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Useful fraction

Efficiency is the ratio of useful output to total input:

$\eta=\frac{E_{useful}}{E_{input}}=\frac{P_{useful}}{P_{input}}$

Choose matching quantities

Use energy ratios for the same process and time interval, or power ratios when input and output are rates. Efficiency is dimensionless and is often reported as a percentage.

Worked example from local Question Bank row 29709

Solar intensity is $240\,\mathrm{W\,m^{-2}}$ over $2.50\times10^4\,\mathrm{m^2}$ , so input power is

$P_{in}=(240)(2.50\times10^4)=6.0\times10^6\,\mathrm{W}=6.0\,\mathrm{MW}$

For a useful output of $1.6\,\mathrm{MW}$ ,

$\eta=\frac{1.6}{6.0}=0.27=27\%$

The remaining input is transferred through non-useful pathways.

Common trap

Do not invert the ratio or use the total output, including unwanted transfers, as the useful output.

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An electric motor of efficiency 75 % raises a mass of 120 kg at a constant speed of $0.50 \mathrm{~ms}^{-1}$ . What is the power input to the motor?

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.3.13—Efficiency

Calculate Efficiency

Retrieve the A.3 Work, Energy and Power Model

A.3.13 Exam Analysis