Video: Gravitational Potential Energy

In this video, we see how gravitational potential energy is found near the Earth and far away, and also discover how it is related to work done on an object.

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Video Transcript

In this video, we’re talking about gravitational potential energy. This is a kind of energy that objects possess based on their position within a gravitational field. To see how this works, say that we have a ball on the ground. Then, say that we pick this ball up, which, as we know, takes some amount of effort. If we let go of the ball, we know from experience what will happen. It will pick up speed and fall back to ground. And when it does, right before the ball hits the ground, we know that it’ll have some amount of kinetic energy. This is energy due to the fact that the ball is in motion. As we consider this energy though, we may recall that energy must come from somewhere. That’s because energy is a conserved quantity. So, where does all this kinetic energy that the ball has right before it hits the ground come from?

Well, let’s imagine that the ball is back in our hand and that when it is, it’s some height — we can call it ℎ — above ground level. We know that this ball has been raised to that height within a gravitational field. We could think of it this way. If this is the Earth and this is our ball some little distance above the Earth, then the ball is being acted on by the gravitational field created by the Earth. The way this field acts is it exerts a force, the force of gravity on objects. And when the ball is being held in our hand, we could say that that force acts like this. It’s in the downward direction, tending to pull the ball back to ground. And we know that this force of gravity is acting on our ball regardless of its position. Even when the ball was at rest on the ground, where it can’t drop any further because it’s contacting Earth’s surface, gravity is acting on the ball.

It’s this force that caused the ball to drop when we released it. And what’s more, this force does work on the ball as it falls. The work that gravity does on the ball while it’s falling shows up in the ball’s kinetic energy. So we’ve partly answered our question. The ball’s kinetic energy came from work done on the ball by the force of gravity. But then how was gravity able to do that work in the first place? Well, the ball had to be lifted up to some height above ground level and then released. And to do that, to move the ball from this position up to this position, we had to do work on the ball against the force of gravity.

Now, when gravity did work on the ball, causing it to descend, we saw that that gave the ball some amount of kinetic energy. On the other hand, when we do work on the ball by lifting it up to a height ℎ above ground level, we also give the ball energy. But it’s not kinetic. Rather, we give it gravitational potential energy, GPE. And now, we’re able to fully answer our question of where the kinetic energy that the ball gained came from. It came from the ball’s gravitational potential energy that we gave to the ball by lifting it up. To recap this cycle, we do work on the ball against the force of gravity, which gives the ball gravitational potential energy. Then, when the ball is released, gravity begins to do work on the ball, and the gravitational potential energy the ball had is converted to kinetic energy. And we see that both of these energy transitions, the ball gaining gravitational potential energy and then that being converted to kinetic energy, involve work being done on the ball.

Now, when we think about how to quantify how much gravitational potential energy an object has, we express it in terms of three factors. The first factor is the mass — we can call it 𝑚 — of our object. The next one is the strength of the gravitational field that our object is in. Typically, this is represented using a lowercase 𝑔, which stands for the acceleration due to gravity that an object would undergo. And the last factor is the height — we can call it ℎ — of our object above some minimum height level. We see in the case of our ball that our height is measured in reference to ground level. If we take these three terms, 𝑚 and 𝑔 and ℎ, and multiply them together, that is equal to an object’s gravitational potential energy. So, if we know an object’s mass, we know the strength of the gravitational field it’s in, and we know its height above some minimum possible height. Then, we can calculate its gravitational potential energy.

The height ℎ is often something we’re given in a problem statement. It’s the vertical distance of our object above some minimum possible level. And again, in the case of our ball, that minimum possible level is ground level. Likewise, object mass is typically given to us or we can solve for it. But the gravitational acceleration 𝑔 is different. This isn’t something that we’re given, but rather it’s a constant that we can memorize. Now, if we look back over at our sketch of the Earth, we can see that all these gravitational field lines are pointing inward, toward Earth’s center. But say that we take a very up close zoomed-in view of our ball just above Earth’s surface. Looking at it from this perspective, the surface of the Earth would seem flat and all the gravitational field lines would be parallel to one another.

Within this framework, 𝑔, the acceleration due to Earth’s gravity, is approximately 9.8 meters per second squared. And so, when we go to calculate gravitational potential energy using this equation, that’s the value that we use for 𝑔. And as we mentioned, this value is worth memorizing because it’s often not given to us in a problem statement. Going back once more to our equation for gravitational potential energy, looking at the left-hand side, we may remember that if we take the product of an object’s mass and the acceleration due to gravity, then the result of multiplying these two values together is called the object’s weight.

If we use a capital 𝑊 to stand for the weight of an object, that’s equal to its mass times 𝑔. And this means we can substitute 𝑊 in for 𝑚 times 𝑔 in our equation for GPE. So sometimes, we’ll see an equation for gravitational potential energy written as weight times height. And if we do, we can recall that this is mathematically equal to 𝑚 times 𝑔 times ℎ. Knowing all this about gravitational potential energy, let’s get some practice with these ideas through an example exercise.

An object with a mass of 15 kilograms is at a point 10 meters above the ground. What is the gravitational potential energy of the object?

All right, so let’s say that this is ground level. And we’re told that our object is above this level a distance of 10 meters. Along with this, we’re told that the mass of our object — what we can call 𝑚 — is equal to 15 kilograms. We want to know, what is the gravitational potential energy of this object? To figure this out, we can recall that the gravitational potential energy of an object — we can refer to it as GPE — is equal to the mass of that object multiplied by the strength of the gravitational field the object is in all times the height of the object above some minimum possible level. For an object, like the one we have here, that’s within 10 meters of Earth’s surface, we can say that 𝑔, the acceleration due to gravity, is exactly 9.8 meters per second squared.

So, when we go to calculate this object’s gravitational potential energy, we know its mass, that’s 15 kilograms. We know 𝑔, that’s a constant, 9.8 meters per second squared. And we also are given ℎ, the height of the object above ground level, 10 meters. When we substitute in these values and then multiply them together, we find a result of 1470 newton meters. This is because a newton is equal to a kilogram meter per second squared. And then, we can recall further that a newton times a meter is equal to the unit called a joule. This is the unit typically used to express energies. So, our final answer is 1470 joules.

Let’s now look at another exercise.

An object held at a point 1.5 meters above the ground has 1176 joules of gravitational potential energy. What is the mass of the object?

All right, in this example, let’s say that this is our ground level and that this shape here is our object. We’re told that this object is 1.5 meters above the ground and that its gravitational potential energy is equal to 1176 joules. Given this information, we want to solve for the mass of this object. To do this, we can recall a mathematical relationship connecting mass, height, and gravitational potential energy. This relationship says that the gravitational potential energy of an object is equal to its mass multiplied by the acceleration due to gravity that the object experiences times its height above some minimum height value. In our case, it’s not GPE we want to solve for though, but it’s the object’s mass 𝑚.

To help us do that, we can divide both sides of this equation by 𝑔 times ℎ. That way, on the right-hand side, a factor of 𝑔 cancels in the numerator and denominator, and so does a factor of ℎ. So then, an object’s mass is equal to its gravitational potential energy divided by 𝑔 times ℎ. Now, in our case, where we have an object that’s 1.5 meters above ground, we can treat the acceleration due to gravity 𝑔 as exactly 9.8 meters per second squared. So, our object’s mass 𝑚 is equal to its gravitational potential energy — which we’re given, that’s 1176 joules — divided by 𝑔 and by ℎ. And for our scenario, ℎ, the height of our object, is 1.5 meters. With all these values plugged into our equation, before we go ahead and calculate 𝑚, let’s work for a bit with the units in these values.

Looking at our denominator, we see we have units of meters per second squared times meters. If we collect these units on the right-hand side of our denominator, what we find is we have meters squared per second squared. And now let’s look at the units in the numerator, joules. A joule is equal to a newton times a meter. And a newton itself is equal to a kilogram meter per second squared. This means a newton meter is a kilogram meter per second squared times meters, or a kilogram meter squared per second squared. But notice now that meter squared per second squared appears in our numerator and our denominator. Therefore, when we calculate this fraction, those parts of our units cancel out. This means that the final unit we’ll end up with is kilograms. And that’s just as it should be since we’re calculating a mass. And when we go ahead and calculate 𝑚, we find a result of 80 kilograms. That is the mass of our object.

Let’s look now at one more example exercise.

An object with a mass of 10 kilograms is positioned 15 meters above the surface of an unknown planet. The object has 1800 joules of gravitational potential energy. What is the acceleration due to gravity at the surface of the planet?

Okay, let’s say that this is our unknown planet. We’re told that we have an object — say that this here is our object — that has a mass of 10 kilograms and is positioned a height — we’ll call ℎ — of 15 meters above the surface of the planet. Along with all this, we know the object’s gravitational potential energy. We’ll call it GPE, and we know it’s equal to 1800 joules. Based on this, we want to solve for the acceleration due to gravity at the surface of this unknown planet. To begin figuring this out, we can recall an equation for gravitational potential energy. This relationship says that an object’s gravitational potential energy is equal to its mass multiplied by the strength of the gravitational field the mass is in times its height above some minimum height level.

Now, typically, we use a specific value for 𝑔. If our object was above Earth rather than above an unknown planet, we would use 9.8 meters per second squared for 𝑔. But since we’re not working with Earth but rather with an unknown planet, we can’t assume this value. Instead, we want to solve for the gravitational acceleration at the surface of this planet using the other information we’re given. To begin doing that, let’s divide both sides of our equation by 𝑚 times ℎ. This makes both those terms cancel out on the right-hand side. So now we see that gravitational acceleration 𝑔 is equal to gravitational potential energy divided by mass times height. And in this example, since we’re given GPE as well as 𝑚 and ℎ, we can substitute in these values to solve for 𝑔. GPE is 1800 joules, 𝑚 is 10 kilograms, and ℎ is 15 meters.

Considering the units in this expression for a moment, we can recall that a joule is equal to a newton multiplied by a meter. And so, if we make that substitution, we can see that the factor of meters in the numerator and denominator cancels out. But then going further, a newton is equal to a kilogram meter per second squared. And when we make that substitution, we see that the factor of kilograms in numerator and denominator cancels out. So, when we complete our calculation, we’ll have an answer in units of meters per second squared. 1800 divided by 10 times 15 is 12. So, 𝑔 is equal to 12 meters per second squared. That’s the acceleration due to gravity at the surface of this unknown planet.

Let’s summarize now what we’ve learned about gravitational potential energy. Starting off, we saw that gravitational potential energy, often abbreviated GPE, is energy possessed by an object that’s due to its position in a gravitational field. We saw that work done against gravity gives an object gravitational potential energy, while work done by gravity on an object takes it away. That is, we saw in the example of our ball that when we lifted the ball up above ground level by doing work against gravity, we gave the ball gravitational potential energy. But then, when we release the ball and allow gravity to do work on it, its gravitational potential energy was lost to kinetic energy.

We learned also that the gravitational potential energy of an object is equal to its mass times the acceleration due to gravity of the field the object is in times its height above some minimum height value. And then we saw that since the weight of an object, 𝑊, is equal to its mass times gravity, the gravitational potential energy of an object is also equal to weight times height. This, then, is a summary of gravitational potential energy.

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