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Elastic store

For a spring within its linear range,

$E_{p,elastic}=\frac12k(\Delta x)^2$

where $\Delta x$ is extension or compression from the natural length.

Area under the graph

The elastic potential energy equals the work done in stretching or compressing the spring. On a force–extension graph it is the area under the graph.

Worked example from local Question Bank row 31357

A spring with $k=100\,\mathrm{N\,m^{-1}}$ is compressed by $0.10\,\mathrm{m}$ .

$E_{p,elastic}=\frac12(100)(0.10)^2=0.50\,\mathrm{J}$

This is the energy available for transfer when the ideal spring is released.

Common trap

Do not use the total spring length as $\Delta x$ , and remember that doubling extension quadruples the stored energy in the ideal model.

[ 1 ]

$\quad 0.25 \mathrm{~J}$ of work is done to compress a spring by a distance of 0.10 m from its unstretched length. What is the spring constant?

Account for energy

Define the system, identify energy stores and describe transfers. Work done by a force transfers energy; total energy is conserved even when mechanical energy is not.

Use the mechanical model

$E_k= rac12mv^2,\quad \Delta E_{p,g}=mg\Delta h,\quad E_{p,elastic}= rac12k(\Delta x)^2$

Conserve their sum only when resistive transfers are absent or included explicitly.

Use rates and ratios

$P= rac{\Delta E}{\Delta t}=Fv,\qquad \eta= rac{E_{useful}}{E_{input}}= rac{P_{useful}}{P_{input}}$

Fuel energy density connects available input energy to a chosen volume.

Final checks

Check the system boundary, signs of work and potential-energy changes, the reference height, extension from natural length, and whether the quantity is energy, power, efficiency or energy density.

A.3.11—Elastic potential energy

Calculate Elastic Potential Energy

Retrieve the A.3 Work, Energy and Power Model

A.3.11 Exam Analysis