Use adaptive quiz-based learning to study this topic faster and more effectively.

# Hooke's law and elasticity

## Elastic and plastic materials

Elastic materials can be deformed by exerting a force on them but they return to their original shape when the force stops.

The rubber in rubber bands is an elastic material.

By contrast, plastic materials stay in their new shape even after the force ceases. Plastic materials are a category of materials and should not be confused with the material "plastic" (e.g. plastic bags).

Porcelain is a plastic material. A force strong enough to deform porcelain would cause it to stay in its deformed state (i.e. break).

No material is infinitely elastic. If an elastic material is stretched beyond the elastic limit, it will be permanently deformed.

A steel spring, for example, is elastic up to a certain degree. Stretching it too much will leave it permanently stretched or even ruptured.

Rubber bands are made of an elastic material.

## Load, restoring force and extension

The load is the force that causes an elastic material to deform.

The force with which an archer pulls back the string of a bow is the load.

The restoring force is the force that causes an elastic material to return to its original shape.

The restoring force is equal in magnitude but opposite in direction to the load.

The force that causes an archer's bow to snap back is the restoring force.

When analysing springs, the displacement $\vecphy{x}$ of the spring's end from its equilibrium position as a result of the load being applied is called the extension.

The direction of the restoring force $\vecphy{F_{\text{s}}}$ depends on whether the spring is compressed (B) or stretched (C).

## Hooke's law

Hooke's law states that the displacement of one end of an elastic material is negatively proportional to the restoring force . This means that the restoring force points in the opposite direction to the displacement.

The magnitude of the restoring force is directly proportional to the magnitude of displacement: $$\Tred{\vecphy{F}}=-\Tblue{k}\Tviolet{\vecphy{x}} \quad \Rightarrow \quad \Tred{F}=\Tblue{k}\Tviolet{x}$$

A spring pulls back against any force that stretches it. The further the spring is stretched, the greater the force.

• $\Tred{\vecphy{F}}$ is the restoring force . The right-hand side of the equation is negative because the force points in the opposite direction to the extension and the load.

• $\Tblue{k}$ is the spring constant , which depends on the material that the elastic material is made of. The value of $\Tblue{k}$ for a particular material is determined empirically. The unit of the spring constant is newton per metre $(\text{N m}^{-1})$.

• $\Tviolet{\vecphy{x}}$ is the displacement of the spring's end from its equilibrium position.

• $\Tviolet{\vecphy{x}}$ can be negative, such as in the case where the spring is contracted. The directions of the restoring and load forces thus have to be reversed.

The spring in a weighing scale obeys Hooke's law. The linear relationship between load and extension makes it simple to convert the extension into a measurement of the load.

## Limitations of Hooke's law

Hooke's law is not a universal principle. The law applies to many materials as long as they are not stretched too far.

Hooke's law ceases to apply beyond a certain extension, and permanent deformation may result if the material is stretched beyond the elastic limit.

Hookean materials are materials for which Hooke's law applies (for a given range of extension).

In the first section, the graph is a straight line (i.e. within the elastic limit). The graph becomes non-linear beyond this limit.

## Elastic potential energy

Elastic potential energy is the energy stored in the form of a reversible deformation of an elastic object.

The spring stores the energy used to extend it. The potential energy is thus given by the work $(W)$ done in deforming the elastic object.

The elastic potential energy of the stretched bow of an archer corresponds to the work done by the archer in stretching the bow.

The potential energy is converted into kinetic energy (movement) as the spring returns to its equilibrium (i.e. unstretched) state.

The elastic potential energy is the mechanical equivalent of the potential of the gravitational field. Both are relative quantities: we only measure differences in potential.

In the mechanical case, the reference potential is taken to be zero (i.e. at zero displacement).

## Elastic potential energy for Hookean materials

For a Hookean material, the elastic potential energy (in $\text{J}$) is given by: $$E_{\text{p}}=\frac{1}{2}kx^2$$

$k$ is the spring constant and $x$ is the magnitude of displacement of the elastic material in $\text{m}$. The potential energy is proportional to the square of the extension.

Recall that the restoring force of an elastic material is related to the displacement by $\vecphy{F}=-k\vecphy{x}$. The elastic potential energy therefore can be expressed in terms of force and the magnitude of displacement: $$E_{\text{p}}=\frac{1}{2}Fx$$

In general, the elastic potential energy is given by the area under the force-extension graph (force on $y$-axis, extension on $x$-axis). This reflects the definition of work as the force times the distance over which the force is applied.

The elastic potential energy is given by the area under the force-extension graph. For a Hookean material, this area is a right-angled triangle.