# How to calculate potential energy

Imagine a ripe mango hanging from a tree under which you are sitting at the park. This delicious fruit could eventually fall and hit you in the head. Furthermore, the heavier the mango, the stronger will be the strike against you.

For physicists, the mango possesses a certain potential energy because it has the ability to fall and hit you. This means it could eventually transform said energy, for example, into movement.

Other situations may be similar. For example, electrons —which are negatively charged— located inside a cloud have a certain potential to fall to the ground, not because of gravity like in our previous example, but due to the Earth locally having a lower concentration of negative charges than the cloud. Due to the electrostatic forces that arise because of that charge difference, electrons inside the cloud can eventually find a path through the isolating air towards the ground, creating a lightning. This type of potential energy is called electric potential energy.

As you can imagine, there are several different types of potential energies depending on the physical phenomena that are involved in a particular scenario. So, what do they have in common? Is it possible to describe them through similar basic principles?

In the first section of this article, you will find concrete procedures to calculate the most common types of potential energies. If you wish to discover the principles that underlie this concept and how to recognize if a specific type of physical phenomenon may be associated with some potential energy, go to the second section. Let’s start!

## Most common potential energies

### How to calculate gravitational potential energy

This type of potential energy exists due to the force exerted by the gravitational field of the Earth upon any mass in its proximity. An object’s gravitational potential energy is calculated by multiplying its mass (m) by the gravity of Earth (g) and its height (h) above a certain reference level, as shown in the following equation:

Where g = 9,8 m/s². If the mass is in kilograms and the height in meters, the potential energy will be in joules.

Do you want to understand what is meant by “a certain reference level”? Then go ahead and read the section titled “What is potential energy?”.

### How to calculate electrical potential energy

This type of potential energy comes from the electrostatic forces exerted by an electric field on a charge. For a point charge located at position r inside an electric field, which experiences an electric potential φ, the electrical potential energy is defined as:

Where q is the charge in coulombs.

### How to calculate elastic potential energy

This type of potential energy takes place when an object stores energy in form of recoverable deformation, which is also called elastic deformation. The most common example is a spring. In this case, the elastic potential energy is calculated by multiplying half of the spring constant k by the square of its difference in length before and after being compressed or stretched, as shown in the following equation:

Here, Δx represents the spring’s length difference before and after compression or stretch.

## What is potential energy?

If an object is moving, it is reasonable to ask yourself why it is moving. For example, a ball rolling on the ground is probably doing so because at some point someone kicked it. In that case, the chemical energy in that person’s leg —stored within its cells— is transformed into movement, meaning there is a certain kinetic energy associated to the kick. Once the kicker’s leg hits the ball, it transfers its kinetic energy to it, causing it to move.

In our previous example we mentioned two key topics in understanding potential energy: energy transfer and energy transformation.  The former refers to the process of transmitting some existing form of energy from one system to another, for example, from one mass to another when crashing. Here, kinetic energy is transferred from one body to the other in form of linear momentum. Another example of this is warming up your hands at a bonfire. You feel the heat of the burning fire without touching it because it is transmitted through air in form of infrared electromagnetic waves.

On the other hand, energy transformation refers to the conversion of one form of energy into another. For example, when you light a match, the chemical energy stored in the molecules that make up the ignition material on its “head” is released in form of heat, one form of electromagnetic energy, due to a violent reaction that is initiated by friction. Another example are wind turbines: the kinetic energy of the air is first transferred to its blades as rotational kinetic energy, which is then converted into electrical energy by an electric generator

Both of these phenomena take place in nature continuously. Let’s use another example: imagine you hold a tennis ball at 1 meter above the ground. While you hold it, the ball is not moving, which means its kinetic energy is initially zero. Regardless of this, you must constantly exert a force on the ball, to prevent it to fall. This force must equal the ball’s weight, which pulls in the opposite direction, as the following image shows.

If you suddenly drop the ball, it will start accelerating downwards due the force of gravity. As it falls, it will gain more and more speed, meaning it will gradually increase its kinetic energy. After a few seconds, the tennis ball will reach the ground at a certain final speed, i.e., with a certain final kinetic energy.

As you have probably heard before, energy cannot be created or destroyed. This is called the law of conservation of energy. So, if the ball had no kinetic energy before you released it, how is it possible it contains a certain amount of it when reaching the ground? This energy gain takes place, as you are probably thinking, due to the force exerted on the ball by the Earth’s gravitational field. This force performs work on the ball, increasing its total energy. Now, regardless of the object’s size, shape, or mass, work will be performed on it by gravity after releasing it from any height.

Another way to explain this scenario, which turns out very useful when dealing with certain physics exercises, is to assume that the ball contains an initial amount of energy before being released. Upon release, this energy gradually transforms into kinetic energy. Furthermore, in order to abide by the law of conservation of energy, the final kinetic energy the ball gains —measured as it reaches the ground— must equal said initial energy.

This new type of energy exists in the tennis ball not due to its movement —since its velocity remains zero while you hold it in your hand—, but due to its position above the ground. Given that the Earth’s gravitational field will perform work on the ball during a free fall, it is sure to say this energy remains stored inside it until you release it.

Since the energy contained in the ball while being held at a certain height can eventually be transformed into kinetic energy, meaning it has the potential to be converted into movement, physicists call it potential energy. Furthermore, since the force responsible for accelerating the tennis ball downwards is gravity, this specific type of energy is called gravitational potential energy

As mentioned before, this type of potential energy exists due to objects’ positions above a certain reference level. Let’s try and understand this better: in our previous example, when the ball is released, it falls and reaches the ground with a final kinetic energy directly proportional to the square of its final speed, meaning the speed it possesses at the moment it touches the floor.

If you initially hold the ball at 1 m above the ground and repeat the exercise a second time by dropping the ball 2 m above the ground, you will find that the final speed in the second case is greater than in the first case. According to the equation of a body in free fall:

Where v is the ball’s instant speed, g the gravity of Earth, and h the free fall distance until the speed is measured. Consequently, if you increase h by 2 times, the square of the ball’s speed will also be 2 times greater. Since the ball’s kinetic energy equals

where m is the ball’s mass; its final value when reaching the ground must also be twice as large as when released from half the height. In conclusion, the amount of kinetic energy the ball can gain when falling is directly proportional to the height from which it is released. Keep in mind that this height is measured with respect to the ground, which in this case is your reference level

Now, once the ball reaches the ground, no further work will be performed on it by the Earth’s gravitational field. The ball has then no potential energy left, which means all of it was converted into kinetic energy during the free fall. Mathematically, we can express this such as:

Where U is the ball’s potential energy before being released. The ball’s final kinetic energy can be obtained by replacing equation 4 into equation 5, assuming h is the total distance travelled by the ball, which equals the height from which it was dropped:

This yields the definition of the gravitational potential energy of a body located at a height h above a given reference level.

Equivalently to our previous case, an electric charge fixed at a certain location inside an electric field will experience a force upon release, causing it to accelerate in a specific direction. Since this interaction leads to an increase in the charge’s kinetic energy, an electrical potential energy is assumed to be present at the beginning.

Similarly, when a spring is either compressed or stretched from its equilibrium length it will exert a force in the opposite direction and will therefore perform work on any mass attached to it. During the compression/stretch, the spring stores a certain elastic potential energy, which equals the work it can perform on the mass upon release in order to regain its original length.

## What do potential energies have in common?

As you could see from our previous examples, potential energy always relates to a force, but not to any type of force. In physics, forces are divided into conservative and non-conservative forces. Work performed by the first type on a mass does not depend on the specific path the mass follows. Let’s see an example:

If you hold a 1 kg brick at 1 m above the ground and drop it, it will accelerate downwards due to gravity. The force exerted on it by gravity is:

Since it falls for 1 m, the total amount of work performed on the brick by the gravitational field is:

Now, if you pick the brick up to a height of 1 m with respect to the ground you have to perform work on it by counteracting its weight. Since the force exerted by gravity goes in the opposite direction of movement, this time, the work performed on it is negative:

If you now drop the brick again, 9,8 J of work —positive, according to our convention— will be performed on it. Now, after letting it initially fall, picking it up and letting it fall again, the total work performed by gravity on the brick is:

This means, that the total work performed on the brick is independent of the path it follows, and only depends on the initial and final position of the brick. This is precisely the property that characterizes conservative forces, which in turn, give place to potential energies.

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