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International A LevelMaths: Mechanics 2EdexcelRevision Notes3. Work and Energy3.1 Work & Energy3.1.3 Energy Principles

Energy Principles (Edexcel International A Level Maths: Mechanics 2)

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Work-Energy Principle

What is the Work-Energy Principle?

  • The Work-Energy Principle has many forms, but is in essence an energy balance
    • the final amount is the initial amount plus any energy put in (or minus any taken out)
    • it's a bit like money in a bank account!
  • The principle can be written as:
    • total final energy = total initial energy ± work done by non-gravitational forces
      • or, using subscripts for final and initial, E subscript f equals E subscript i plus-or-minus W D
    • "total energy" here means the sum of Gravitational Potential Energy and Kinetic Energy
      • e.g. E subscript i equals G P E subscript i plus K E subscript i equals m g h subscript i space end subscript plus 1 half m v subscript i to the power of 2 space end exponent
    • Non-gravitational forces are any external forces that are not related to gravity
      • e.g. frictions, tensions, driving forces, etc
      • but wouldn't include weight, m g, because work done against gravity has already been considered in the GPE part
    • Use + for work done by forces that "help" the object to move forwards
      • e.g. tension in a string pulling forwards, a driving force, etc
    • Use - for work done by forces that "hinder" (resist) the object from moving forwards
      • e.g. friction, tension in a string pulling backwards, a resistance force, air resistance, etc
  • Some situations may have more than one form of work done
    • add or subtract each one, depending whether they help or hinder

How else can the Work-Energy Principle be written?

  • You can write the Work-Energy Principle in terms of gains or losses in KE and GPE
    • but this method can cause a lot of sign errors!
  • Write out each term in the original version, "total final energy = total initial energy ± work done by non-gravitational forces"
    •  use subscripts for final and initial
      • m g h subscript f space end subscript plus 1 half m v subscript f to the power of 2 space end exponent equals m g h subscript i space end subscript plus 1 half m v subscript i to the power of 2 space end exponent plus-or-minus W D
    • group together KEs and GPEs as an overall change ("final - initial")
      • 1 half m v subscript f to the power of 2 space end exponent minus 1 half m v subscript i to the power of 2 space end exponent equals negative open parentheses m g h subscript f space end subscript minus m g h subscript i close parentheses plus-or-minus W D
    • change in KE = - change in GPE ± WD
      • you can read this as gain in KE = loss in GPE ± WD
      • but some situations lose KE, so the gain is negative (you need to be really careful with the signs and what "loss" means!)
  • The first method (energy balance) works for every situation, but this "gain-loss" method needs adapting for each situation

Can I use the Work-Energy Principle on an inclined plane?

  • Yes, and remember to calculate work done as "the component of force in the direction of motion" × "distance moved in direction of motion"
    • in this case, the direction of motion is parallel to the slope
  • You may also need to calculate the final vertical height for GPE
    • e.g. using trigonometry, with the distance in the direction of the slope as the hypotenuse

Examiner Tip

  • If a question asks you to find something "using the work-energy principle", don't use Newton's 2nd Law with SUVAT!

Worked example

A dog uses a constant force of P N to push its toy of mass 0.5 kg up a rough slope inclined at 20°  to the horizontal. The force, P, acts parallel to the slope. The toy starts from rest and, after moving 10 metres up the line of greatest slope, it is travelling at 1 ms-1.  The coefficient of friction between the toy and the slope is 0.1.

Use the work-energy principle to find P.

worked example showing how to use the work-energy principle on an inclined plane with friction

fm1-2-4-2-work-energy-principle-we-solution-2-pf-for-mc

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Conservation of Energy

What is Conservation of Energy?

  • Conservation of Energy is a special case of the Work-Energy Principle when there is no work done by non-gravitational forces
    • so total final energy = total initial energy
    • total energy here means GPE + KE
    • or, using subscripts for final and initial:
      • m g h subscript f plus 1 half m v subscript f to the power of 2 space end exponent equals m g h subscript i plus 1 half m v subscript i to the power of 2 space end exponent
  • This could be because there are no non-gravitational forces
    • e.g. a particle falling freely under gravity from a fixed height above the ground
  • Or this could be because all non-gravitational forces are always perpendicular to the direction of motion 
    • e.g. the vertical reaction force on an object moving along a horizontal surface contributes no work done (the force has no horizontal component)
      • recall work done is the "component of force in the direction of motion" × "distance moved in direction of motion"

Examiner Tip

  • In practice, you can apply the Work-Energy Principle as before, but it'll be slightly easier as there's no work done by external forces

Worked example

A particle of mass m kg is fired horizontally off a vertical cliff at 8 ms-1 and travels freely under gravity. The cliff is 30 metres high and the ground below is horizontal.

Use energy considerations to find the speed of impact of the particle with the horizontal ground below.

worked example showing how to use conservation of energy for a projectile

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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.