Physics can explain not only how things move, but why. The explanation for why things are able to move often starts with energy. Energy is what gives an object the ability to do something. In this post, we’ll be looking specifically at what potential energy is, the two main types of mechanical potential energy, and how to calculate potential energy using the gravitational potential energy formula and the elastic potential energy formula.

Potential Energy

Back in a previous post, we created a working definition of energy in reference to kinetic energy. In physics, energy is the ability to exert a force on an object. We also briefly discussed potential energy. Recall that potential energy is the energy stored in something – the energy it has the potential to use.

Summary and Types of Potential Energy

As stated, potential energy refers to the energy stored in an object because of its position or its structure. This will make more sense as we get into the different types of potential energy. Let’s focus on two forms commonly seen in mechanical physics: gravitational potential energy and elastic potential energy. However, before we dive into the types and how to calculate potential energy, let’s make sure we know how to talk about energy.

Units of Measurement

Physicists measure potential energy – and every other form of energy – in Joules (J). One joule is equivalent to one newton-meter or one kilogram-meter squared per second squared. If we look at that first equivalent unit, one newton-meter, we can get a better idea of how we can relate to potential energy. Specifically, we can see how it relates to work.

The Relationship Between Potential Energy and Work

Recall that work tells us how much effort – how much energy – was needed to exert a force on a given object as it moved a given distance. As stated above, potential energy has units of force times distance. This should create an immediate connection between the two ideas. While energy and work are not equivalent, potential energy is the potential to exert a force and work is the measure of what that force can do over a given distance.

Gravitational Potential Energy

We started this post by explaining that there are two really important types of mechanical potential energy – gravitational potential energy and elastic potential energy. So, what is gravitational potential energy?

What is Gravitational Potential Energy?

Gravitational potential energy is related to an object’s position. As we’re on Earth, we generally define this relative to the Earth. You can think of this as the energy needed to raise an object to a given height. Once you’ve used energy to raise an object, that energy has been transferred to the object which now has the potential to do something, mainly to fall. The higher you’ve raised the object, the more potential energy it will have. If you were to be on Mars, you would probably measure this based on position relative to Mars, and the same would be true of anywhere else you went. For now, we’ll focus on Earth-centered gravitational potential energy.

How to Calculate Gravitational Potential Energy

The formula for gravitational potential energy is

Gravitational Potential Energy Formula E_{P}=mgh

In this formula, E_{P} is potential energy (you could also add a G to be more specific if you wanted to), m represents mass, g is gravitational field strength, and h is height – usually specifically height above the ground.

An important factor to consider is the gravitational field strength. You may be familiar with it being referred to as gravitational acceleration, which is understandable. In most cases, the gravitational field strength is 9.81\text{ m/s}^{2}, as we typically find ourselves on Earth near its surface. This is equivalent to the gravitational acceleration on our planet. However, if you were to travel to other celestial bodies such as Mars or the Moon, you would have to use the gravitational field strengths for those locations. In the event that these standard values are no longer applicable, you would have to determine a new value to use for g. Although this may not be covered in an introductory physics course, it is essential to know if you plan to delve further into the field.

A Special Case

Furthermore, gravitational field strength can become zero. Because all of the values in our equation are being multiplied together, you may think you’ll always increase the overall potential energy by increasing any of the values. While that is true to some extent, if you were to increase the height enough, the gravitational field strength would drop to zero. For example, a planet in a galaxy on the edge of the distant universe has a gravitational potential energy of effectively zero, relative to the Earth. This isn’t likely to come up often but is an important conceptual point to keep in mind as it can sometimes simplify problems drastically. If you’d like to deep dive and learn more about gravitational potential energy at a large distance, you can do that here.

Now, let’s look at a gravitational potential energy example you’re likely to see during an assignment.

Example of Gravitational Potential Energy

A 3\text{ kg} bird is flying 100\text{ m} above Earth’s surface. What is the bird’s gravitational potential energy? Assume gravitational field strength to be 9.81\text{ m/s}^{2}.

Solution:

• m=3\text{ kg}
• g=9.81\text{ m/s}^{2}
• h=100\text{ m}

• E_{P}=\text{?}

• Formula for gravitational potential energy: E_{P}=mgh

E_{P}=mgh

E_{P}=3\text{ kg} \cdot 9.81\text{ m/s}^{2} \cdot 100\text{ m}

E_{P}=2{,}943\text{ J}

Elastic Potential Energy

Now that we know how potential energy can depend on position, we need to understand how it can depend on the structure of an object. In the next section, we’ll talk about elastic potential energy.

What is Elastic Potential Energy?

You’ll mostly encounter elastic potential energy with springs during physics courses, but you can also think about them with elastics. Consider a rubber band. If you stretch the rubber band out, it will reshape itself as soon as you let go. If you manage to compress one, the same will happen. When the elastic (or spring) is stretched or compressed, it has elastic potential energy. Thus, this is the potential for something to reshape itself. There can also be a potential for the object to affect another, like launching a clicky pen off of a desk.

How to Calculate Elastic Potential Energy

The formula for elastic potential energy is

Elastic Potential Energy Formula E_{P}=\frac{1}{2}kx^{2}

The first thing to notice is that k is a spring constant. Effectively, this measures how much force it takes to stretch or compress the elastic object. This will generally be given to you in the problem or will be the value you need to find in the problem. The spring constant is specifically measured in units of newtons per meter. The x value is the compression or expansion distance. It measures how far the object has been stretched or compressed from its resting position.

Unlike gravitational potential energy, this equation is as straightforward as it seems. Increasing the spring constant or the distance will increase the elastic potential energy. Similarly, decreasing both values will decrease the elastic potential energy. It is worth noting that because the distance is squared and the spring constant is not, a change in distance will have a greater impact on the elastic potential energy than a change in the spring constant. Now, let’s look at an example of how to calculate elastic potential energy.

Example of Elastic Potential Energy

A spring is compressed a distance of 0.25\text{ m} and has a spring constant of 14\text{ N/m}. What is the elastic potential energy in the spring?

Solution:

• k=14\text{ N/m}
• x=0.25\text{ m}

• E_{P}=\text{?}

• Formula for elastic potential energy: E_{P}=\frac{1}{2}kx^{2}

E_{P}=\frac{1}{2}kx^{2}

E_{P}=\frac{1}{2} \cdot 14\text{ N/m} \cdot (0.25\text{ m})^{2}

E_{P}=0.44\text{ J}

How to Find the Spring Constant

In many instances, the spring constant is already given. But if not, it can easily be found experimentally in a lab by applying Hooke’s Law. In the experiment, a spring is hung from a fixed point and different known masses are added to it. When the masses are added to the spring, it extends as a result. According to Hooke’s Law, the force exerted by a spring is proportional to how far it extends from its rest position. This force can be expressed as F = -kx, where k represents the spring constant and x is the extension. Repeat the lab with different masses and graph the force vs extension. The slope of this graph will be equal to k. You can then use this value in the elastic potential energy formula.

You can review more about the spring force in our post on Types of Forces.

Conclusion

In conclusion, energy is one of the fundamental concepts that allows us to understand why the universe moves the way it does. Whether an object has gravitational potential energy, elastic potential energy, or both, knowing what these values mean will help you predict what the object can do and how it can influence the world around it.