In this explainer, we will learn how to calculate changes in the energy of an object in a gravitational field using the definition of the gravitational potential energy, .
It is an extremely familiar observation that an object released from rest at some point above the surface of Earth will move toward Earth; in other words, it will fall.
For an object released from above Earth to reach the surface of Earth, the object must move. An object that is at rest when it is released is not moving. The velocity of such an object must therefore increase for the object to fall.
When an object increases velocity, it accelerates. A body that accelerates must have a nonzero net force acting on it. In the case of a falling object, this force is its weight.
The following figure shows a falling object that is uniformly accelerated by its weight, . The distance that the object travels in one second increases each second as the object accelerates.
An object that moves a distance in the direction of a force has work done on it by the force. The work done on a falling object by its weight increases the kinetic energy of the object.
Energy is a conserved quantity. The total energy of a system does not change. This means that an increase in kinetic energy cannot occur without an accompanying decrease in some other energy. The work done by the weight of the object requires energy to be transferred from some other energy source.
This other energy, which kinetic energy is transferred from when an object descends, is gravitational potential energy.
Energy transfers from gravitational potential to kinetic when an object descends. The opposite occurs when an object ascends.
If the motion of a falling object is reversed, so that an object at the surface of Earth is projected upward away from the surface, then the object is uniformly accelerated toward Earth by its weight. The acceleration is in the opposite direction to the initial velocity of the object, so it will decrease in velocity. Eventually, the object will come instantaneously to rest and then start to fall back to the surface of Earth.
As the projected object ascends and slows, there is a decrease in kinetic energy. To conserve energy, the gravitational potential energy must increase.
The following figure shows how the kinetic and gravitational potential energies change when an object is projected vertically upward from a height of zero to the greatest height it reaches, where its velocity is zero.
It is important to note that the graph shows the changes in energy with height, not with time.
The following figure shows how the kinetic and gravitational potential energies change when an object initially at rest falls vertically downward to a height of zero where its velocity is greatest.
In both cases, we can see that the sum of the kinetic and gravitational potential energies is constant, as the following figure shows.
This is to be expected considering that energy is a conserved quantity.
An object that increases in height must have work done on it, which is transferred to gravitational potential energy. The work done, , by a force, , is given by the formula where is the distance moved by the object in the direction of the force.
An object that moves vertically upward a distance of 1 metre increases the height, , of its position by 1 metre. The work done on an object to move it from the surface of Earth to a height is given by
The vertically downward force on the object as it ascends is the weight of the object. The weight of an object can be expressed as the product of the mass of the object and the gravitational field strength at the position of the object. A force equal to the weight of an object can therefore be written as where is the mass and is the gravitational field strength.
An object that moves upward at constant velocity must have balanced upward and downward forces acting on it. If either the upward or the downward force was greater than the other, the object would accelerate upward or downward rather than move at constant velocity. The acceleration of the object would mean an increase in kinetic energy.
We see then that the force on an object that increases the height of the position of the object without increasing kinetic energy must be equal to .
The work done to move an object from the surface of Earth to a height without an increase in kinetic energy is therefore given by
This value of work is the gravitational potential energy transferred to increase the height of the position of the object.
Definition: Gravitational Potential Energy
The gravitational potential energy, GPE, transferred to increase the height of the position of an object by is given by where is the mass of the object and is the gravitational field strength at the position of the object.
If is measured in kilograms, in metres, and in newtons per kilogram (N/kg) or metres per second squared (m/s2), then GPE is measured in joules.
On Earth, the value of is approximately 9.8 N/kg or 9.8 m/s2.
It is helpful to clarify what is meant by “on” Earth. This does not mean on the surface of Earth; it includes heights above the surface of Earth.
The following figure shows most of Earth and a small square region, one side of which is on the surface of Earth.
The next figure shows a small part of the surface of Earth and the square region, and also a smaller square region inside the larger square region.
The gravitational field strength decreases with height above the surface of Earth, but this decrease is very gradual. The magnitude of the change of the gravitational field strength with height can be compared to the extent to which the surface of Earth is noticeably curved rather than flat.
The corners of the base of the purple square noticeably do not quite touch the surface of Earth. This means that at the height equal to the top of the purple square, the gravitational field strength would be noticeably not quite equal to 9.8 N/kg.
For the red square, there must be some very small gap between the corners of the base and the surface of Earth, as the base is straight and the surface of Earth is curved. The gap is too small to notice, however, and this means that the amount by which the gravitational field strength at the height of the top of the red square is different from 9.8 N/kg is also too small to notice. Questions about gravitational potential energy that do not state otherwise are assumed to occur sufficiently close to Earth to be like the space inside the red square, in which any variation in is too small to notice.
It is important to understand that the curve that would be compared to the base of the square is the curve of the entire Earth, not some small part of it.
Consider the following figure that shows a part of the surface of Earth that does not curve uniformly as there is a local distortion in the shape of Earth, such as a mountain range.
The base of the red square is clearly misaligned to the surface of Earth around the distortion, but the surface of the entire Earth must be misaligned to the base of the red square for there to be noticeable change in the value of from 9.8 N/kg within the square. This means that does not noticeably change because of the presence of local features, even those the size of mountains.
The following figure shows sea level and the heights of various objects above it. The distances are to scale.
The value of at the top of the Burj Khalifa is 9.8 m/s2 to two decimal places.
The value of at the top of Mount Everest is 9.78 m/s2 to two decimal places.
The value of in a passenger jet at its cruising altitude is 9.77 m/s2 to two decimal places.
To one decimal place, the value of is 9.8 m/s2 at sea level, at the top of the Burj Khalifa, at the top of Mount Everest, and in a passenger jet at its cruising altitude. All of these positions can be considered near the surface of Earth.
The value of on the International Space Station is 8.86 m/s2 to two decimal places.
Only at very great heights, such as that of the International Space Station, is significantly less than 9.8 m/s2. Only points at such great heights as these would be correctly described as not near the surface of Earth.
We see then that in all normal cases it is reasonable to assume that a value of 9.8 m/s2 can be used for .
Gravitational potential energy can be considered zero at the surface of Earth. This means that the gravitational potential energy of an object can be considered proportional to its height above the surface of Earth. This can be more simply expressed as height above the ground.
Let us now look at an example of determining gravitational potential energy.
Example 1: Determining Gravitational Potential Energy
An object with a mass of 15 kg is at a point 10 m above the ground. What is the gravitational potential energy of the object?
Answer
We can define the gravitational potential energy of the object as the energy transferred to it to change its position from being on the ground to being 10 metres above the ground. This energy can be found using the formula where is the mass of the object, is the height of the object above the ground, and is the gravitational field strength in the region that the object moved through, in this case from the ground to 10 metres above the ground.
We use 9.8 N/kg or 9.8 m/s2 for . We have then that
We can imagine a situation in which there is a known energy transferred to an object that increases the height of its position, and from this we can determine the height that the object increases its position by. Let us look at such an example.
Example 2: Determining a Change in Height from a Change in Gravitational Potential Energy
An object held at a point above the ground has 2 352 J of gravitational potential energy. The object’s mass is 20 kg. How far above the ground is the object?
Answer
We can define the gravitational potential energy of the object as the energy transferred to it to change its position from being on the ground to being at a point some height above the ground. This energy can be found using the formula where is the mass of the object, is the height of the object above the ground, and is the gravitational field strength in the region that the object moved through, in this case from the ground to some height above the ground.
We use 9.8 N/kg or 9.8 m/s2 for .
The formula can be rearranged to make the subject by dividing the formula by :
Substituting the known values, we obtain
The value of is not the same on all planets. Let us now look at an example where a value of different from its value on Earth is used.
Example 3: Determining Gravitational Field Strength from a Change in Gravitational Potential Energy
An object with a mass of 10 kg is positioned 15 m above the surface of an unknown planet. The object has 1 800 J of gravitational potential energy. What is the acceleration due to gravity at the surface of the planet?
Answer
We can define the gravitational potential energy of the object as the energy transferred to it to change its position from being on the ground to being at a point some height above the ground. This energy can be found using the formula where is the mass of the object, is the height of the object above the ground, and is the acceleration due to gravity in the region that the object moved through, in this case from the ground to 15 metres above the ground. This acceleration in metres per second squared (m/s2) is equal to the gravitational field strength in newtons per kilogram (N/kg).
The formula can be rearranged to make the subject by dividing the formula by :
Substituting the known values, we obtain
This is the value of the acceleration due to gravity at the surface of the planet and at heights near the surface.
Let us now look at an example in which the weight of an object is given rather than its mass.
Example 4: Determining a Change in Height from a Change in Gravitational Potential Energy
A bird flying over the sea has a weight of 15 N and has a constant 765 J of gravitational potential energy. How far above the sea does the bird fly?
Answer
We can define the gravitational potential energy of the object as the energy transferred to it to change its position from being on the ground to being at a point some height above the ground. This energy can be found using the formula where is the mass of the object, is the height of the object above the ground, and is the gravitational field strength in the region that the object moved through, in this case from the ground to some height above the ground.
The question does not state the mass of the bird but does state its weight. The weight of an object is related to its mass of by the formula where is the force—the weight of the bird— is the mass of the bird, and is the gravitational field strength in the region that the bird moved through, in this case from the ground to some height above the ground.
We can see that the formula is therefore equivalent to the formula where is the weight of the bird.
This formula can be rearranged to make the subject by dividing the formula by :
Substituting the known values, we obtain
So far in this explainer, gravitational potential energy at some height has been compared to gravitational potential energy at the ground, which has been defined as zero.
In defining gravitational potential energy at the ground as zero, we have said that objects on the ground cannot transfer any gravitational potential energy to kinetic energy as they cannot move vertically downward. If an object can decrease the height of its position, then it has nonzero gravitational potential energy relative to the point that it could fall to.
We can consider examples where the positions that objects can fall to are not necessarily the same as each other. In these cases, we can determine changes in gravitational potential energy without defining a single point at which we say that the gravitational potential energy is zero.
Let us now look at such an example.
Example 5: Determining Changes in Heights from Changes in Gravitational Potential Energy
A room contains a book, a chair, a bookshelf, and a hatch that leads to a basement, as shown in the diagram. The book has a weight of 7.5 N. The seat of the chair is 0.45 m above the floor of the room, the bookshelf is 1.5 m above the floor of the room, and the basement floor is 2.2 m below the floor of the room. The book is placed in various positions and its gravitational potential energy changes depending on its position.
- By how much will the gravitational potential energy of the book decrease if the book is released from a point on the floor of the room to the left of the hatch?
- By how much will the gravitational potential energy of the book decrease if the book is released from a point on the floor of the room to the right of the hatch?
- By how much will the gravitational potential energy of the book decrease if the book is released from a point at the height of the chair seat?
- By how much will the gravitational potential energy of the book decrease if the book is released from a point at the height of the bookshelf?
- The hatch is opened, and the chair is moved next to it. A reader sits on the chair and holds the book at the height of the chair seat, but over the open hatch. How much can the gravitational potential energy of the book decrease if the book is released from this point?
- The chair is moved so that it sits on the basement floor. The reader sits on the chair and holds the book at the height of the chair seat. How much can the gravitational potential energy of the book decrease if the book is released from this point?
- The hatch is opened, and the book is moved vertically from being on the floor of the basement to being at a point that is at the same height above the room’s floor as the bookshelf. How much does this increase the gravitational potential energy of the book?
Answer
Part 1
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is on the floor to the left of the hatch, it cannot fall any distance and so it will not decrease in GPE. The answer is 0 joules.
Part 2
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is on the floor to the right of the hatch, it cannot fall any distance and so it will not decrease in GPE. The answer is 0 joules.
Part 3
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is released from the height of the seat of the chair, it will fall 0.45 m to reach the floor, so its gravitational potential energy will decrease by
Part 4
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is released from the height of the bookshelf, it will fall 1.5 m to reach the floor, so its gravitational potential energy will decrease by
Part 5
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is released from the height of the seat of the chair but over the open hatch, it will fall 0.45 m to reach the hatch and a further 2.2 m to reach the floor of the basement, so its gravitational potential energy will decrease by
Part 6
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is released from the height of the seat of the chair when the chair is in the basement, the book will fall 0.45 m to reach the basement floor, so its gravitational potential energy will decrease by
Part 7
We determine the decrease in gravitational potential energy using the formula where is the weight of the book, which is 7.5 N, and is the distance that the book can fall before it is supported by some surface that prevents it from falling further. The value of can be determined from the figure in the question, shown below.
If the book is initially on the floor of the basement and is taken to the bookshelf, the height of the position of the book will be increased by the height of the basement ceiling from the basement floor, which is 2.2 m, and by the height of the bookshelf from the floor of the room, which 1.5 m. The gravitational potential energy of the book will increase by
Let us now summarize what has been learned in this explainer.
Key Points
- The gravitational potential energy, GPE, transferred to increase the height of the position of an object by
is given by
where is the mass of the object and is the gravitational field strength
at the position of the object.
If is measured in kilograms, in metres, and in newtons per kilogram (N/kg) or metres per second squared (m/s2), then GPE is measured in joules. - The value of is approximately constant near the surface of Earth, with a value of 9.8 N/kg or 9.8 m/s2.
