Energy Principles (Cambridge (CIE) A Level Maths): Revision Note

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,

  • "total energy" here means the sum of Gravitational Potential Energy and Kinetic Energy

    • e.g. 

  • 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, , 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

    • 

  • group together KEs and GPEs as an overall change ("final - initial")
    

    • 

  • 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

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:

    • 

• 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"