Elastic Forces

Elastic force constant

The elastic force constant, spring stiffness or spring constant is a constant of proportionality between the force on a spring and the resulting extension/compression. It is a representation of a spring's capacity to resist an external force; the stiffer the spring, the greater the effort required to compress or stretch it. A spring with stiffness or spring constant of$10\mathrm{N}/\mathrm{m}$would require$10\mathrm{N}$to displacing it by$1\mathrm{m}$.

So, given the spring constant how do we calculate this elastic force that returns the object back to its original shape? Well, we can use the formula discussed earlier.

$F=ke$

or in words,

$\mathrm{Force}=\mathrm{spring}\mathrm{constant}\times \mathrm{extension}$

• $F$ is the force expressed in Newtons$\left(\mathrm{N}\right)$.

• $k$ is the stiffness of the spring or the elastic object, expressed in Newtons per meter$(\mathrm{N}/\mathrm{m})$.

• $e$ is the displacement of the spring, expressed in meters$\left(\mathrm{m}\right)$.

Now as you can infer from the above formula, the extension of the spring or any elastomer is directly proportional to the force applied. However, this is only true up to a point called the limit of proportionality.

This relationship and condition is stated by Hooke's law. Hooke's law, also known as the law of elasticity, states that the deformation is directly proportional to the deforming load or force that applies up until the limit of proportionality.

Now let's understand this process of stretching an elastic object using a graph. The graph records the force applied on the Y-axis and the extension of the material on the X-axis.

The force vs extension curve for a spring shows the linear and non-linear relationship between them, Ranjit Boodoo CC-BY-SA-4.0

The figure above shows the relation between the force applied to two different springs versus the extension of the springs. We can see that the force is directly proportional to the displacement of both springs A and B up to a certain point only. This point is the limit of proportionality; once the force exceeds this limit, the relationship between extension and force becomes nonlinear. In many cases, including in the graph shown above, after the limit of proportionality, the extension increases at a higher rate than before for the same increment in force. Now can you say something about the spring constant for springs A and B just by looking at the graphs? Well, we can see that the graph of A is much steeper than B, this means that for the same magnitude of force, spring B gets deformed more than spring A. Hmm, so what does this mean? Well, refer back to the definition of the spring constant. Now you'll be able to say that spring A has a higher spring constant or stiffness than spring B as it requires a greater force to achieve the same deformation. Now that we have a good understanding of how elastic force works, let's see why this force is produced.

Elastic energy meaning

Let's next take a closer look at the meaning of the term elastic energy.

Work needs to be done to compress a spring or bring about any kind of deformation. By the principle of the conservation of energy, this work is then stored in the compressed object as elastic potential energy. When the external force is removed this elastic potential energy is released and converted into kinetic energy.

The quantity of elastic potential energy (or just elastic energy) stored in such a device is proportional to its extension; the greater the extension or deformation, the greater the elastic potential energy stored.

Elastic energy formula

The following elastic energy formula may be used to calculate the amount of elastic potential energy contained in a stretched spring:

or in words,

• $k$is the spring constant expressed in Newtons per meter$(\mathrm{N}/\mathrm{m})$.

• $e$is the extension of the spring expressed in meters$\left(\mathrm{m}\right).$

• $E_e$is the elastic potential energy expressed in Joules$\left(\mathrm{J}\right)$.

Elastic spring force example

Here we provide an elastic spring force example problem that you can use to test your understanding.

Elastic force merge of two springs

The elastic effect provided by springs can be useful for a number of real-world applications, from industrial processes to home appliances and children's toys. However, if we only have one spring to hand, we are somewhat limited in the force that we can generate. If we had two springs, we could combine them in more than one way to produce multiple different outcomes which we can use in different circumstances depending on our needs. The following example contains two springs in parallel and series. At this point, you might be scratching your head and wondering why terms from electronics are being used in an explanation about springs. Springs can be configured in just the same way that electrical components can in electrical circuits.

Identical springs placed in parallel have an effective spring constant which is equal to twice the spring constant of one spring. This is because, in order to extend both of the springs by some amount, twice as much force is needed because two springs have to be pulled against instead of one.

Two springs of equal spring constant in parallel will produce an effective spring constant of twice a single spring, Wikimedia Commons

Identical springs placed in series have an effective spring constant which is equal to half the spring constant of one of the springs. This is because in order to extend the strings by some distance you only have to provide half the force in order to gain the same extension that you would produce with one spring.

Two identical springs in series have an effective spring constant that is equal to half the spring constant of one spring, Wikimedia Commons