For the next couple of problems we're going to be talking about gravitational potential energy.
Now, we talked quite a bit about potential energy due to gravity before,
but we had a few assumptions when we had that discussion,
Particularly, the fact that we were dealing with small distances.
So up to now, we've been treating acceleration of falling objects as a constant which we call (g)
and for this to be the case, we found that the force of gravity on an object has to be equal to (mg).
and we also found that the potential energy due to gravity is equal to (mgh),
and it turns out that this equation actually only works well over small distances
and why is that? Well, we only get this equation based on the assumption
that acceleration is constant.
In the last problems; however, we found that (g) is actually not constant
and that the force of gravity varies depending on where you are in the earth.
In particular, we found that the force of gravity more generally,
is actually proportional to 1 over the distance away from the center of the earth squared.
The force of gravity actually gets weaker the further away you get
from earth and stronger the closer you get.
This assumption that (g) is constant everywhere, and therefore that the potential energy
is this equation, is just not valid when we change our altitude too much.
Describing the gravitational force asf (mg) and the gravitational potential energy as (mgh),
works great over small distances especially if we're near the surface of the earth.
But as we change our distance away from the center of the earth,
(g) is no longer constant and these equations no longer become as accurate.
To figure out an equation for force and potential energy that works at all distances,
let's start by looking at this relationship we found here. That the force of gravity
is proportional to 1 over the distance away from the center of the earth squared.
So right now, this is a proportionality but we can always turn proportionalities into equations
by adding a constant of proportionality.
I can turn this proportionality into this equation over here with this constant of proportionality
which I'm going to call (G), and this (G) here is really to make sure
the units work out correctly in this problem, so that we can get a force,
even though we've really got 1 over a distance squared.
All right, but this doesn't seem quite right either.
We know that the force of gravity actually depends on mass.
And there's no mass in this equation yet, but we have to ask
whose mass does this gravitational force depend on?
Well, it should depend on the mass of the object which I've been calling (m),
and that should depend on a mass of the earth which I'm going to write as (ME).
Let 's add those into our equation.
All right, so now we've got a slightly more complicated-looking equation
but it's got all the essentials.
It got this constant of proportionality to make sure our units are right.
It got the mass of both the earth and the mass of the object we're looking at,
and the force is proportional to 1 over the distance between the object
and the center of the earth squared.
And it turns out this equation actually is correct and it has a very special name.
It's called Newton's Law of Universal Gravitation.
It's a much more general way of talking about the gravitational force all throughout the universe,
not just on earth too.
All right, so now we can describe the gravitational force at any scales we want.
We don't just have to stay close to the earth's surface
and we don't have to deal with small distances.
What about the gravitational potential energy more generally?
Well unfortunately, to drive the general potential energy formula,
we need a bit of calculus, so I'm just going to give it to you.
It's equal to negative G times mass of the earth times mass of the object over the distance
between the object and the center of the earth, no square this time.
So it looks quite a bit like this equation, but that's not that surprising.
These two equations look quite similar too.
Let's take just a moment to make sure this equation makes sense.
This equation had units of force and to get from this equation to this equation,
we basically multiply by distance to get rid of the square and the denominator.
This must have units of force times distance and force times distance
is equivalent to units of energy which is great because we're dealing with potential energy.
The units make sense, but what's the deal with this negative sign?
That seems kind of strange, so to see what this negative is doing here in the equation,
let's consider two points at distances R1 and R2 away from the earth.
We expect the potential energy of 0.2 over here to be greater
than the potential energy of 0.1 over here.
We saw when we were using (mgh) that as the distance away from the earth increased,
the potential energy increased as well.
So we should expect the same thing to happen here,
with (mgh) however, the distance away from the earth was in the numerator,
but in this case, the distance away from the earth is in the denominator.
The behavior is actually quite different.
You can imagine that as we get further away from the earth, this value of hour is going to increase
which will cause the potential energy to decrease which is the opposite
of what we want to have happened.
We need the negative sign here to make sure that as hour gets larger,
the potential energy gets larger as well and that can only be done by having a negative here.
Now, I know that's a rather confusing topic, so please feel free to discuss this on the forums.
Okay, now that we know more general formula for gravitational potential energy,
we can do problems relating to the earth and space and at large distances
without having to worry about the air induced by using (mgh).