AcademyWork, Energy, and Power
Academy
Power and Rate of Work
Level 1 - Physics topic page in Work, Energy, and Power.
Principle
Power is the rate of work or energy transfer.
Notation
\(P\)
power
\(\mathrm{W}\)
\(W\)
work or transferred energy
\(\mathrm{J}\)
\(t\)
time
\(\mathrm{s}\)
\(\vec{F}\)
force
\(\mathrm{N}\)
\(\vec{v}\)
velocity
\(\mathrm{m\,s^{-1}}\)
Method
A work-time graph turns power into slope: steeper parts transfer energy faster.
Instantaneous power follows from dividing the small work done by the small time taken.
Average rate
\[P_{\mathrm{avg}}=\frac{W}{\Delta t}\]
Instantaneous rate
\[P=\frac{dW}{dt}\]
Small work
\[dW=\vec{F}\cdot d\vec{r}\]
Force power
\[P=\vec{F}\cdot\frac{d\vec{r}}{dt}=\vec{F}\cdot\vec{v}\]
Angle form
\[P=Fv\cos\theta\]
Rules
Average power
\[P_{\mathrm{avg}}=\frac{W}{\Delta t}\]
Instantaneous power
\[P=\frac{dW}{dt}\]
Force power
\[P=\vec{F}\cdot\vec{v}\]
Energy rate
\[P=\frac{dE}{dt}\]
Examples
Question
A motor does
\[600\,\mathrm{J}\]
of work in \[4\,\mathrm{s}\]
Find its average power.Answer
\[P_{\mathrm{avg}}=\frac{W}{\Delta t}=\frac{600}{4}=150\,\mathrm{W}\]
Checks
- Watts are joules per second.
- Perpendicular force gives zero power.
- Positive power adds mechanical energy.
- Negative power removes mechanical energy.