Quantitative and qualitative application of energy conservation to…

The idea

Energy conservation means the total energy in a closed system stays constant: energy may transfer between stores but is never created or destroyed. When a spring or wire is stretched elastically, work done by the stretching force transfers energy to the elastic potential store; the stored energy equals the area under a force–extension graph, and for a straight-line region obeying Hooke's law it is ½FΔL or ½k(ΔL)². If the material is deformed plastically, some transferred energy is dissipated by internal friction and becomes thermal energy, so the energy recovered on unloading is less than the energy put in. Quantitatively, equate the elastic strain energy to other stores, for example ½k(ΔL)² = mgh for a catapult launching a mass, and solve for the unknown. Qualitatively, compare loading and unloading curves: a hysteresis loop encloses an area representing energy dissipated per cycle.

What earns the marks

• State that energy is conserved: the work done in stretching equals the energy stored plus any energy dissipated.
• Calculate elastic strain energy from the area under a force–extension graph, using ½FΔL or ½k(ΔL)² only for the straight-line Hooke's law region.
• Explain that plastic deformation dissipates energy as thermal energy, so the unloading curve lies below the loading curve and less energy is returned.
• Apply conservation quantitatively by equating elastic strain energy to kinetic, gravitational potential or thermal energy and solving for the unknown quantity.
• Interpret a force–extension graph qualitatively, identifying the elastic limit and recognising that beyond it the material does not return all the stored energy.

Common mistakes and how to put them right

• Using ½FΔL for a curved, non-linear region of the graph.

  Put it right: The correct approach is to estimate the area under the curve, for example by counting squares.

• Assuming all the work done stretching a wire is recoverable.

  Put it right: Beyond the elastic limit some energy is dissipated, so the recovered energy is smaller.

• Forgetting to convert units such as centimetres to metres or grams to kilograms before substituting into ½k(ΔL)², which produces an answer wrong by a power of ten.

Exam tips

• Sketch the force–extension graph and shade the area representing stored or dissipated energy before doing any calculation.
• When a question says 'qualitative', give a written comparison of energy transfers rather than a numerical answer.
• Check that your final energy value has the unit J and that the magnitude is physically sensible for the context described.