### Video Transcript

In this video, we’re going to learn about gravitational potential energy. We’ll learn what this energy is, how to calculate it, and how it relates to work done on an object. To start out, imagine that you are a contestant on the game show “Guess What,” where contestants win prizes for making correct guesses. The game show host leads you to an area where a number of interesting looking objects are resting on pedestals. The host explains that your challenge is using the floor as reference point to guess which of these objects is the most energetic.

On the face of it, this question seems strange. After all, none of the objects are in motion. To understand it, we’ll want to know something about gravitational potential energy. When we talk about gravitational potential energy, or GPE for short, we can zoom way out and consider the Earth as a whole as a start point. Because the Earth is a massive object, we know that it creates a gravitational field around itself. This field is attractive to other masses, say such as an orbiting satellite.

Because of the force of attraction that exists between the Earth and other masses, those other masses have a tendency to move. Because of this tendency, objects like the satellite have potential energy. If we call the potential energy of this object capital 𝑈, then we can write that the magnitude of that gravitational potential energy is equal to the universal gravitational constant times the mass of the Earth multiplied by the mass of the smaller object divided by the distance between their centers.

This expression for gravitational potential energy is useful to us over large distances where the acceleration due to gravity is not constant. But let’s say that we zoom in from this astronomic scale view to a perspective that’s closer to Earth’s surface. In this perspective, which is more like the perspective of our everyday life, the acceleration due to gravity — we call 𝑔 — is essentially a constant value, 9.8 meters per second squared downward.

In this environment, where we can treat 𝑔 as constant, say we decide to lift this box of mass 𝑚 from rest on the ground to placing it at rest on a tabletop. In order to lift the box up, we’ll need to exert a lifting force on it, at least as great as the force of gravity acting on the box, 𝑚 times 𝑔. And then if the height of the table is ℎ, we can say that we’ve moved the box that distance.

If we were to calculate the work we’ve done in moving this box up a height ℎ, the force we used in moving the box was equal to 𝑚 times 𝑔. And the distance against gravity we moved it is the height ℎ. So the work we did on the box was 𝑚 times 𝑔 times ℎ. This is energy that we’ve given to the box by raising it up in a gravitational field. It’s a potential energy of the box because the box is at rest. And we symbolize it PE sub 𝑔 for a gravitational potential energy.

So the gravitational potential energy of an object is equal to its mass times the acceleration due to gravity times its height. Like we mentioned before, this formulation for GPE assumes that 𝑔, the acceleration due to gravity, is constant. In other words, we’re not working over astronomically large distance scales.

Another thing to notice about this equation is that the height ℎ implies a reference point or a zero point to measure with respect to. In our example of the box and the table, our reference point was ground level. That was where we set height equals zero.

It’s an interesting fact of gravitational potential energy that we can choose our height reference to be anywhere. We could choose height to equal zero at the level of the tabletop or at the top of the box when it’s on the table or the level of the ground or anywhere in between. It’s up to us. Often because we’re used to thinking of it that way, we choose ground level to be where ℎ equals zero.

One last thing to notice about gravitational potential energy is when we calculate a change in GPE, all that matters is the start and end point of our object. Here’s what that means. We know our box mass 𝑚 started here on the floor and ended here on the table. From the perspective of gravitational potential energy, it doesn’t matter how the box moved from its start point to its end point. Those are the only points that matter. The box could take virtually any path and moving from its start to end point. And the final change in gravitational potential energy would still be the same.

Let’s get some practice working with GPE through a couple of examples. The Great Pyramid of Giza has a mass of 6.45 times 10 to the ninth kilograms. And its center of mass is 34.7 meters above the surrounding ground. How much gravitational potential energy is stored in the pyramid? We can call this quantity PE sub 𝑔. And to start on our solution, we can recall that gravitational potential energy is equal to an object’s mass times 𝑔, the acceleration due to gravity, times its height.

In the case of this pyramid, we’re told its total mass. We’re also told the height of its center of mass above ground level. And we can assume that the acceleration due to gravity is exactly 9.8 meters per second squared. When we enter these values into our equation for PE sub 𝑔 and calculate their product, we find the result of 2.19 times 10 to the 12th joules. That’s the gravitational potential energy stored in the pyramid.

Now let’s look at an example involving work and GPE. A cruise ship is docked in a harbor. The level of the dock is 2.15 meters above the waterline. A woman on the cruise ship stands on a deck 55.5 meters above the waterline. She pushes a 36.0-gram mass pebble off the deck. A man standing on the dock holds out a net 1.20 meters above the top of the dock and catches the pebble in the net. Assume that vertically upward motion corresponds with positive displacement.

How much work is done by gravity on the pebble? How much does the gravitational potential energy of the pebble change? Taking gravitational potential energy at the waterline to be zero, what is the gravitational potential energy of the pebble on the deck of the cruise ship? Taking gravitational potential energy at the waterline to be zero, what is the gravitational potential energy of the pebble when it is caught in the net? Taking gravitational potential energy at the waterline to be 21.5 joules, what is the gravitational potential energy of the pebble on the deck of the cruise ship? Taking gravitational potential energy at the waterline to be 21.5 joules, what is the gravitational potential energy of the pebble when it is caught in the net?

In this exercise, we want to solve first for the work done on the pebble by gravity and moving it from the deck of the cruise ship into the net. Then we wanna solve for the change in potential energy the pebble experiences over that descent. And finally, we wanna calculate the gravitational potential energy of the pebble under four different conditions. We’ve called these PE sub 𝑔 one, two, three, and four.

Let’s start on our solutions by drawing a diagram of the situation. In this scenario, we have a ship docked at harbor. The height of the ship’s deck above the waterline — we’ve called ℎ sub 𝑠 — is 55.5 meters. The dock next to the ship is a height we’ve called ℎ sub 𝑑 of 2.15 meters above the waterline. And on the dock, a man stands holding a net a height we’ve called ℎ sub 𝑛 above the height of the dock. It’s 1.20 meters.

We’re told that a woman on the deck of the ship kicks a pebble, which falls down into the net the man is holding. In part one, we want to solve for the work done by gravity on the pebble as it falls from the deck of the ship into the net. We can recall that work is equal to force dotted with displacement.

In our case, because the force of gravity and the displacement of the pebble are in the same direction, we can write work as a simple product between these two terms. The pebble’s distance traveled is equal to the height of the ship minus the sum of the height of the dock plus the height of the net above the dock. The gravitational force on the pebble is equal to its mass times 𝑔.

In our statement, we’re told that the pebble’s mass is 36.0 grams. And we can treat 𝑔 as exactly 9.8 meters per second squared. When we plug in for these values, we’re careful to convert our mass from units of grams to units of kilograms. When we enter these values on our calculator, we find a result of 18.4 joules. That’s how much work gravity did on the pebble as it fell.

Next, we wanna calculate the change in potential energy the pebble undergoes in this transition from the deck of the ship into the net. We know that, in general, gravitational potential energy is equal to an object’s mass times the acceleration due to gravity times its height ℎ.

In the case of our falling pebble, the change in potential energy is equal to 𝑚𝑔 times the quantity its final height minus its initial or starting height. The final height of the pebble is ℎ sub 𝑑 plus ℎ sub 𝑛. And its initial height is the height of the ship, ℎ sub 𝑠. Plugging in for these values, again converting our mass to units of kilograms, we find a result of negative 18.4 joules. That’s the change in potential energy of the pebble.

Next, we wanna solve for the gravitational potential energy of the pebble under different conditions. Recall that, to calculate this gravitational potential energy, we’ll want to set a zero point for our elevation as a reference. When we calculate PE sub 𝑔 one, our reference point is the waterline, where at that level the gravitational potential energy is zero joules.

In this scenario, our pebble is on the deck of the ship. So PE sub 𝑔 one is equal to the pebble’s mass times 𝑔 times the height of the ship, ℎ sub 𝑠. Plugging in for these values, again using kilograms for units of our mass, we find a result, to three significant figures, of 19.6 joules. That’s the pebble’s gravitational potential energy.

Next, we want to solve for PE sub 𝑔 two. This is when the pebble has fallen into the net the man is holding. And our reference point is still the waterline at zero joules of gravitational potential energy. In this case, PE sub 𝑔 two is 𝑚 times 𝑔 times ℎ 𝑑 plus ℎ 𝑛. When we plug in for these values and calculate, we find a result of 1.18 joules. That’s the pebble’s gravitational potential energy under these conditions.

When we move on to calculate PE sub 𝑔 three, now our reference level, still the waterline, has a gravitational potential energy of 21.5 joules. Our pebble is back on the deck of the ship. So PE sub 𝑔 three is equal to 𝑚 𝑔 ℎ sub 𝑠 plus 21.5 joules, the offset to account for our new reference level. We know that 𝑚 times 𝑔 times ℎ sub 𝑠 was already calculated. That’s the result we found for PE sub 𝑔 one. When we add that result with 21.5 joules, we find a result of 41.1 joules. That’s the gravitational potential energy of the pebble on the deck with this new reference level.

And finally, we go on to calculate PE sub 𝑔 four. This is when the pebble is back in the net. And our reference level at water level is 21.5 joules of gravitational potential energy. This means that PE sub 𝑔 four is 𝑚𝑔 times the quantity ℎ sub 𝑑 plus ℎ sub 𝑛 plus 21.5 joules. We’ve already calculated 𝑚𝑔 times the quantity ℎ 𝑑 plus ℎ sub 𝑛. That’s PE sub 𝑔 two. So when we add that result to 21.5 joules, we find it’s equal to 22.7 joules. So we see that the gravitational potential energy of the pebble depends on its height. And it also depends on the energy level of our reference point.

Let’s summarize what we’ve learned so far about gravitational potential energy. First, we’ve seen that gravitational potential energy is energy due to an object’s vertical position with respect to a reference level. As an equation, we can write this as PE sub 𝑔 is equal to 𝑚 times 𝑔 times ℎ. We’ve also seen that gravitational potential energy is related to the work done on an object by gravity and moving it from one at rest position to another. And finally, we’ve seen that changes in gravitational potential energy only depend on the start and end point of an object, not on the path taken between those points.