# Elastic Potential Energy and Combination of Springs

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5.2Elastic Potential Energy and Combination of Springs

In these revision notes for Elastic Potential Energy and Combination of Springs, we cover the following key points:

• What is elastic potential energy and how it is calculated?
• What does the area under the elastic force vs deformation graph represent?
• In which kinds of energy, the elastic potential energy can be converted?
• What does the Law of Energy Conversion say?
• How we combine the springs?
• How to find the total spring constant in a certain spring combination setup?
• How the spring constant changes when we combine two or more springs?

## Elastic Potential Energy and Combination of Springs Revision Notes

The area under the elastic force vs deformation graph represents the work done to extend or compress a spring but at the same time, it also represents the work done by the spring on objects in contact when it is released and tries to turn back to the unstretched position. This work contributes in changing the potential energy of the spring, which is otherwise known as Elastic Potential Energy, EPE.

Giving that the area of a right triangle as the one represented by the graph, is

Area = Base × Height/2

Substituting the relevant quantities, we obtain for the elastic potential energy EPE of a spring:

EPE = k × x2/2

As a stored energy, the elastic potential energy can be converted into other forms of energy as well. Some of the most widespread examples in this regard include:

1. From elastic to kinetic energy and vice-versa
2. From gravitational potential energy to elastic potential energy and vice-versa
3. From elastic into thermal energy
4. etc.

The Law of Conservation of Energy (as one of fundamental laws in physics) states that:

"Energy can neither be created, nor destroyed; it only can be converted from one form of energy to another."

To change the level of stiffness in springs, we can combine two or more spring in two basic ways:

### a. In series

In this combination, the springs are placed one after another. The formula for the spring constant of such a system is

1/ks = 1/k1 + 1/k2

where k1 and k2 are the constants of each spring respectively.

### b. In Parallel

When two or more springs are connected in parallel, they distribute the load amongst them and as a result, they will extend less than if there was only one spring available. Such a combination creates a system of springs with a higher stiffness. The equation for the spring constant of the system of two parallel springs is

kp = k1 + k2

where kp is the spring constant of the parallel setup, while k1 and k2 are the constants of the individual springs.

We can combine the above two basic setups of springs to obtain another setup.

### c. Mixed combination of springs

In this case, we start by calculating the constant of the parallel part and then, we think as having a single spring in that part. It is combined in series with the rest of the springs to get the total spring constant of the entire system.

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