Power 2006-11-16 19:41:01 2006-11-17 11:55:33 Definition Added a unit table % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \section{Power} Power is the rate of energy transfer. Since there are several forms of energy, there are several ways of describing power. In general terms of energy, power is defined as $$P = \frac{dE}{dt}.$$ \subsection{Mechanical Power} The energy transfer in mechanical systems where work is done by an applied force $$ P = \frac{dE}{dt} = \frac{dW}{dt}.$$ Using the relation between work and force $$ dW = {\bf F} \cdot d{\bf r}$$ and then differentiating to get power, $$P = \frac{dW}{dt} = {\bf F} \cdot \frac{d{\bf r}}{dt} = {\bf F} \cdot {\bf v}.$$ The corresponding form of power in rotation is $$P = {\bf M} \cdot {\bf \omega},$$ where ${\bf M}$ is the torque and ${\bf \omega}$ the angular velocity vector. \subsection{Electrical Power} Since energy is transfering from a device storing electrical energy to another device in the circuit that converts to another form of energy, power is the rate of change of electrical potential energy. For a DC circuit $$P = \frac{dU}{dt} = i V.$$ \subsection{Units} The SI unit of the power is one joule per second, which is called {\em watt}: $$\frac{\mathrm{J}}{\mathrm{s}} := \mathrm{W}.$$ The watt is equal to $\mathrm{kg \cdot m^2/s^3}$ in the base units. The english units of power are $$ 1 \left [horsepower \right] = 1 \left [hp \right] = 550 \left [\frac{ft \, lb}{s} \right]$$