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Elastic Potential Energy (Edexcel A Level Further Maths: Further Mechanics 1)

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Mark

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Mark

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Elastic Potential Energy

What is Elastic Potential Energy?

  • 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

How do I calculate Elastic Potential Energy?

  • Let an elastic string (or spring) of natural length l metres and modulus of elasticity lambda N be stretched to a new length of open parentheses l plus x close parentheses metres, where x metres is the extension
    • The formula for elastic potential energy is fraction numerator lambda over denominator 2 l end fraction x squared
      • Sometimes written 1 half open parentheses lambda over l close parentheses x squared (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
    • x 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 T equals lambda over l x can be found by integration to give fraction numerator lambda over denominator 2 l end fraction x squared
      • Or by noting it's a triangle of base x and height lambda over l x then using 1 half× base × height

Examiner Tip

  • Be careful when selecting which formula to use, as they are very similar!
    • T equals lambda over l x is Hooke's Law, for finding a force
    • E P E equals fraction numerator lambda over denominator 2 l end fraction x squared is for finding the elastic potential energy

Worked example

An elastic string of natural length 1.5 metres and modulus of elasticity 60 N is stretched to a total length of 3.5 metres.

Find the energy stored in the string.

worked example showing how to find the elastic potential energy stored in a string

Work-Energy Principle with Elasticity

How do I include Elastic Potential Energy in the Work-Energy Principle?

  • The Work-Energy Principle is an energy balance
    • Total final energy = total initial energy ± work done
        • or, using subscripts for final and initial, E subscript f equals E subscript i plus-or-minus W D
  • "Total energy" can now include elastic potential energy (EPE)
    • e.g. the total initial energy is E subscript i equals G P E subscript i plus K E subscript i plus E P E subscript i equals m g h subscript i space end subscript plus 1 half m v subscript i squared space plus space fraction numerator lambda over denominator 2 l end fraction x subscript i to the power of 2 space end exponent
  • 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 (m g) and no longer tensions that use Hooke's Law (lambda over l x)
      • These are both already accounted for in the GPE and EPE parts

How do I know when to use the Work-Energy Principle?

  • 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

Examiner Tip

  • The maximum speed for a particle on an elastic string occurs when it's acceleration is zero, which is when it passes through its equilibrium position
    • You may need to find this position using Newton's 2nd Law + Hooke's Law in equilibrium

Worked example

 A particle of mass m kg is attached to one end of a light elastic spring of natural length l metres and with modulus of elasticity 4 m g N. The other end of the spring is attached to the point at the top of a rough slope of length 3 l metres at 30° to the horizontal. The particle is held on the bottom of the slope and released from rest. The coefficient of friction between the particle and the ramp is fraction numerator square root of 3 over denominator 4 end fraction.
 
Find, in terms of g and l, the speed of the particle as it passes through the point halfway up the slope.

worked example showing how to use the work-energy principle in conjunction with elastic potential energy

part 2 of worked example showing how to use the work-energy principle in conjunction with elastic potential energy

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    Mark

    Author: Mark

    Expertise: Maths

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.