Calculating gravitational potential energy | Modeling energy | High school physics | Khan Academy

In previous videos, we have introduced the idea of energy as the capacity to do work, and we have talked about multiple types of energies. We've talked about kinetic energy, energy due to motion. We've talked about potential energy, which is energy by virtue of position.

When we're talking about potential energy, we're talking about it relative to some other position. In particular, in this video, we're going to talk about gravitational potential energy, which is potential energy due to position in a gravitational field.

So let's say that this is the surface of the Earth. Let's say that I have a 5-kilogram mass right over here, and let's say that it is 10 meters above the surface of the Earth. My question to you is: how much more potential energy does it have in this position than when it is in this position, when it is sitting on the surface of the Earth, 10 meters lower? Pause the video and try to think about that.

All right, now let's work on this together. So our gravitational potential energy is going to be equal to our mass times lowercase g, which you can view as the constant for Earth's gravitational field near the surface of Earth. The reason why I say near the surface of Earth is, as you get further and further from Earth, this thing could actually change. But near the surface of the Earth, we assume that it is roughly constant.

Then you multiply that times your height. So calculating this is pretty straightforward as long as you know what g is. We can approximate g as 9.8 meters per second squared. So when you multiply all of this out, this is going to be equal to your mass, which is 5 kilograms, times the gravitational field constant, so times 9.8 meters per second squared, times your height, which in this situation is 10 meters.

So times 10 meters, and so this is going to be equal to 5 times 9.8, which is 49, times 10, which is 490. We have kilograms, and then we have meters times meters, so times meter squared per second squared. These might seem like strange units, but you might recognize this as also the units of force times distance, which we could also express in terms of joules.

So this is 490 joules, which is our unit for both energy and our unit for work. Now let's make sure that this makes intuitive sense. Well, one way to think about it is: how much work would it take to go from here to here? Well, you're going to be lifting it a distance of 10 meters.

As you're lifting it a distance of 10 meters, what is the force you're going to have to apply? Well, the force you're going to have to apply is going to be the weight of the object. The weight is its mass times the gravitational field. So in order to put it in that position from the ground, you're going to have to do its weight times the height, or 490 joules of work.

You can do 490 joules of work to get it there, and then you can think about it as that energy being stored this way, and now it can then do that work. How could it do that work? Well, there's a bunch of ways you could do it. You could have this attached to maybe a pulley of some kind, and if then if it had another weight right over here, and let's just for simplicity assume it has the same mass.

Well, if you let this first purple mass go, it's going to go down, and if you assume that this pulley is completely frictionless, this mass is going to be lifted by 10 meters. So if you have a 5-kilogram mass that is lifted by 10 meters in Earth's gravitational field, near the surface of the Earth, you would have just done 490 joules of work.

So hopefully this makes sense why you're just really taking the weight of the object times its height, and hopefully it also makes sense that it then has the capacity to do that amount of work. In this case, we said relative to sitting on the ground.