Mechanical efficiency measures how much of the input energy or input power becomes useful mechanical output.
No real machine is perfectly efficient because some input is lost as heat, sound, deformation, vibration, or friction.
1. General efficiency formula
The basic efficiency relationship is
\[
\eta=\frac{\text{useful output}}{\text{total input}}\times100\%.
\]
Efficiency is often written as a percentage, but in rearranged formulas it is usually used as a decimal.
For example,
\[
72.9\% = 0.729.
\]
2. Power-based efficiency
Use the power-based formula when the machine or motor operates continuously or at steady state:
\[
\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times100\%.
\]
Here \(P_{\mathrm{in}}\) is the input power supplied to the system, and \(P_{\mathrm{out}}\) is the useful mechanical
output power.
The power lost inside the system is
\[
P_{\mathrm{loss}}=P_{\mathrm{in}}-P_{\mathrm{out}}.
\]
3. Energy-based efficiency
Use the energy-based formula when analyzing a process over a finite time or event:
\[
\eta=\frac{W_{\mathrm{useful}}}{E_{\mathrm{in}}}\times100\%.
\]
Here \(E_{\mathrm{in}}\) is the total energy supplied, and \(W_{\mathrm{useful}}\) is the useful mechanical work
produced.
The energy lost during the event is
\[
E_{\mathrm{loss}}=E_{\mathrm{in}}-W_{\mathrm{useful}}.
\]
4. Rearranged formulas
If efficiency and input are known, the useful output is
\[
\text{useful output}=\frac{\eta}{100}\times\text{input}.
\]
If efficiency and useful output are known, the required input is
\[
\text{input}=\frac{\text{useful output}}{\eta/100}.
\]
| Unknown |
Power-based formula |
Energy-based formula |
| Efficiency |
\(\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times100\%\) |
\(\eta=\frac{W_{\mathrm{useful}}}{E_{\mathrm{in}}}\times100\%\) |
| Useful output |
\(P_{\mathrm{out}}=\frac{\eta}{100}P_{\mathrm{in}}\) |
\(W_{\mathrm{useful}}=\frac{\eta}{100}E_{\mathrm{in}}\) |
| Input |
\(P_{\mathrm{in}}=\frac{P_{\mathrm{out}}}{\eta/100}\) |
\(E_{\mathrm{in}}=\frac{W_{\mathrm{useful}}}{\eta/100}\) |
| Loss |
\(P_{\mathrm{loss}}=P_{\mathrm{in}}-P_{\mathrm{out}}\) |
\(E_{\mathrm{loss}}=E_{\mathrm{in}}-W_{\mathrm{useful}}\) |
5. Physical interpretation
A normal mechanical system should satisfy
\[
0\%\le\eta\le100\%.
\]
If a calculation gives \(\eta>100\%\), the input and output are not physically consistent for an ordinary machine.
Common causes include missing input energy, comparing different quantities, measuring output incorrectly, or using
non-steady values in a steady-state formula.
6. Worked example
Suppose a motor receives \(8500\ \mathrm{W}\) of input power and delivers \(6200\ \mathrm{W}\) of useful output power.
Its efficiency is
\[
\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times100\%.
\]
Substitute:
\[
\eta=\frac{6200}{8500}\times100\%.
\]
Therefore,
\[
\eta\approx72.9\%.
\]
The lost power is
\[
P_{\mathrm{loss}}=8500-6200=2300\ \mathrm{W}.
\]
So the motor converts about \(72.9\%\) of the input power into useful mechanical output, while about
\(27.1\%\) is lost.
Formula summary
| Concept |
Formula |
Use |
| General efficiency |
\(\eta=\frac{\text{useful output}}{\text{input}}\times100\%\) |
Core definition of efficiency. |
| Power efficiency |
\(\eta=\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\times100\%\) |
Steady motor, engine, or machine operation. |
| Energy efficiency |
\(\eta=\frac{W_{\mathrm{useful}}}{E_{\mathrm{in}}}\times100\%\) |
Finite process or event. |
| Power loss |
\(P_{\mathrm{loss}}=P_{\mathrm{in}}-P_{\mathrm{out}}\) |
Power not converted into useful output. |
| Energy loss |
\(E_{\mathrm{loss}}=E_{\mathrm{in}}-W_{\mathrm{useful}}\) |
Energy lost during a process. |
Efficiency is not the same as output. A high-output machine can still be inefficient if it requires much larger input.
