![[SI Units]] ## Dimensions Every quantity has dimensions that can be expressed as a product of the dimensions of its base units raised to the appropriate powers. For example: $ \text{Volume, } m^3 = L^3 $ $ \text{Density, } kg \cdot m^3 = M \cdot L^{-3} $ $ \text{Force, }N=M\cdot L\cdot T^{-2} $ Where $M$ is **mass**, $L$ is **length**, $T$ is **time**. Adding or subtracting quantities is only meaningful if their dimensions are the same. The dimensions of a product are the product of the dimensions. ### Worked example A pascal, $Pa$ is defined as the pressure caused by a force of $1N$ over $1m^2$. What are its dimensions? Expressed in terms of base units: $ 1\ Pa=1\ N/m^2=1(kg \cdot m\cdot s^{-2})/m^2=1\ kg\cdot m^{-1}\cdot s^{-2} $ By inspection, this has dimensions $M\ L^{-1}\ T^{-2}$. Alternatively: $ \frac{\text{Dimensions of force}}{\text{Dimensions of area}}=\frac{M\ L\ T^{-2}}{L^2}=M\ L^{-1}\ T^{-2} $ ## Dos and don'ts | Do | Don't | | ------------------------------------------------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------------- | | Leave a space between a number and associated unit (e.g $5\ \text{kg}$) | Capitalise units, even if they're names: newton, pascal etc | | Use italic fonts for quantities and upright for units (e.g $m=5\ \text{kg}$) | Abbreviate units or combinations of units; $2\ \text{seconds}$ or $2\ \text{s}$ but not $2\ \text{sec}$ | | Use a space for multiplication and a solidus or negative powers for division (e.g $5\ \text{kg m/s}$ or $3\text{ kg m s}^{-1}$) | Use more than one solidus in a unit symbol | | Use parentheses to avoid confusion where necessary, e.g $2\text{ N/(m s)}$ or (preferably) $2\ \text{N m}^{-1}\text{ s}^{-1}$ | Modify units for particular versions of quantities; $5\text{ V}$ not $5\text{ VDC}$, even if the voltage is DC | ## Precision, accuracy and uncertainty No measurement is exact. The **accuracy** of a measurement described how close the measurements are to the true value. The **precision** of a measurement describes how close those measurements are to one another. The **uncertainty** of a measurement is the margin of doubt that exists about the result, given that the true value is usually unknown. ## Decimal places and significant figures Most calculates provide mode prevision than is justified. Give most results to three decimal places, rounding to the nearest value. For higher numbers, you can give to three **significant figures**, or the number of significant figures provided in the question.