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