-Good morning.
We have already worked with gravitational potential energy.
Today, we are going to introduce universal gravitational potential energy.
Flipping Physics
Bobby, what is the equation we already have for gravitational potential energy?
- Gravitational potential energy equals the mass of the object times acceleration due to gravity times h,
where h is the vertical height above the horizontal zero line.
Yeah, that's right and remember we have to decide
where to set the horizontal zero line in order to use this equation.
- Correct, and Bo where is this equation applicable? - Where is this equation applicable?
- Yes, when can we use this equation?
- We have always used this equation for gravitational potential energy.
So, I don't really understand your question.
- That is a fair point. How about this instead?
Billy, if this equation represents your gravitational potential energy,
then this is the gravitational potential energy which exists between you and what other object?
- That is the gravitational potential energy between me and … well, little g
is the acceleration due to gravity here on planet Earth, so that must be the gravitational potential energy
which exists between me and the Earth.
- Correct. So now back to the question for you Bo, where is this equation applicable?
- Oh, okay. So we can use gravitational potential energy equals the mass of the object times the acceleration due
to gravity times the vertical height above the horizontal zero line when the object is on the surface of a planet.
- That is correct. More generally speaking, we can use this equation whenever the gravitational field
is constant.
When you look at the surface of a planet on a local level, like where we are right now, I am assuming,
the acceleration due to gravity has a constant magnitude of, on planet earth,
9.81 meters per second squared down.
We live our lives in a constant downward gravitational field which we defined in our previous lesson.
When we look at the Earth from a non-local, or a global perspective,
you can see the gravitational field is not constant.
As we get farther from the Earth the magnitude of the gravitational field decreases and,
therefore, the gravitational field lines in the drawing get farther apart.
The equation for Universal Gravitational Potential Energy is the negative of the quantity,
big G or the universal gravitational constant which has a value of 6.67 times ten to the negative 11 newtons
times meters squared divided by kilograms squared, times mass 1, times mass 2, all divided by r,
the distance between the centers of mass of the two objects.
- Isn’t that Newton’s Universal Law of Gravitation?
- What’s up with the negative?
- Where did the capital U for gravitational potential energy come from?
I thought it was capital PE sub g.
- Okay. So, one thing at a time.
I use a capital U sub g for universal gravitational potential energy, because that is what is typically done.
I don’t know why it is a capital U. I am sorry.
Second, no, the equation for universal gravitational potential energy
is not Newton’s Universal Law of Gravitation, however, it is similar.
I think it would be helpful to compare these two equations for gravitational potential energy
to the two equations we have for the force of gravity.
Remember, we have an equation for the force of gravity which is planet specific,
which means it is valid where the acceleration due to gravity is constant,
just like we have an equation for gravitational potential energy where the gravitational field is constant.
And then we have Newton’s Universal Law of Gravitation
which is the force of gravity which exists between any two objects
and is an equation we can always use for the force of gravity.
Now, we also have Universal Gravitational Potential Energy
which is the equation we can always use for the gravitational potential energy
which exists between any two objects.
Now let’s talk about the negative sign.
I think, the easiest way to understand the negative sign is through a graph.
Let’s graph the gravitational potential energy which exists between an object and the Earth.
On the y-axis, we have the gravitational potential energy and on the x axis we have, r,
the distance between the centers of mass of the two objects.
Let’s start with an object on the surface of the planet.
Bobby, what would the gravitational potential energy which exists between the Earth and an object
which is on the surface of the Earth be equal to?
- Using the equation,
it would be equal to the negative of the universal gravitational constant times the mass of object one,
which would be the object, times the mass of object two, which would be the Earth, all divided by, r,
the distance between the center of mass of the object
and the center of mass of the Earth, which is the radius of the Earth.
- And, Bo, how about the universal gravitational potential energy of the Earth-object system
when the object is infinitely far away from the Earth?
- The object is infinitely far away from Earth?
- Yes.
Then r, the distance between their centers of mass would be infinitely large
and anything over infinity is zero, so the universal gravitational potential energy would be zero.
- And because the equation is proportional to one over r,
the curve would have a concave downward shape and look like this.
Notice that, for universal gravitational potential energy,
we do not need to set the zero line because the zero line is already set for us.
The zero line is where both objects are infinitely far away from one another.
A reason the zero line is located infinitely far away is to make Newton’s Universal Law of Gravitation
and universal gravitational potential energy both be zero when the objects are at the same locations,
when they are infinitely far from one another.
Also, class, can universal gravitational potential energy ever be positive?
[Class]: No.
- Universal gravitational potential energy will never be positive.
This is because the zero line is where the two objects are infinitely far away from one another.
This predetermined location for the zero line
is why the negative is in the equation for universal gravitational potential energy.
But please realize the change in
universal gravitational potential energy can be positive.
It is the value itself of universal gravitational potential energy, which will always be negative.
But the change in universal gravitational potential energy can be positive.
One other item to notice about the gravitational potential energy
which exists between an object and the Earth when the object is on the surface of the planet,
in other words, when r equals the radius of the earth.
At this point the acceleration due to gravity equals the universal gravitational constant
times the mass of the Earth divided by the radius of the Earth squared.
We derived this equation twice previously
and you definitely need to show that derivation whenever you use this equation.
Also, we can multiply the universal gravitational potential energy by the radius of the Earth
over the radius of the Earth because that equals 1
and then we have the equation for the acceleration due to gravity on the surface of the Earth
inside the parenthesis.
So we can substitute in little g and we get that the universal gravitational potential energy
of an object on the surface of the planet
equals the negative of the mass of the object times the acceleration due to gravity times the radius of the Earth.
Note the similarity between that
and the gravitational potential energy equation we use in a constant gravitational field.
[Class]: Coolio. Whoa. Wow.
- Yeah.
Three things to be careful of:
1) Please do not forget the negative. It’s easy to forget and, sadly, often is.
2) Remember it requires two objects to have gravitational potential energy: object 1 and object 2.
A single object, by itself, cannot have gravitational potential energy.
And 3) The variable “r” is not squared in universal gravitational potential energy.
“r” is squared in Newton’s Universal Law of Gravitation, which means
students have a tendency to add a square, it is not there, beware.
Again, do not add a square, it is not there, beware.
Thank you for learning with me today, I enjoyed learning with you.