This video is about Gravitational Fields.
We already encountered Gravitational Fields in Subtopic 6.2.
There we learned about General Gravitational Field Concepts, Gravitational Field Strength
and Gravitational Potential Energy.
In fact we already worked with Gravitational Potential Energy in Subtopic 2.3.
In this video I will rely on this previous learning, so feel free to go back and watch
some of the videos from Subtopics 6.2 to help you refresh some of these concepts.
In this video we'll learn about Gravitational Potential at a point and Gravitational Echipotentials.
Let's begin with Gravitational Potential at a point.
So far we have been using potential and potential difference interchangeably for both Gravitational
and Electric Fields.
Even though potential and potential difference are closely connected, they mean different
things.
Let's explore this difference.
We'll start by defining Gravitational Potential Difference which is the work done when moving
a unit mass between two points.
In our system a unit mass means one kilogram.
We can express this definition by using an equation.
So Delta Vg Gravitational Potential Difference is equal to W, work done, divided by M, mass.
You can find the rearranged version of this formula in the IB Physics Data Booklet where
the variables are defined like this.
Now that we have clearly defined Gravitational Potential Difference between two points, let's
discuss Gravitational Potential at a point.
Logically, if a Gravitational Potential Difference exists between two points, then Gravitational
Potential must be a certain value at one of the points and a different value at the other
point.
Since we can calculate the Gravitational Potential Difference between any two points in the
universe, we should also be able to find the Gravitational Potential at any given point.
However, in order to do this, we have to choose a reference point where Gravitational Potential
is equal to zero.
Recall from Subtopic 2.3 that when we work with Gravitational Potential Energy, we often
choose the ground as our zero reference point.
For example, we often say that when an object is on the ground, its height is zero, so its
Gravitational Potential Energy is zero.
This also means that Gravitational Potential on the ground is equal to zero.
When we are working with objects on Earth, we could take C level to be our zero Gravitational
Potential reference point.
However, when we leave Earth, so for example if we wanted to find the Gravitational Potential
at the surface of Mars, this reference point would not make much sense.
So we are back to the question of where this zero Gravitational Potential reference point
should be chosen.
The solution that physicists came up with is to define Gravitational Potential to be zero
at infinity.
Of course, infinity is not a real place and we cannot actually go there to check whether
Gravitational Potential there is really zero, but we are able to work further with Gravitational
Potential based on this definition.
Recall from Subtopics 6.2 that Newton's law of gravitation tells us that the Gravitational
Force, F, is inversely proportional to the square of the distance between two objects.
As we increase the distance between two objects, so R to infinity, 1 over R squared becomes
infinitely small, so practically zero.
This means that when two objects are at an infinite distance, the Gravitational Force
between the objects is zero, so the Gravitational Potential of the system is also zero.
By system we mean the system composed of the two objects.
As the objects start to approach each other and the distance between them becomes finite,
there will be gravitational attraction between the objects due to the Gravitational Force.
If we want to move the objects back to an infinite distance again, work must be done
on the system.
In other words, we must supply energy to move the objects apart.
Therefore when the two objects are closer to each other than infinity, the Gravitational
Potential of the system is negative.
Therefore Gravitational Potential at any given point in the universe is also negative.
We'll explore this idea in more detail in subtopic 10.2.
For now, let's go ahead and define Gravitational Potential at a point.
It is the work done per unit mass when moving a test mass from infinity to the point.
Now that we have defined Gravitational Potential difference and Gravitational Potential at a
point, we are ready to discuss Gravitational Equipotentials.
As a start, let's see a simple example.
Imagine that you are in a room.
There is the ground and three shelves.
You can picture a simple bookshelf.
The shelves, as you see on the diagram, are 1, 2 and 3 meters above the ground.
As we learned earlier, we usually take Gravitational Potential to be 0 at infinity.
However, we also mentioned that we are free to choose a different zero reference point,
if it makes sense, in the context of the given situation.
So in this example, just as we have done in subtopic 2.3, when working with Gravitational
Potential energy, we will choose the ground as our zero reference point.
So we can write that at the ground, Vg is equal to 0.
Let's find the Gravitational Potential of the three shelves.
We saw earlier that Gravitational Potential difference, or change in Gravitational Potential,
is equal to W over M, where W is the work done to move an object from one point to another,
and M is mass.
The work done when we move an object from the ground level to one of the shelves is
equal to the Gravitational Potential energy gained by the object.
As we learned in subtopic 2.3, we can calculate this as M times G times delta H.
G is the Gravitational Acceleration, and delta H is the change in height.
We still have M in the denominator.
After cancelling M, we end up with G times delta H, where G is 9.81, and when we are
moving an object from the ground to the first shelf, delta H is 1 meter.
This gives us 9.81 or approximately 10.
The unit that we use for Gravitational Potential is based on its definition, so work done divided
by mass.
For our unit here is Joules per kilogram.
Let's add this to the diagram.
Logically, since the second shelf is 2 meters and the third shelf is 3 meters above the
ground, the Gravitational Potential energies here will be 20 and 30 Joules per kilogram.
You might be asking, how come that the Gravitational Potential values here are positive, when a
few minutes ago we said that Gravitational Potential is always negative.
The reason is that instead of infinity, here we chose the ground to be our zero reference
point.
So when we move an object from the ground to one of the shelves, we are supplying energy
to the system.
When the object moves back to the ground, so to the zero potential, the work is done by
the Gravitational field.
Recall that when a mass is moved from a specific position to infinity, work must be done on
the object against the field.
So this is the reason why we have positive Gravitational Potential values here.
You have probably noticed that no matter where we are on a given line, so for example at
this point, this point or this point, Gravitational Potential is always the same.
Because of this, the lines you see here are called Gravitational Equipotentials.
Based on what we just discussed, we can write that Equipotentials represent points where
the potential is the same.
When we are working in two dimensions, so for example on a screen or a piece of paper,
we have Equipotential lines, while in three dimensions we have Equipotential surfaces.
So for a shelf, as we have seen here, the Equipotential surface is the entire horizontal
flat area of a given shelf.
When we learned about fields, we saw how to draw electric field lines, but we haven't
drawn Gravitational field lines.
The concept is very similar.
Gravitational field lines are lines along which a point mass would move when released
in a Gravitational field.
To visualize the lines near the surface of the earth, just take a piece of string and
hang a small mass at the end of it.
The direction of the string shows you the direction of the Gravitational field lines.
Let's add field lines to our diagram.
Since field lines show the direction in which a mass would move, when released in the Gravitational
field, logically the field lines here are pointing downwards.
Based on this, we can also conclude that when a mass is released in a Gravitational field,
it moves from a higher to a lower potential.
Finally, let's add what we already see on the diagram, that Equipotentials are perpendicular
to field lines.
Finally, let's see how Equipotentials and field lines look like around a point mass
or sphere, for instance a planet.
So Equipotentials and field lines in such a situation look like this.
Notice that the direction of the field lines is towards the point mass or sphere and that
the Gravitational field gets stronger as we get closer to the mass or sphere.
This is also the reason why the Equipotentials here are not equally spaced.
The potential difference between the Equipotentials that are shown by the dashed lines is the same,
but the distance between the Equipotentials increases as we move away from the mass or
sphere.
A logical question to ask here is why are the Equipotentials on the left equally spaced
if they represent Equipotentials near the surface of a planet?
Theoretically these lines are curved and are not equally spaced, but the curvature and
the differences are so small that we can simply ignore them when drawing the Equipotentials.
Let's summarize what we have learned in this video.
We began by defining Gravitational potential difference between two points.
Then we learned a formula to calculate Gravitational potential difference along with the variables.
Moving on to Gravitational potential, we said that it is defined to be zero at infinity
and recalled Newton's law of gravitation, which tells us that the gravitational force
between two objects is inversely proportional to the square of the distance between the
objects.
From these two statements we deduced that when two objects are at an infinite distance,
the gravitational force between the objects is zero, so the gravitational potential of
the system is also zero.
When the objects are moved so that the distance between them becomes finite, work must be
done in order to move the objects again to an infinite distance from each other.
As a result, the gravitational potential of such a system is always negative.
Finally, we summarized these statements in the definition of Gravitational potential
at a point.
Next, we discussed Gravitational Equipotentials and stated that Equipotentials represent points
where the potential is the same.
We added that when a message released in a gravitational field, it moves from a higher
to a lower potential and that Equipotentials are perpendicular to field lines.
We finished up by drawing Equipotentials and field lines near the surface of a planet and
around a point mass or a planet.
This completes our discussion of Gravitational fields.
In the next video, we'll learn about electrostatic fields.