Question

Derivation of elastic potentialenergy

Answer 1

Elastic Potential Energy

Potential energy is the energy an object has stored in it due to its position. When we think of potential energy often the first thing that comes to mind is an object high in the air and just starting to fall. It has potential energy stored in it due to its height, and that energy will be turned into kinetic energy as it falls. However, this is not the only situation in which an object has potential energy. It is a specific type of potential energy called gravitational potential energy.

Another common type of potential energy is elastic potential energy. This is the energy an object has in it due to being deformed. Any object that can be deformed and then return to its original shape can have elastic potential energy. Objects that this would apply to include things like rubber bands, sponges, and bungee cords, among many others. When you deform these objects they move back to their original shape on their own. As a counter-example, an object that would not be affected by elastic potential energy would be something like a sheet of aluminum foil. If you crumple a sheet of it into a ball it won't change back into a sheet when you let go.

Hooke's Law

One of the most common objects to look at when discussing elastic potential energy is a spring. Springs can be deformed in two different ways in which they return to normal afterwards. They can be stretched, and they can be compressed.

In order to find the formula for elastic potential energy of a spring we first need to look at something calledHooke's law. This law states that the force needed to stretch a spring is proportional to the displacement of the spring. The displacement of the spring is how far the spring has stretched or compressed from its original shape.

Mathematically, Hooke's law can take the following forms.

F = - kxF = kx

We often see the formula with the negative sign in order to represent that Hooke's law is a restoring force, but the positive version is a valid representation as well. Here x is the displacement of the spring, and k is something known as the spring constant. This constant is the measure of the stiffness of a spring, and it is unique to each spring. The spring constant depends on factors such as what material the spring is made of and the thickness of the coiled wire, among others.

Finding Elastic Potential Energy

So, why do we need to know all this to find the elastic potential energy? Well, that's because the potential energy is equal to the work done by the spring, and work is a force multiplied by a distance. So Hooke's law gives us our force. For the distance, we use the displacement of the spring. You might assume we would get the formula for elastic potential energy as follows.

PE = Work = force * distance

So:

PE = (kx) * x

This then simplifies to:

PE = kx^2

However, this turns out to be wrong. To see the correct equation for elastic potential energy we need to look at a force vs. displacement graph.