- A-Level Further Maths

# Elasticity & Collisions

According to **Hooke's Law, tension, T, is directly proportional to extension, x,** in elastic strings and springs:

T = kx

The constant of proportionality, k, depends on two things: the **modulus of elasticity, λ**, of the material, and its unstretched, **natural length, l**:

**Note:**natural length is noted with a lowercase 'L' - do not confuse this with an uppercase 'i'. The font of these note sheets is not massively helpful here, so sorry!

Tension, T, is a force so is measured in Newtons. Extension and length are both distances, measured in metres. This gives the modulus of elasticity must also be measured in Newtons.

**Strings can be compressed as well as stretched.** In this instance, Hooke's law still applies but the force in the spring, thrust (helpfully also written as T), acts the other way.

### Modelling

For all questions, strings and springs are modelled as light. This means you do not take into account its weight. See the __notes sheet on modelling in mechanics__ for single maths - the same rules apply.

## Elastic Potential Energy

When an elastic string or spring is stretched or compressed, **it stores elastic potential energy** within it. **This is equal to the work done** in stretching/compressing the string/spring.

The work done is represented by the area under a force-extension graph:

Using the equation for the area of a right-angled triangle, A = ½ab, we can find the equation for the elastic potential energy stored in an elastic system: