we are back and we are going to keep talking about section 3.2 and we are talking about potential energy right now the idea of potential energy is simple because it's the same as the definition of work so that makes it a straightforward application what's a little bit less simple is what we left off talking about in the last lecture the reason why we do this the reason why we separate out the force of gravity and say the work done by gravity that's going to be potential energy but the work done by friction that's not so we need to keep pursuing this idea make sure we understand that distinction so we're still on the slide i kept it on the same slide here with the definition potential energy due to gravity is force mg the weight of the object times h and i said a couple of times now that using the h as the variable gives us that solid mental picture that we have to be talking about vertical but here's the place where that notation is maybe a little bit weak because what it doesn't do for us using that h what it doesn't do is give us the [Music] necessary this is necessary idea that your potential energy must be measured with respect to something so the idea of a relative measurement so what do we mean by relative we mean compared to so take a look at that roller coaster on the picture on the left side of the slide and when you are thinking about the potential energy it's very natural to think about the ground or ground level being zero the height is equal to zero if the thing is sitting on the ground it's at height equal to zero that feels intuitive and it feels correct and i'm not saying that it's incorrect but what i am saying is that in the context of solving different problems it may not be the best way saying that the ground is h equal to zero might not have anything to do with the problem itself that's why we're looking at this roller coaster here so you can see that we've got the hill that swoops up high and then we do something over and we can't see what happens and we come around and we're swooping down back over the water and you can see that the track of the roller coaster well it doesn't hit the water it's above the water and it doesn't hit the ground by comparison it looks like if we look over to where the trees are at the edge of the water it looks like we're a little bit above the ground as well and so here is that exact point that i'm making what difference does it make where the water level is to the roller coaster none the roller coaster is bound to follow the path of the track what difference does it make where the literal ground level where the trees are what difference does it make where that is located up or down high or low it doesn't make a bit of difference to the roller coaster so for a reference this is the really powerful idea you get to look at your problem and you get to decide where do you want to draw a line where do you want to draw a line and say it makes sense right here if i say h my height is equal to zero right here that makes sense why does it make sense well because my object doesn't ever go below that point my object maybe the water is not relevant maybe the ground is not relevant and maybe you pick the point right here where the roller coaster car is coming down and bottoms out and that's as low as it goes might as well make that be your zero point so the mgh for the equation it doesn't break down because it's a valid equation but our our thinking has to be a little bit more sophisticated to recognize that we need to specify always where are we going to set up a reference level if you are setting your books or you came in with a bet a lot of grocery bags and you set them on the table top you had to lift them up and you set them on the table top well the motion of the bags whether you raised them or lowered them or the books whether you raised them or lowered them is relative to the table because that's where the bags and the books finally ended up if you're setting things down on the table the floor doesn't matter it doesn't matter whether the floor is 30 inches below the table top or 36 inches below the table top the location of the floor doesn't matter in the context of the problem so you always have to specify and then the next thing that you're always thinking about is that even though we're writing the equation mgh we need to think about it like mathematically a delta a delta h so remember our delta notation and sometimes we get a little bit relaxed with our mathematical notation and we need to recognize here that delta notation means change in so delta h would be h final minus h initial and if we get to pick h initial and call that zero then h final minus h initial is always going to just be h which is why we tend to get a little relaxed with that notation however the meaning of it is really important that it's always going to be a change in position because we get to pick our reference level we hopefully will be looking at our problem and making that decision based on the problem and the situation and the object that we're analyzing and make that decision to make the problem be as easy as possible so that's the next thing that i wanted to look at is setting up a reference level and looking at comparing a couple of different reference levels to see what's going to be easy and what's going to be maybe less easy and what's going to be obvious and maybe what's less obvious so this is one of your pop quiz questions comparing reference levels so we've got a roller coaster here just part of the ride just the simple loop so notice we've got a couple of points marked off here point number one at the very bottom of the loop and then point number two we're about halfway we're halfway around the loop point number three we're at the top of the loop and then the car is gonna continue around the loop and head on towards point number four but we're interested in points one two and three so i'm gonna skip these multiple choices and you're gonna come back to those multiple choices when you're looking at pop quiz but what i'm gonna do on the next slide is we're going to analyze what difference does it make if we set a reference level at point 1 and then what changes if we set our reference at point 2. so we're going to analyze this and then you'll know what the answer to this quiz question is so let's analyze this okay so top to bottom what i've done is looked at setting a reference level and then calculating the potential energy of the car the roller coaster car at the other two points so for example the very top picture notice i've got the line labeled at point one position one the low point of the problem i've got that labeled as h equals zero and so if point one is h equals zero then my potential energy at point one is going to be zero as well so i only need to calculate my potential at points two and three okay but now let's take a look at the arrows because i want you to see how potential energy your gravitational potential energy i want you to see how that could be either positive or negative so let's look at these arrows carefully i've got the force of gravity on the far left where the roller coaster cars at the higher up the the ramp i've got that force of gravity labeled mg and that points down so no choice there gravity only works one way and that way is pointing down toward the ground okay so that's obvious but i'm putting it there anyway because we want to be able to make a comparison now if i am at position two or position three my h m g h here's the thing this is again where the [Music] the visual image might not match what we really need to do and how we really need to express it mathematically because what is natural what would feel really natural is if you're standing at point one where h is equal to zero your feet are on the ground and you look up to 0.2 or 0.3 your literal visual arrow of looking up points an arrow in the up direction and you can see on my picture i've got two arrows pointing down to the ground and that might not feel intuitive to you but mathematically that's what we're doing so the direction of those arrows why are the h2 and h3 and they are vectors remember that's a displacement so they are in fact vectors why are they pointing down they're pointing in the direction toward the reference level so we are looking at where are you compared to your reference level so from wherever the object is to wherever you've defined as your reference level so notice and you can just scan down all three of these pictures notice that the arrow heads the pointy parts of the arrow are always on the h equals zero line so now mathematically i'm not putting any numbers in because right now i don't want numbers i just want the concept to make sure i get why my potential energy is going to be positive or why my potential energy is going to be negative okay so starting at the top picture with my reference level at the lowest point of the track point one and if h is equal to zero right there then both my h2 and my h3 those arrows are pointing down so i've made mathematically i've made this as explicit as i can so that we can see really carefully that we can see what's going to happen excuse me gravity that force pointing down so that's down in the negative direction and then it doesn't matter if i'm looking at position two or position three both of those arrows are pointing down as well so my force is pointing down and my displacement is pointing down and a negative times a negative is a positive number now let's make one more connection here go back to section 3.1 and the definition of work that was exactly what we said when the force and the displacement point in exactly the same direction whether they're both pointing up or they're both pointing down or they're both pointing whatever direction but if it's the same the work done is positive so positive work would be done by gravity to get from position three to position one positive work would be done by gravity on the roller coaster car to move from position two to position one okay so go down to the middle picture and we see that i've shifted and labeled that position 2 as our reference level and with position 2 at as the reference level to get from position 1 to position 2 you've got to move in the up direction and that makes h1 positive but my mg that weight due to gravity h is positive but gravity is negative so notice that at position one you have negative potential energy but notice that at position three again your arrow points from position three where are you to the reference level position two so gravity down h3 vector displacement down as well negative times negative is positive so in this case if we label position 2 as our reference level any point that is below that red line at the midpoint through the loop so if you are below position two then your potential energy is going to be a negative value but if you're at any point above position two any point up here or maybe you're over here on the straight part going up to the rest of the coaster if you're above position two then the potential energy will be positive okay and last but not least we went from position one taking the lowest point and saying h equals 0 right there to the bottom picture going to the highest point and saying okay let's let h equal to 0 right here and again focus on the direction arrows gravity down doesn't change never changes no choices no options it only works one way got it but now you are drawing this h vector this displacement vector from where did you start position one or position two ends at what you have defined as h equal to zero and both of those displacement vectors are now pointing in the up direction so what we see here and this is exactly what happened in the previous problem any location that is below that is lower than your reference level is going to result in negative potential energy if you have wherever your reference level is and we're talking about gravitational pe right here at or near the surface of the earth wherever your level is anything that is above h equals zero has positive potential energy anything that is below h equal to zero has negative potential energy now the thing we didn't do because we didn't put numbers in the thing we didn't do was compare the quantity but that's a pretty straightforward visual image here because you can see pretty easily in the top picture that h3 is significantly larger than h2 so when we multiply the same weight and g when we multiply the same weight by h2 we're going to get a smaller number whatever that number is and when we multiply it by h3 we're going to get a bigger number you will have positive potential energy in both locations but you're going to have more potential energy at position 3 than you are at position 2. same thing with go down to the very bottom image again the h1 vector is much larger than the h2 vector so your potential energy is going to be a bigger number at position one and a smaller number at position two but the sign on both of them in that case is going to be negative okay we are going to end right here and then we have a little bit more in section 3.2 to talk about because the next thing up is kinetic energy and we'll see in a bit