Efficiency calculations: output, input and percentages

By the end of this lesson

By the end you can use $\text{efficiency} = \dfrac{\text{useful output}}{\text{total input}}$ in any direction: to find the efficiency, the useful output, the total input, or the energy dissipated — in fractions or percentages.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$\eta$	efficiency

One equation, three questions

Efficiency questions come in three shapes, and all of them use the same equation:

$\text{efficiency} = \dfrac{\text{useful energy output}}{\text{total energy input}}$

efficiency given (or asked): divide;
useful output asked: multiply the input by the efficiency;
total input asked: divide the useful output by the efficiency.

Here is the same equation rearranged for each job.

Definition

The efficiency equation rearranged

Writing $\eta$ (the Greek letter eta) for efficiency:

$\eta = \dfrac{E_\text{useful}}{E_\text{in}}$

Rearranged for each energy:

$E_\text{useful} = \eta\, E_\text{in} \qquad E_\text{in} = \dfrac{E_\text{useful}}{\eta}$

The dissipated energy is always what is left:

$E_\text{dissipated} = E_\text{in} - E_\text{useful}$

Remember to divide a percentage by $100$ before using it in these equations: $45\% = 0.45$ .

Worked example: Find the efficiency

An electric motor is supplied with $2.0\ \text{kJ}$ of energy. It transfers $1.4\ \text{kJ}$ of kinetic energy to the machine it drives.

What is its efficiency, as a percentage, and how much energy is dissipated?

Have a go first?

Quick check

CalculateQuestion 1 · Recall

CalculateQuestion 2 · Apply

Answer the 2 questions above to continue.