- Good morning.
Bo, what are the two equations we have for the force of gravity acting on an object?
Flipping Physics
- Well, we have force of gravity equals mass times acceleration due to gravity.
- Which we can use when the acceleration due to gravity is constant,
like on the surface of the Earth where the acceleration due to gravity is 9.81 meters per second squared.
- Yeah, thanks Billy.
Um, the other equation is Newton’s Universal Law of Gravitation
where the force of gravity equals the universal gravitational constant
times mass one times mass two
all divided by the square of the distance between the centers of mass of the two objects.
And we can always use that equation, even if the acceleration due to gravity is not constant.
- Now we are going to introduce the concept of the gravitational field.
let's start with the simpler of the two equations.
If we divide the equation for the force of gravity by mass,
we get the acceleration due to gravity equals the force of gravity divided by mass.
Billy, what are the units for little g here?
- That would be force in newtons divided by mass in kilograms, so newtons over kilograms.
- I thought acceleration was in meters per second squared.
- It’s the same thing because newtons are kilograms times meters per second squared,
so if you divide that by kilograms, you get meters per second squared.
I get the units, but I don’t know what this is…
- Okay, so first off. Yes, Newton’s per kilogram is the same as meters per second squared.
Second, the acceleration due to gravity equals the force of gravity divided by mass is the equation for a
gravitational field when the gravitational field has a constant value, like it does on planet Earth.
- Hold up, did we not recently prove the acceleration due to gravity at the top of Mount Everest is
9.77 meters per second squared
and therefore show the gravitational field on the surface of the Earth is actually not constant?
- Okay, sure, right. It does vary ever so slightly, however, for our purposes
we can assume the acceleration due to gravity on the surface of our planet is constant
and therefore there is a constant gravitational field which exists on the surface of planet Earth.
We illustrate it this way....
These lines here are called field lines, more specifically, gravitational field lines.
The strength of the field is represented by how close the lines are to one another.
Because all these lines are parallel and down,
the distance between the lines is constant and they illustrate a constant downward gravitational field.
In other words, we live our lives in a constant downward gravitational field
which has a magnitude of 9.81 meters per second squared.
Any object we drop while standing on the surface of this planet will, in the absence of air resistance,
increase its velocity by 9.81 meters per second down every second.
The concept of a gravitational field essentially removes the mass of the object
which has the force of gravity acting on it
and identifies that the mass of the object is irrelevant as far as the acceleration due to gravity is concerned.
- No, I’m sorry.
I'm still confused. What ... what is a gravitational field?
- How about this?
If Mr.P were holding a book, the book would have a force of gravity acting on it.
The same would be true for a medicine ball and an umbrella.
Each object has a different mass and therefore a different force of gravity acting on it.
However, because the gravitational field equals the force of gravity divided by mass,
each object has the same ratio of 9.81 newtons per kilogram
and therefore the same gravitational field acting on it.
But imagine if Mr.P were not holding anything.
Even though there is no mass in that location,
that region of space still contains a gravitational field with a value of 9.81 newtons per kilogram.
We could put a mass there and it would result in a corresponding force of gravity acting on the object.
That is what it means to have a gravitational field.
The empty space contains the possibility of a force of gravity;
all you need to add is a mass to experience the force of gravity.
- Wow, thanks Bobby.
- You're welcome.
- Okay, so that is the constant gravitational field which exists when we look at the planet on a local level .
However, if we look at the planet from a non-local level,
in other words, from outer space, the gravitational field is no longer constant.
Bobby, can you derive the equation for the gravitational field there,
when we look at the planet from outer space?
- Uh. Maybe?
OK, so. Going back to the beginning here,
we need to set our original force of gravity equation,
force of gravity equals the mass of the object times the acceleration due to gravity
equal to Newton’s Universal Law of Gravitation
or the universal gravitational constant times the mass of the object times the mass of the Earth
all divided by the distance between the centers of mass of the two objects and …
[Class]: Everybody brought the mass of the object to the party!
Everybody brought mass!
- Correct. We have derived this equation before, however,
realize now we can identify this as the gravitational field which surrounds and is caused by the Earth.
And, more generally, if we replace the mass of the Earth in this equation with the mass of any object,
you get the gravitational field that surrounds and is caused by any object which looks like this....
Notice, according to the equation, because the gravitational field is proportional to one over r squared,
the gravitational field decreases as we get farther from the planet.
And notice I have added a beautiful drawing of Mt. Everest to my Earth drawing.
The peak of Mt. Everest is a bit farther out in the gravitational field
and therefore the drawn field lines are just a bit farther from one another
and therefore the gravitational field is just a bit weaker at the top of Mount Everest.
- That’s right. Previously we determined the acceleration due to gravity on the International Space Station was
8.6 meters per second squared.
You can add that to your drawing and the field lines are even a bit farther apart there,
again showing the gravitational field is weaker as you get farther from the planet.
- Correct. The farther apart the gravitational field lines are in the gravitational field drawing,
the weaker the gravitational field.
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.
- Are those lines actually real?
- Okay. So, no. These gravitational field lines in the drawing
are a way for us to visualize the concept of a gravitational field.
It's our way of illustrating what the acceleration of any object would be if it were placed in a location
in the gravitational field. But no, it’s not like these lines exist in the real world.
They are a visual construct to help us understand the physics.
Oh, also notice the equations for gravitational fields we have here are the magnitudes of those fields.
For the constant gravitational field, the direction is downward, toward the center of the planet.
For the gravitational field which exists around any object, the direction is toward the center of mass of that object.
- So the gravitational field is always toward the center of mass of the object?
- Right. So, the direction of the gravitational field is always toward the center of mass of the object
creating the gravitational field because the force of gravity is always a pull.
Thank you very much for learning with me today. I enjoyed learning with you.