Gravitational potential energy at large distances | AP Physics 1 | Khan Academy

Let's do a little bit of review of potential energy and especially gravitational potential energy because in this video we're going to get a little bit more precise.

So, let's say that I have an object here. It has a mass of m, and I were to change its position in the vertical direction. We're assuming that we are on Earth, where the gravitational field is g, and we have a change in the vertical direction of delta y. So this is its new position, the mass of the object of mass m.

So, what is its change in gravitational potential energy? Well, we have seen this before. Our change in potential energy due to gravity is going to be the mass of our object times the gravitational field times our change in position in the vertical direction.

Now, this is all good, but there's a couple of things that we can get a little bit more precise or that we can think about in this video. One is we assumed a constant gravitational field, and that's fine if we're near the surface of a planet, and we also are just doing everything on a relative basis.

But is there a way to come up with an absolute number on what is the gravitational potential energy? Well, to think about that, let's just go back to Newton's law of gravity. Newton's law of gravity tells us that if I have two objects, so this object has mass m1, and this object has mass m2, and the distance between their center of masses is r, that the magnitude of the force of gravity, and if the force of gravity is acting in this direction on that object, it would have an equal and opposite direction on that object right over there.

So, the magnitude of the force of gravity is going to be equal to the gravitational constant times the product of the two objects' masses divided by the radius squared. Or, another way to think about it, if we wanted to know the gravitational field created by m2, well then you would just divide both sides by m1.

So, if you divide both sides by m1, you get that the gravitational field actually isn't constant. It is equal to the gravitational constant times, these cancel out, m2, the mass of the object generating the gravitational field divided by r squared.

And so, this can actually help us come up with an absolute formula for gravitational potential energy. We can say that gravitational potential energy, and this is typically denoted with a capital G over here, is equal to, let's say, the mass of the object, and let's just call that m1, the mass of the object times the gravitational field, so that's g times the mass of the object creating the field divided by r squared, times the height that this is above the center of mass of the object generating the gravitational field.

Well, what's that? Well, if this was Earth, you would view r as, well, how far am I above the center of mass of Earth? And so, you would multiply that times r times r, and so this r and one of the r's in the r squared will cancel out. So, that will just go away, and then this will just become an r.

And now, one thing to think about is what should the potential energy be as r goes to infinity? Well, what is the gravitational field as we go to infinity, as r gets really, really, really large? Well, as r gets really, really, really large, the gravitational field goes to zero, but at the same time, the further you get away, the higher the potential energy you should have.

And so, the way that we've thought about this is, we say, hey look, let's just put a negative out front. You could just put it right over here, and then it all works out that gravitational potential energies are always going to be negative, but they get less negative the further out you get to. And at infinity, it gets to zero.

So, let me rewrite this. The potential energy due to gravity can be written as the negative of the gravitational constant times the mass of the first object times the mass of the second object divided by r. You can see this looks an awful lot like Newton's law of gravity.

What the difference is? We have a negative in front of the gravitational constant; instead of dividing by r squared, we only divide by r. But this is really, really, really, really valuable now because now we have an absolute sense of what gravitational potential energy is, and it meets all the properties we need.

It would still meet this idea that the further you get away, you will have an increase in gravitational potential energy because it's going to become less negative. But at the same time, as you go to infinity, as r gets infinitely far, your potential energy is going to approach zero.

So, I will leave you there. In future videos, we will apply this.