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Video transcript

I want to talk to you about Lowell diagrams that's right I said Lowell diagrams these are a great way to visualize conservation of energy and even better they force you to think about what's part of your energy system and what isn't part of your energy system and if you don't know what an energy system is maybe that's where we should start an energy system is an object or a collection of objects whose energies we're going to keep track of and we're going to keep track of them in these two charts and notice it looks like an L and then an O and then an LS where they get their name from this isn't laugh-out-loud this is an energy chart a circle where we're going to define our system and then another chart so to understand what these Lowell diagrams mean let's just look at an actual example let's say you took a mass m and you released it from rest from a height H and this mass falls down the first thing we should do is choose what is going to be part of our energy system whose energies are going to keep track of and whose are we not going to keep track of the way people typically do this problem when they're doing conservation of energy they would say it starts with potential energy and then it turns into kinetic energy and if you're doing that what you're really doing is you're saying I'm taking my mass that's going to be part of my system so I'm going to take this mass here and make that part of my system so we'll keep track of the energy of this mass and then if you're talking about gravitational potential energy here's the weird thing you're also talking about the earth that earth is part of your system so this is the earth right here it's got continents and got California and Mexico and South America and there's Florida and there's all kinds of stuff here so now that we've selected our system we can go ahead and start charting what the energies look like so what kind of energy was there initially in this system there was gravitational potential energy because this mass M started a height H above the earth I'm going to represent that with this bar this is going to be a bar chart I'm going to just go up four units so I'm gonna say there were four units of gravitational potential energy you might be like how do you how do you know there were four how do you know they weren't like three or five or 4.2 well it doesn't really matter too much the real value in this lowell diagram is being able to conceptualize what's happening to the energies no matter what you draw over here you just want to be consistent when you then draw this diagram over here and I'll show you what that means in so for now let's just say there were four units of gravitational potential energy to start with was there any kinetic energy note because it was released from rest so if we drop this mass it starts with no kinetic energy and this you as here is going to be any spring energy we won't have that for this example because there aren't any Springs so there won't be any spring or elastic energy that's it we just had gravitational potential energy to start and then what did that energy turn into if this was the initial position what kind of energy do we have when the mass gets right above the surface of the earth right before it's going to hit the earth what kind of energy is there there's going to be kinetic energy so this mass is going to have kinetic energy and we know it's going to have kinetic energy because it's going to be moving right before it hits the earth and kinetic energy is the energy and object has because of its motion so it's going to have kinetic energy how much kinetic energy is it going to have well energy is going to be conserved in our system and since the energy is going to be conserved if I started with four units of gravitational potential energy I should end with four units of kinetic energy now you might be confused you might wonder why wasn't there any gravitational potential energy well I was assuming that this ground was the H equals zero line and we know that the gravitational potential energy is mg H where H equals zero the potential energy is zero so when this mass gets down to the final point which I'm representing in this diagram it no longer has any potential energy since the H would be zero and since it has no potential energy all of this gravitational potential energy I had initially has to turn into kinetic energy in order for the energy of our system to be conserved now if I were you I might object and say wait a minute we learned earlier that when work is done energy gets transferred and the total energy of an object might change and and this earth is pulling down on this mass right so the earth is exerting a gravitational force downward the objects moving downward through some displacement that means the Earth's doing some positive amount of work on this mass giving it energy doesn't that mean the energy of the system is going to change and the answer is no and the reason is this earth is also part of our energy system so the earth did do work on the mass and this mass gained kinetic energy but since the earth and the as we're both part of the system this work was internal to the system and internal work never changes the total energy of the system and if that was confusing think about it this way let's say you and your friend instead of the earth and the mass there's you and your friend and instead of energy we'll talk about money you the earth give your friend the mass ten dollars instead of ten joules of energy you give them ten dollars and ask you how much total energy is there between you and your friend well there's still the total amount you lost ten dollars your friend gained ten dollars but between the two of you you still have the total amount of money you had when you started and the same is true with energy whenever work is done internally between objects in your system there's no change in the total energy from one chart to another that's why I had to draw the kinetic energy as four units because no energy entered or exited our system and something that's really useful about these Lowell diagrams is that you can translate them straight into a conservation of energy equation because whatever initial energy that you start with plus any external work that's done has to equal the final energy that you end with and the reason is external work is how much energy gets transferred into the system so if you start with ten joules of energy and you transfer five joules of energy into the system you've got to end with 15 joules of energy so we can just plug straight into this what did we have for our particular scenario we started with potential energy so our initial energy was M G H plus external work there was no external work there was only internal work so this would be zero and that's got to equal the final energy the only type of energy we had to end with was kinetic energy one-half MV squared and then if we had numbers we can plug into this formula and just solve for the height or the speed or whatever we wanted to solve for so what would an example look like where energy wasn't conserved in other words where there was external work done on the system we could use this same example I just need to take the earth I'm gonna stick this earth outside of our system so now our system is just the mass M what that means is that the work done by the earth on the mass is now going to be external work and since there's external work done on the system the system's energy has got to change now it's not going to stay the same you're not going to see the energy of the system remain conserved since something's giving it energy the earth is giving this mass energy this time because it's doing external work so how would that change our Lowell diagram this mass M still ends with kinetic energy I mean just by imagining the earth as not part of the system that doesn't change what actually happened the mass still has to get down here with four units of kinetic energy it's still moving just as fast as it did before but our system which is this mass has to be gaining energy in the way that's possible is that the mass does not start with any gravitational potential energy so people don't like this people like away.what of course this mass starts with gravitational potential energy it started up here but technically speaking gravitational potential energy is an energy that exists between two masses takes two to tango and it takes two masses to have gravitational potential energy if I stick one of these masses out here it can do work on the other mass we wouldn't say that it started with any gravitational potential energy so this would now be zero but we would say that there was external work done so that's where this energy comes from the earth is now doing how much work well if the mass gained four units of kinetic energy then the work done had to be four units of work so if you stick the earth outside of the system the system no longer has gravitational potential energy the earth is doing external work on the mass and this might bother you know you might be like weighted but it's the same situation that's happening all we did was change whether we consider the earth part of our system so how can that change the math it didn't really I mean if you think about it instead of the four units of energy existing initially as this potential energy and having no work we're just saying that there was no potential energy to start and there were four units of external work done it's just a different story the numbers will come out the same and it's all going to be consistent but depending on whether you stick that earth outside of your system or inside of your system will determine whether you say that there was initially gravitational potential energy or whether you say there was external work done on the mass so let's look at another example let me get rid of this let's instead consider this example where a mass starts compressed against a spring from rest that will be our initial point and it fires the mass and the mass goes up this incline and comes over here and then it's moving with some velocity over here we'll make this our final point so if we want to make a lull diagram we first have to pick what's part of our energy system because if we don't know it's part of our system we don't even know what's going to be eligible to have an energy in here so again let's make the mass part of our system so the mass will be part of our system let's make the spring part of our system so we'll put the spring in here and let's again put the earth into our system so what would this Lowell diagram look like for this scenario well since we start from rest we don't start with any kinetic energy and even though the earth is part of our system which means our system could have gravitational potential energy I'm just going to assume that the mass starts at H equals zero so let's make the ground H equals zero that way this potential energy term MGH starts as zero so there'd be no gravitational potential energy to start with and the only energy we would start with is the fact that this spring is coiled up ready to explode and fire this mass forward all this energy is stored up as spring or elastic energy so we will have elastic or spring energy to start with let's just say we started with five units of spring energy again this is somewhat random for a particular problem with numbers you could solve for exactly how much spring energy you started with because remember spring energy can be found using the formula one-half KX squared where X is the compression of the spring and K is the spring constant but to keep things simple let's just say we start with five units of spring energy so what would our final energies look like if this mass is up here moving if it's moving we know it's got kinetic energy so I could say there's kinetic energy how much again to figure out exactly how much you'd have to solve for your particular problem and know this height but to give you an idea of what conservation of energy says let's just say there were three units of kinetic energy that you ended with up there what are the kinds of energy do you have up here we're gonna have gravitational potential energy because this mass starts above the H equals zero line since the mass is higher up in the air it's got gravitational potential energy so now if I say there's three units of kinetic energy and I started with five units of spring energy if energy conserved I better end up with two units of gravitational potential energy so I could draw two units and now I'd make sure that if I started with five units of energy I've ended with a total amount of five units that would say energy is conserved let's check is energy of our system going to be conserved we can determine that by just figuring out whether there were any external sources of work did anything external to our system do any work if not the total energies got to stay the same this five units has got to turn into five units over here and if we assume that these surfaces here are frictionless and there was no air resistance or no loss in energy due to dissipative forces there will be no external work done because everything's part of our system so like work was done in here the spring did work on the mass giving it energy there was internal work done but since there was no external work the energy had to be conserved we can write this in an equation again by saying that the total energy we start with the spring energy which would be 1/2 K x squared plus any external work done but the external work done was zero since everything is internal has to equal the final energy and the final energy here is the kinetic energy which is one-half MV squared plus the potential energy which is going to be plus MGH and now if you had numbers for a particular problem you could just plug those into here and solve for whatever variable you were looking for but you might be wondering what if there is friction between the mass and the ramp what if there is friction over here what would that do well there's two ways we can handle it now you know there's two ways we can include these surfaces as part of our system or we could say that those surfaces are outside of our system if the surfaces are outside of our system we would represent that as them taking energy away now there's external work done but instead of the external work giving our system energy the external work is taking energy away this friction is taking energy and turning it into thermal energy that's what this extra spot is here for thermal energy but since the surfaces are now not part of our system we wouldn't show that over here we would just include that as part of the external work done so there would be some negative amount of work done maybe maybe the surfaces did negative 1 unit of external work with that mean is that if we started with five units of energy and there was negative one unit of external work done we've got to only end up with four units of energy but my potential Energy's got to be the same I mean I ended up this high so that can't really change so that means my kinetic energy is now going to be smaller and that makes sense the friction slowed this mass down I end with four units of energy even though I started with five because there was negative one unit of external work done by the friction on these surfaces that's one way to handle this problem and that's all consistent the other way to do it we could just say that our surfaces which we said we're external are now internal to our system that way this external work done is no longer external this work done is now going to be internal the surface took the energy it still did negative work but since it's inside of our system that's now going to be part of our energy system so how would we represent that it no longer be a negative one here we wouldn't say this is done we'd essentially conceptually move that negative one over by adding one to both sides and that means we would gain some thermal energy I'm going to write that as Delta e thermal because there's going to be some gain and the thermal energy of our system and I would represent that now not as an external work done but as one unit of thermal energy generated during this process so recapping Lowell diagrams are a great way to visualize what we're talking about when we say conservation of energy and since we define our system there a nice way to visualize what we mean when we say that the total initial energy of our system plus any external work done on our system has to equal the total final energy of our system