- [Tutor] Let's do a little bit
of a 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 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,
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,
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's
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 wanna 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 rs 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,
the gravitational potential energies
are always gonna 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
time the mass of the second object divided by r
and 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 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.