Elasticity and the spring constant

In a nutshell

Objects may be deformed when multiple forces are applied to them. Elastic objects return to their original shape when the forces are removed. The amount an elastic object extends when a force is applied to it is proportional to the strength of the force.

Equations

Variable definitions

Elasticity

Applying multiple forces to an object may cause it to stretch, bend or compress. Objects are elastically deformed if they return to their original shape once the forces are removed. Objects are inelastically deformed if they don't return to their original shape once the forces are removed. Objects that are elastically deformed are called elastic objects (e.g. a spring). 

When a force stretches or compresses an object, energy is transferred into its elastic potential energy stores. For elastic objects like springs, all the work done on the object goes into increasing the elastic potential energy stores. For inelastic objects, some of the work done goes into breaking bonds between the particles in the system. 

The amount an elastic object extends when a force is applied to it is directly proportional to the strength of the force. The formula is:

​$$force=spring\space constant \space \times \space extension \\ \ \\ F=k\times x$$​​

where the variables are defined above. This is known as Hooke's law. 

Definition

The spring constant $$k$$ (also called 'stiffness') is the amount of force required to extend an elastic object by $$1\,m$$. It depends on the material - a stiffer spring has a larger spring constant. 

This formula holds for elastic objects only. Objects are only elastic up to a limit, called the elastic limit. When the applied force is big enough, the object will become inelastic and will permanently deform as a result. This is shown in the force-extension graph below:

A Force B Extension 1. Elastic limit

The straight line part of the graph above is the elastic deformation of the spring. In this section of the graph, the spring obeys Hooke's law, and the stiffness is found by taking the gradient of the graph. The curved part of the graph represents the behaviour of the spring beyond the limit of elasticity, during which the spring does not obey Hooke's law, and behaves inelastically.  

Example

A weight of $$20\,N$$ is applied to an elastic spring, causing the spring to extend by $$10\,cm$$. What is the stiffness of the spring?

Write down the relevant information provided in the question (converting to base units where appropriate): 

$$F=20\,N \\ \ \\ x=10\,cm=0.1\,m$$

Write down the relevant formula: 

$$F=k\times x$$

Rearrange the relevant formula: 

$$k=\frac{F}{x}$$

Substitute the values into the correct formula: 

$$k=\frac{20}{0.1}\\ \ \\ k=\underline{200N/m}$$.

So the stiffness of the spring is $$\underline {200\, N/m}$$.