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Systematic unit conversion

To convert from a certain unit system to another (like the SI system), there is a structured method that uses the fact that we can multiply quantities by the number 1 (dimensionless) without affecting the quantity.

For a description of quantities, see the measurement theory page. The method of using multiply by one is explained based on an example.

To make 5 gallons of beer, a recipe says that we need 2.0 ounces (oz.) of malt. Because ounces are a weight unit and gallon a volume measure, we know that in SI units we should express it in kg per m3. We start with

\begin{equation} malt=\frac{2.0 oz.}{5 gallon}. \label{eq:Malt} \end{equation}

Next we look up that one U.S. gallon is equal to 3.785 liters and one ounce is 28.350 gram. A liter is 1/1000 m3. We write the conversion units as a dimensionless normalized ratio \begin{equation} 3.785\frac{l}{gallon}=1 \end{equation} with \begin{equation} 1000\frac{l}{m^{3}}=1 \end{equation} and \begin{equation} 28.350\frac{gr}{oz.}=28.350\frac{0.001 kg}{oz.}=1 \end{equation}

respectively. Because these are equal to one without any unit, we can simply multiply equation \eqref{eq:Malt} with them (or with the inverse if needed): \begin{equation} \begin{aligned} malt&=\frac{2.0 oz.}{5 gallon}\\ &=\frac{2.0 oz.}{5 gallon} \left ( 3.785\frac{l}{gallon} \right )^{-1} \left ( 1000\frac{l}{m^{3}} \right ) \left ( 28.350\frac{0.001 kg}{oz.} \right ). \end{aligned} \end{equation}

As a result, the non-SI units cancel out. So our equation simplifies to \begin{equation} malt = \frac{2.0}{5} \left ( 3.785 \right )^{-1} \left ( 1000\frac{1}{m^{3}} \right ) \left ( 28.350 \cdot 0.001 kg \right ) \end{equation}

which can be calculated as \begin{equation} malt = \frac{2.0}{5} \cdot \frac{28.350 \cdot 1000 \cdot 0.001 kg}{3.785 m^{3}} = 2.996 \frac{kg}{m^{3}}. \end{equation}

Sensor Technology TOC

These are the chapters for the Sensor Technology course:

theory/sensor_technology/sta_easyunitconversion.txt · Last modified: 2017/10/10 18:04 by glangereis