's just a matter of plugging in the actual numbers that's the easy part, I'm going to round gravity as simply 10 approximately 10 and then the initial height is 100 and when you do this you get that the final velocity is I have it here.... Actually, I don't have it here I believe this is roughly 45, cool? And that's it for this one, one of the benefits of using the energy equation you might have noticed is that everything is always going to be positive or at least the majority of the time you don't have to worry about signs gravity just plugged in as a positive 9.8 or sometimes we'll round it to 10 and you don't have to worry about the directions of positive or negative and all that stuff, OK? The other benefit is that you don't have to pick any equations it always starts from here and then you just cut out whatever energies you don't have until you get to whatever variable you're looking for, alright? That's it for this one.

**Example:** Using the Energy Equation

Hey guys, now that I have introduced the conservation of energy equation I want to show you a few examples of using it so let's check it out so one really important point about the energy equation is that energy problems will always include two points or more and I'll show you that in the example, OK? Here I have summarized a lot of the equations we've talked about kinetic energy, potential energy is a combination of both gravitational potential energy is MGH this is the total, the complete conservation of energy equation which is the most important of all of these and then the work by non-conservative forces is a combination of the work done by applied forces or by U and then the work by frictional forces which we're not going to have to worry about in this page, alright? So, the conservation of energy we're going to use it's going to be very useful when the speed and/or the height of an object are changing and one way to remember this is kinetic energy depends on speed and potential energy has to do with height so if these numbers are changing you want to use the conservation of energy equation, OK? The most important task as you go into the conservation of energy equation and solving these problems is to identify which types of energies exist at the beginning and the end? These problems are problems of 2 or more points so we're going to have initial and final as you can see also in the equation, initial and final so I'll show you what I mean let's get started here so you have a 4 kilogram object that you're going to launch to directly up so let me a little 4 kilogram object here you're going to throw it up here, I'm going to say that you're going to give it some sort of initial velocity of 40 it's here on the floor and you're going to throw it up and we're going to use conservation of energy to find the maximum height? What happens at the maximum height is that over here the final velocity is 0 at the maximum height but I want to know what is the height at that point so I want to know the final height because I'm going from initial to final over here so I want to know what is the final height and by the way the initial height is 0, OK? You didn't necessarily have to list all this you're going to sooner or later run into it when you write the energy equation so when I say use conservation of energy I mean to say that you are going to solve this using the big energy equation right here, some problems won't tell you to do it this way but you would still do it like this, OK? So that's the big equation let's go through each element and try to determine which ones we have kinetic energy has to do with speed and this has a velocity or speed in the beginning so we do have kinetic energy, potential energy has to do with height and at the beginning this thing is at the floor I'm launching it from the ground so the height is 0, work non-conservative is a combination of the work done by you plus the work done by friction, there is no air resistance so we're going to ignore that so there's no sort of friction here so that's just 0 the work done by you in this case it's tricky and I want to address that, you might think that if you're throwing something then you certainly doing work on this object and you are when you bring the object from a speed of 0 to a speed of in this case 40 you're doing work but in this situation we consider the initial part of the motion, initial here means once it leaves your hand with 40, notice that I said the initial velocity is 40 meters per second so it going from 0 and then leaving your hand at 40 that's certainly has some work done by you but that happened before the initial part of the problem and because it happened before this point here you are not doing any work between initial and final the work done by you here would only counts if you're doing work while this thing is moving after its left your hand and then until it gets to the top, OK? So, you're sort of doing the work before the point where it would count, OK? So, if I throw something in a certain speed and you use the initial speed then the work done by you is 0, a little tricky but it leaves your hand and you're not doing really anything anymore one way to think about it again is to just thing well am I doing any work between once it starts moving and then the end of the motion here? And the answer there is no, OK? So, I will explain that in details so we don't have to worry about that in future problems so the work done by U is 0 there's no non-conservative work. Kinetic energy at the end, at the end you reach a speed of 0 you start momentarily so the kinetic energy is 0 and potential and has to with height you do have a height so you do have potential energy, if you look here the only two types of energies we have are these two and I can write kinetic initial equals potential final again this an equation left equals right but it's also a statement of conversion of energies, all the kinetic energy became potential energy all the speed in the beginning became height at the end, OK? Let's expand this and what I mean by that is let's write what kinetic energy stands for which is 1/2MVsquared and this initial and potential energy stands for MGH and its final height because its final potential energy, notice that I can cancel the masses very often we're going to do this and since I'm looking for the final height I just have to solve here, H final is V initial squared divided by 2 divide G on both sides of this and up here so this is 40 square squared, I'm going to put plug gravity in as a 10 just to make the math a little bit easier on your test unless you professor says otherwise you will plug in a 9.8 so if you do this you get the 80 meters if you use 9.8 you get something like 81.6 I believe, OK? That's it for this one I do have a practice problem for you guys very similar and I want to try this out let's give it a shot.

**Concept:** Using the Energy Equation

Alright, I hope you try this one we have an object that's 6 kilograms it's being thrown down from a height of 20 meters so here's a 6 kilogram objects you toss it down with some sort of initial velocity from a height of 20 meters it says the object reaches the ground with 30, so here's the ground right before touching the ground the object has a V final of 30 and we want to know what is the V initial that you used to throw this object, we're going to use conservation of energy and we're going to ignore air resistance so once again K initial+U initial+ Work non-conservative= K final + U final, that's how we're always going to start which kinds of energies do we have? Is there kinetic initial? Well in the beginning there is a speed so there is kinetic energy is there potential initial? There is because there is a height in the beginning so I do have this type of energy what about work non-conservative work non-conservative is the work done by you plus the work done by friction and the work done by friction is 0 because we are ignoring air resistance as we should and the work done by you is 0 as well because even though you know as we discussed earlier you do not do any work from the instance it leaves your hand to the point where it hits the ground you did work before it left your hand and that doesn't count, OK? So, this is 0 because you don't do work in between, is there kinetic energy at the end? There is kinetic energy at the end because you do have a speed...You have a speed of 30 there and the potential at the end is 0 because the final height is 0 because you are about to hit the ground, OK? Now we have to expand all of these which means replace K with 1/2MV squared, this is in V initial and all the other ones and then solve for our variable, MGH initial= 1/2MV final squared this is just me replacing all the energies with their equations, notice that the mass cancels and very often it will and you should always look out for that and now I'm going to start plugging the numbers 1/2 V initial is what I'm looking for so I'm going to kind of circle that, gravity we're going to use just a 10, the initial height is 20, 1/2 the final velocity is a 30 so I'm going to do 30 squared, this is V initial + 200=450 so now we just have to sole for V initial and I'm going to have 1/2V initial squared, this is 450-200 is going to be 250 and if you solve V initial you get that the V initial should be...skip a few steps here V initial will give you 22.4 meters per second that's what you should have gotten 22.4 meters per second and that's it for this one hopefully you got it.

**Example:** Using the Energy Equation

Alright so the previous two problems the example in the practice I gave you we talked about problems we had problems where you're going from one point to another those are very simple initial to final write the energy equation and work with it but in some problems we're going to have more than two points and when that happens you have to figure out which two points out of three or four let's say should I work with because the energy equation is always from one point to another and you have to figure out which two points should pick one, we'll you will always pick the given point and the target point and I'll show you with that means it's basically known and unknown we'll see that in this example here you launch an object directly up from the ground so it's kind of draw an object here notice that you're not given the mass but the masses will probably cancel so it doesn't really matter, you're going to launch it from the ground so you toss it up and this is the object is going up with 20 at 30 meters so at a height of 30 meters the object is still going up with a velocity of 20 meters per second, OK? That means that the object will keep moving up a little more it still has a speed up or velocity up so it's still going to go a little more until it gets to its highest point at the highest point its velocity will be 0, OK? Now you might notice here I have three points of interest at the beginning, this point B here which is the given point I gave you some information about that point and then the highest point here, the beginning of the end is always important and any other point that gets mentioned between is important as well so we label it like that so I'm going to call this V at point B and then V at point C, OK? And the questions you're asked are what is the launch speed? So, I'm asking for the velocity or speed at this point what is Va? And what is the maximum height? Maximum height is the height at point C, so I want to know what is Va? And what is Hc? heights are changing, speeds are changing we're going to use conservation of energy equation, now when I'm looking for Va there are three points here so in solving for Va I could write a conservation of energy equation from A to B, from A to C or from B to C those are the three possibilities. Now when I say write the energy equation from A to B, I mean that A would be the initial and then B would be the final, right? These are the three possibilities but really since we're looking for Va it's our target or unknown it has to be part of the interval so it has to be one of these two, OK? Now if you look a little closer I know information about B I know everything there is to know about B, I know the velocity of B is 20 and I know the height of B is 30, so I know a lot more about B than I know about C I don't know the height of C so it's better to work with B in fact you can't do this otherwise so the only choice you really have is to write an energy equation from the A to B, A being the target or the unknown and B being the given or known part of the equation, OK? So, A to B which means we're going to write something like this, Ka+Ua+work non-conservative=Kb+Ub, A takes the place of initial B takes the place of final because they are initial and final, right? So, is there kinetic energy in the beginning? Well if this thing went up there has to be a velocity at the bottom for this thing to go up so there is kinetic at the bottom. Is there potential at the bottom? No because at the bottom your height is 0, OK? Is there work non-conservative? Work non-conservative is the work done by you plus the work done by friction there is no air resistance we already talked about how when you throw something you don't do work, the work done by you in this case is 0 because you didn't do any work on the way up you just did work prior to it leaving your hand, OK? So, it doesn't count, alright kinetic energy a point B? I have a kinetic energy point B because I have a speed and I have potential point B because I have a height, OK? Now let's replace these with their equations, 1/2MVa squared (this was zero as well) = 1/2MVb squared a point B plus this is MGH, OK? So it's important you know that kinetic energy is 1/2MV squared and potential energy is MGH because you're going to use this all the time, the masses cancel and they almost always do, now we're going to solve this a little bit differently from before, I want to simplify this equation as much as possible plug in the numbers at the end that's one way to do it, notice that I have 1/2 here and 1/2 here so one thing I can do I really hate fractions so I'm going to multiply this entire thing by 2 to get to the fractions and I think that's helpful so if I multiply 2 times a 1/2, this 1/2 goes away and this becomes simply Va squared which is much simpler, this half goes away as well and this becomes Vb squared but this GH becomes 2GH, OK? So now I'm looking for Va and all we have to do is plug in numbers so Va will be the square root of the stuff here, Vb is 20, gravity is 10 and the height is 30, OK? And if you do this and I solve this earlier the answer is approximately 26.5 meters per second, I'm going to use gravity as a 10 just so it's a little bit faster but unless otherwise stated you would probably do a 9.8 during the test, OK? So that's Va got that, now that I know Va IÕm going to go ahead and put it here, Va is 26.5 notice how I now know the two variables that there are to know about point A.

Let's keep going to part B, Part B says What is the maximum height? What is Hc? Here again I can go from A to B, A to C or B to C but we already discussed that since I'm looking for Hc, C has to be part of the equation so this guy can't be it, in this case since we now know more information about the problem I know everything there is to know about B and A you could actually choose between those two, in this first part there is only one option but now I have more information so I can simply pick whichever one will be the easiest or whichever one is the simplest or easiest same thing and that's going to be A to C, the reason why A to C is simpler is because if you look at the variables this has a velocity and a height this only has a height because the velocity here is 0 more things will cancel or one more element will cancel in the equation you're going to have less things your equation will be simpler, OK? So, the simplest one is A to C because one of the numbers will cancel because the speed is 0, you could have done it with B would have worked just as well but this is a little bit simpler so we're going to go from A to C, Ka + Ua + work non-conservative = Kc + Uc. Is there a kinetic energy at point A? Yes, there is because I have a speed. What about potential energy at point A? there is potential......There is no potential energy at point A rather because you are at the ground there is no work non-conservative we already talked about that. Is there kinetic energy at point C? Is there speed at point C? There is no speed at point C so there is no kinetic energy there and what about potential energy point C? There is potential energy at point C because I have a height so you have to go through this, this is the most important part if you butcher this you're going to get the wrong answer but if you get this right now it's just algebra, right? So, 1/2MVa squared is the only thing I have here and then the only thing I have on the other side is MGHc, notice that the masses cancel and we're solving for Hc I'm going to move everything to the other side so Va squared has a 2 down here and then I have to divide both sides by G you get this, Va was roughly 26.5 squared and this is 2, gravity I'm going to use a 10 just rounding it and this gives your approximately 35 meters, OK? So that's it for this one hopefully made sense let me know if you guys have any questions.

The potential energy between two identical atoms has the form U(x) = A / x ^{10} − B / x^{5} where x is the separation distance between the atoms and A and B are constants with appropriate units. The two atoms, initially very far apart, are released from rest. What is the maximum kinetic energy that each of the two atoms can have?

1. K_{max} = (A^{2} / 2B)

2. K_{max} = (B^{2} / 4A)

3. K_{max} = (A^{2} / 8B)

4. K_{max} = (A / 4)

5. K_{max} = (B^{2} / 2A)

6. K_{max} = 0

7. K_{max} = (B / 8)

8. K_{max} = (B^{2} / 8A)

9. K_{max} = (A^{2} / 4B)

10. K_{max} = (B)

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You and your friend, who weighs the same as you, want to go to the top of the Eiffel Tower. Your friend takes the elevator straight up. You decide to walk up the spiral stairway, taking longer to do so. Compare the gravitational potential energy of you and your friend, after you both reach the top.

A) It is impossible to tell, since the times you both took are unknown.

B) It is impossible to tell, since the distances you both traveled are unknown.

C) You friend's gravitational potential energy is greater than yours, because he got to the top faster.

D) Your gravitational potential energy is greater than that of your friend, because you traveled a greater distance getting to the top.

E) Both of you have the same amount of gravitational potential energy at the top.

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Swimmers at water park have a choice of two frictionless water slides, as shown in the figure. Although both slides drop over the same height *h*, slide 1 is straight while slide 2 is curved, dropping quickly at first and then leveling out. How does the speed v_{1} of a swimmer reaching the bottom of slide I compare with v _{2}, the speed of a swimmer reaching the end of slide 2?

A) v_{1} > v_{2}

B) v_{1} < v_{2}

C) v_{1} = v_{2}

D) The heavier swimmer will have a greater speed than the lighter swimmer, no matter which slide he uses.

E) No simple relationship exists between v_{1} and v_{2}.

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Is it possible for a system to have negative potential energy?

A. Yes, as long as the total energy is positive.

B. Yes, since the choice of the zero of potential energy is arbitrary.

C. No, because this would have no physical meaning.

D. Yes, as long as the kinetic energy is positive.

E. No, because the kinetic energy of a system must equal its potential energy.

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A 150 g object starts on the ground, is moved 1.2 m in the +x direction over a frictionless surface, is then moved upwards 50 cm, then moved in a straight line back to its starting position. How much work does gravity do during this motion?

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A 12 kg mass is moving down a frictionless incline under the influence of gravity. If the incline has a height of 1 m, and an incline angle of 35°, how much work is done by gravity as the mass slides down the surface? What is the kinetic energy of the mass at the bottom of the incline?

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When an object moves from A to point B, gravity does positive work on the object. When the object from point A to point B, its gravitational potential energy

A) stays the same

B) increases

C) decreases

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