Elastic Potential Energy (Edexcel A Level Further Maths) : Revision Note

It takes putting in energy to stretch an elastic string beyond its natural length

• This is the work done in stretching it

Once it is stretched and held in position, the energy put in is now stored in the string

• This is a type of potential energy, because the string has the potential to contract and release it

This stored energy is called the elastic potential energy (EPE)

• Note that the "work done to stretch it" and the "elastic energy stored in it" have the same value

  • Questions may use either phrase

It also takes energy to compress a spring from its natural length

• The compressed spring has energy stored in it, ready to "spring" open

• This is also elastic potential energy

Let an elastic string (or spring) of natural length metres and modulus of elasticity  N be stretched to a new length of metres, where  metres is the extension

• The formula for elastic potential energy is 

  • Sometimes written (in a form like kinetic energy)

  • The units are in Joules

  • It's also the same as "work done by stretching"

This formula works for the compression of a spring

• represents the length of compression from its natural length

The formula can be derived by knowing that, in general, work done is the area under a force-distance graph

• The area under the graph of can be found by integration to give

  • Or by noting it's a triangle of base and height  then using × base × height

The Work-Energy Principle is an energy balance

• Total final energy = total initial energy ± work done

  • or, using subscripts for final and initial,

"Total energy" can now include elastic potential energy (EPE)

• e.g. the total initial energy is

The ± work done terms now refer to any non-gravitational and non-elastic forces

• e.g. friction (-), driving force (+), air resistance (-), etc

• But not weight () and no longer tensions that use Hooke's Law ()

  • These are both already accounted for in the GPE and EPE parts

You can use it when a particle moves from one position to a different position, for example:

• when it has been pulled down and released from rest

• when it has been projected at a certain speed

• when you need to find the distance it has moved

• This would not include a particle hanging at rest (in equilibrium)

  • Here, you could use Newton's 2nd Law + Hooke's Law

You can use it when a question involves finding speeds

• Speeds can be found from kinetic energy

  • Not from Newton's 2nd Law or Hooke's Law

You should not use it if asked to find the initial acceleration

• Acceleration is part of Newton's 2nd Law 

It is not required if asked to find the equilibrium extension

• The particle is not moving, so Newton's 2nd Law + Hooke's Law works

SUVAT equations cannot be used for particles on elastic strings or springs as their acceleration is not constant

• They can only be used if particles detach from the elastic string (e.g. the string breaks, or becomes slack, etc) 

  • The particle then becomes a projectile under gravity

  • Don't forget that the Work-Energy Principle can also be used on projectiles, and in some cases (e.g. finding speeds) it's much quicker than SUVAT