Calculating Efficiency

Efficiency measures how much of the total energy input is transferred to useful output. It is a ratio with no unit, often expressed as a percentage.

The equations

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

$\text{efficiency} = \frac{\text{useful power output}}{\text{total power input}}$

Multiply by 100 to express as a percentage.

Key facts

• Efficiency is always less than 100% for real machines (some energy is always dissipated as thermal energy).
• Efficiency can never exceed 100% (conservation of energy).
• Both energy and power can be used in the equation, but do not mix them (both numerator and denominator must use the same quantity).

Worked example

A motor uses 500 J of electrical energy and transfers 350 J as kinetic energy. Calculate the efficiency.

$\text{efficiency} = \frac{350}{500} = 0.70 = 70\%$

The remaining 150 J (30%) is transferred to thermal energy of the motor and surroundings.

Worked example: finding useful output

A 2000 W engine is 40% efficient. Calculate the useful power output.

$\text{useful power} = \text{efficiency} \times \text{total power} = 0.40 \times 2000 = 800\text{ W}$

Common errors and how to correct them

Dividing total input by useful output (inverting the fraction). The useful output is always the numerator; total input is the denominator. An efficiency above 100% means the fraction is inverted.

Using wasted energy instead of useful energy. The numerator must be the useful output, not the wasted output. If given the wasted amount, subtract it from the total to find the useful output first.

How to apply this in an exam

State the equation, substitute the correct values, calculate, and give the answer as a decimal or percentage (match the question). If the answer exceeds 100%, check for an inverted fraction or a unit mismatch.