- Elastic limit
- Maximum force one can apply to the end of an elastic object for it to behave elastically.
- Plastic deformation
- Object permenantly changes shape

## Elastic objects

- Obey Hooke's law
- Return to their original state ($e=0$)

## Hooke's law

Hooke's law states that force is proportional to the extension of an elastic object : $F\propto e$
Therefore:
\begin{align}F=ke\end{align}
\begin{align}[k]=[\frac{F}{e}] = Nm^{-1}\end{align}
Where K is the **Stiffness constant, elasticity constant or spring constant**

### Experiments

To measure $k$, one should rearrange the formula into $y=mx+c$ format: $F=ke+0$
One should then measure extension ($e$) in meters, mass in Kg, $F$ in N. The gradient will be $k$, and thus the **spring constant**

### The behaviour of rubber?

Rubber is strange and annoying, and doesn't follow Hooke's law. As a rubber band is streached, it warm up, releasing thermal energy. As a result, it take extra time to return to their original length. It form a graph like the one below. The area under the loading line is **work done by the masses on the band, and thus a gain in potential energy**. The area under the unloading line is work done by the band on the surrondings. The remaining energy is **heat to the surrondings**. The **net gain** is the difference between the two areas, like on a Lorenz curve.

[insert diagram of rubber band graph]

### The problems with Hooke's law

- The spring constant depends on the dimensions of the object.

## A better way: The Young Modulus

### Tensile stress - $\sigma$

\begin{align}\sigma=\frac{F}{A}\end{align} where F is tensile force, A is cross sectional area \begin{align}[\sigma]=Nm^{-2}=Pa\end{align} (pressure)

### Tensile strain - $\varepsilon$

\begin{align}\varepsilon=\frac{e}{L}\end{align} where e is extension, L is original length \begin{align}[\varepsilon] = dimensionless\end{align}

### Young modulus

\begin{align}E=\frac{\sigma}{\varepsilon}\end{align} \begin{align}[E]=Pa\end{align} One should expect a value around $10^9$ (gigapascals).