1

concept

## Gravitational Potential Energy

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Hey guys. So for the studio, want to take a closer look at gravitational potential energy. Remember this is one of the main types of energy that we have within potential energy which is really a type of mechanical energy. We're gonna see some pretty interesting relationships between the work done by gravity and this new type of energy. Let's check this out. Remember that the whole point of the whole idea behind the work that's done by gravity is that objects fall, gravity does work on them and if they start to fall faster, meaning their speed increases and then their kinetic energy is going to increase. We've seen this type of situation before, but let's take a look at this example here. So the whole idea is that we have a 5.1 kg box that's falling from an initial height of and it falls to a height of four. Which means that it has some displacement from Delta Y, which we can calculate as -6. And if we want to calculate the work that's done by gravity, right as this thing is falling, we're just going to use our equation negative MG delta. Why? So this is negative 5.1 times 9.8 times this is gonna be negative six. So what you're gonna have is the negatives cancel and you get the work done by gravity is 300 jewels. This makes sense that the work done by gravity is positive because it's basically helping the box get faster as it falls. Gravity does positive work if it helps or goes along with the motion. Right? So remember that this work done really changes the kinetic energy of the box here. The initial velocity is zero, which means that the kinetic energy is equal to zero. But when it's down here it has some final velocity and the kinetic energy is really just gonna be whatever the work that was done was. So first it goes from zero jewels and now it's going to have 300 joules of kinetic energy because gravity did 300 jewels of work on this box. So the whole point of this video is that this energy has to come from somewhere. It can't just be created or destroyed. We know that. And the whole idea is that this kinetic energy has to be transferred from other types of energy called gravitational potential energy. So we've talked a little bit about potential energies before we say that they're basically stored energy and that's exactly what this potential energy is. Gravitational potential energy, which we write as you G is stored energy basically due to an object heights. So, the fact that this object is at some distance of 10 m means it has some potential energy to some stored energy that then that can then become kinetic energy as gravity does work on it. All right, so, the formula that we use for Yugi is just gonna be M. G. Y. Now some textbooks might also write this as MGH. So if you see that, it means the same thing as M. G. Y. So let's go ahead and get to part A here. In part. They were going to calculate the initial gravitational potential energy. So that's really just gonna be you G. Initial. So if you G is M G. Y, then we're just going to calculate this by doing M. G. Y. Initial. We have all the numbers for this 5.19 point eight. And then the initial height is 10. You go ahead and work this out, you're gonna get 500 jewels. So basically because this box is 10 m above the surface, it has some initial stored energy of 500 jewels. So let's take a look at part B and part B. Now. We want to calculate the final gravitational potential which we're just gonna use em gy final. So this is going to be 5.1 times 9.8 times four now and this is going to be 200 jewels. So This makes sense because now when you're at 4m you are you have less height above the ground, so you should have less stored energy. You're U. G. Final is equal to 200 jewels because you've lost some height and now in part C we want to calculate the change in the gravitational potential. Remember change just means final minus initial. So delta Yugi. So delta is just gonna be final minus initial. So this Yugi final minus Yugi initial. And we just calculated those things. Remember Yugi final is 200 ug initial is 500 so basically we have 200 -500 and this equals a change of negative 300 jewels. Now this one makes, this should make some sense that it's negative because as you're falling and losing height, you are losing some of that stored energy. So your delta, you should actually be negative 300 jewels here. But there's actually another way to kind of understand what's going on here. So what I want to do is actually want to rewrite this Yugi final and you G initial and basically replace them with their equations, this is M G Y final minus MG why initial? So I can group together the MGs and I can say that this is MG why final minus Y initial. Now remember that final minus initial? It's just delta. So really this delta U. G is equal to MG times delta Y. So if you take a look at the equations here, we can see is that this change in gravitational potential energy which is MG times delta. Why is the negative of the work that is done by gravity, which is negative MG delta. Why these two equations are opposite of each other. And so basically what we can say here is that the work that is done by gravity is always the change on the second. Sorry about that. Some some funky happened. So the work that's done by gravity is the change specifically the negative change in gravitational potential energy. So W. G. Is negative delta. You. So really what we can do here is we can now come up with a relationship between the work that is done by gravity, with the change in the kinetic energy and the change in the potential energy. And basically what we can say is that if there is no other forces that are acting on an object, then as objects fall, than what we've seen, is that the work that's done by gravity is going to be positive because gravity pulls things down, it makes them go faster and it's going along with the motion. Your kinetic energy is going to increase because your speed is going to increase. But as you fall, you're losing height. Which means that your potential energy is going to decrease, your delta Y. Is going down when you're rising, when objects start to rise, then it's the opposite. The work that's done by gravity is gonna be negative. Your kinetic energy is going to decrease because your speed is going to decrease as you're going up. But because you're going higher, your potential energy is going to increase so your delta Y. Is going up. So really what happens here is that when you have objects that are falling, the kinetic energy really gets transferred to gravitational potential energy and vice versa. And the thing that does that transfer is really the work that is done by gravity. So if the force of gravity does work, and it's really just a transfer between gravitational potential and kinetic energies. All right, so that's it for this one. Guys. Let me know if you have any questions.

2

example

## Gravitational Potential Energy is "Relative"

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Hey guys, let's work out this problem together. So here we have a two kg ball that's initially six m above the ground. So let's go ahead and take a look at part a here. We want to calculate the change in gravitational potential energy. So that's delta U. G. We're gonna be using this equation right here for duty for two different cases. What happens here in part is that we have this ball that is six m above the ground. So I know that this sort of level right here. This why value is equal to six. That's my initial The ball then is going to fall down to another height. It's going to fall down to a height of three m. So here I have Y equals my wife. Final equals three. What I want to do is I want to calculate the changing gravitational potential energy. So you've lost some heights and so therefore you've lost some gravitational potential energy. So how do we calculate this? What happens is we're just going to use MG times delta. Why? But what happens in energy problems is a gravitational potential energy is always calculated relative to an arbitrary reference point. What happens in part A is that we're choosing the ground which is why equals zero to be where the gravitational potential energy equals zero. So we're doing here is we're saying if this is six and this is three, the ground level is why equals zero. And this is where you g. Is equal to zero notice. If you plug in zero into this equation here, you're just gonna get zero. So what happens is we can actually just choose our ground level to be wherever we want. So what happens we're going to calculate the delta U. Which is going to be MG times delta Y. But I'm actually gonna write it out in the longer way. So I'm gonna actually write it out as MG. Why? Final minus Y. Initial? That's what DELTA Y. Means. So what happens? We're going to get a massive two. We're gonna have a G. Of 9.8 and then my final minus initial is going to be three minus six. Amending a blower than I started. So I should get a negative number here. So we've got three minus six. When you plug this all in, you're gonna get negative 58.8 jewels. Just makes sense that you get a negative number because as you're falling, you should be losing gravitational potential energy. All right, So let's take a look at part B. Now. In part B. We want to calculate the same variable. It's the change in the gravitational potential. But now we're going to choose our reference points. This arbitrary reference point to be somewhere else. So now what happens is we have the floor like this, we have the ball that's still six m above the ground. But now what we're doing is we're sort of choosing this height here to actually be zero. This is where my Y. Equals zero and therefore this is where my gravitational potential energy is going to equal zero. The ball is still going to fall three m and so it's still going to end up at some height. But now we want to calculate the gravitational potential energy. Right? So what happens here as it falls to a height of why equals negative three? So it's going to fall three m. The delta. Why the change is the same, no matter how you set the numbers. So the change is still three. So we want to calculate the delta. You. So now what happens is my delta U. Is going to be MG. And then why final minus Y. Initial? So what happens? We're just gonna get two times 9.8 times negative three. Because what happens is we're gonna get negative three minus zero. So that's ry initial. So what happens when you calculate this is you're gonna get negative 58-8 jewels again? So it turns out that in energy problems, whenever you're calculating the change in the gravitational potential, only the change in the height is important. So you can choose your arbitrary reference point, your relative, you know where y equals zero to be wherever you want in the problem, that's actually not going to change what happens to your changing gravitational potential because it doesn't depend on the initial or the final heights. What only matters is the difference between these two points right here. So the delta Y was equal to negative three in both of the cases here. And so that's why we end up with the same negative gravitation potential. So usually one good rule of thumb is that if you know delta, why if you actually know the change in the height, you can set the ground level, right? Where we set our ground level are arbitrary reference point to be wherever you want and that's where you're U. G. Is going to be equal to zero. Usually what you want to do is you want to pick the lowest point of the problem because it's gonna make your calculations a lot simpler. Alright, So that's it for this one guys. I mean if you have any questions.