SI Units and Measurement

Cheatsheet Content

SI Base Units The International System of Units (SI) defines seven base units from which all other units are derived. Quantity Unit Name Unit Symbol Length meter m Mass kilogram kg Time second s Electric Current ampere A Thermodynamic Temperature kelvin K Amount of Substance mole mol Luminous Intensity candela cd SI Prefixes Prefixes are used to denote decimal multiples and submultiples of SI units. Prefix Symbol Factor yotta Y $10^{24}$ zetta Z $10^{21}$ exa E $10^{18}$ peta P $10^{15}$ tera T $10^{12}$ giga G $10^9$ mega M $10^6$ kilo k $10^3$ hecto h $10^2$ deca da $10^1$ deci d $10^{-1}$ centi c $10^{-2}$ milli m $10^{-3}$ micro $\mu$ $10^{-6}$ nano n $10^{-9}$ pico p $10^{-12}$ femto f $10^{-15}$ atto a $10^{-18}$ zepto z $10^{-21}$ yocto y $10^{-24}$ SI Derived Units (Examples) Units derived from the base units, some with special names and symbols. Quantity Unit Name Unit Symbol In Base Units Area square meter $\text{m}^2$ $\text{m}^2$ Volume cubic meter $\text{m}^3$ $\text{m}^3$ Velocity meter per second $\text{m/s}$ $\text{m} \cdot \text{s}^{-1}$ Acceleration meter per second squared $\text{m/s}^2$ $\text{m} \cdot \text{s}^{-2}$ Force newton N $\text{kg} \cdot \text{m} \cdot \text{s}^{-2}$ Pressure pascal Pa $\text{kg} \cdot \text{m}^{-1} \cdot \text{s}^{-2}$ (or $\text{N/m}^2$) Energy, Work, Heat joule J $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-2}$ (or $\text{N} \cdot \text{m}$) Power watt W $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3}$ (or $\text{J/s}$) Frequency hertz Hz $\text{s}^{-1}$ Electric Charge coulomb C $\text{A} \cdot \text{s}$ Electric Potential volt V $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-1}$ (or $\text{J/C}$) Resistance ohm $\Omega$ $\text{kg} \cdot \text{m}^2 \cdot \text{s}^{-3} \cdot \text{A}^{-2}$ (or $\text{V/A}$) Measurement Principles Accuracy vs. Precision Accuracy: How close a measurement is to the true value. Precision: How close repeated measurements are to each other. Significant Figures Non-zero digits are always significant. Zeros between non-zero digits are significant (e.g., $1005 \text{ m}$ has 4 sig figs). Leading zeros are not significant (e.g., $0.0025 \text{ kg}$ has 2 sig figs). Trailing zeros are significant only if the number contains a decimal point (e.g., $100. \text{ g}$ has 3 sig figs; $100 \text{ g}$ has 1 sig fig). In scientific notation, all digits are significant (e.g., $1.23 \times 10^4 \text{ s}$ has 3 sig figs). Rules for Calculations with Significant Figures Addition/Subtraction: The result should have the same number of decimal places as the measurement with the fewest decimal places. Example: $12.11 \text{ cm} + 18.0 \text{ cm} + 1.013 \text{ cm} = 31.123 \text{ cm} \rightarrow 31.1 \text{ cm}$ Multiplication/Division: The result should have the same number of significant figures as the measurement with the fewest significant figures. Example: $2.5 \text{ m} \times 3.42 \text{ m} = 8.55 \text{ m}^2 \rightarrow 8.6 \text{ m}^2$ Uncertainty and Error Absolute Uncertainty: The range within which the true value of the measurement is expected to lie. Often expressed as $\pm \Delta x$. Relative Uncertainty: The ratio of the absolute uncertainty to the measured value, often expressed as a percentage. Formula: $\text{Relative Uncertainty} = \frac{\Delta x}{x} \times 100\%$ Propagation of Uncertainty: Addition/Subtraction: If $Q = x + y$ or $Q = x - y$, then $\Delta Q = \sqrt{(\Delta x)^2 + (\Delta y)^2}$ (for independent uncertainties). Multiplication/Division: If $Q = x \cdot y$ or $Q = x / y$, then $\frac{\Delta Q}{Q} = \sqrt{\left(\frac{\Delta x}{x}\right)^2 + \left(\frac{\Delta y}{y}\right)^2}$. Powers: If $Q = x^n$, then $\frac{\Delta Q}{Q} = |n| \frac{\Delta x}{x}$. Types of Errors: Random Error: Unpredictable fluctuations in measurements, affecting precision. Reduced by taking multiple readings. Systematic Error: Consistent, repeatable error in all measurements due to faulty equipment or experimental design, affecting accuracy. Unit Conversions Use conversion factors to change between units. A conversion factor is a ratio equal to 1. Example: Convert $50 \text{ km/h}$ to $\text{m/s}$. $$ 50 \frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = 13.89 \frac{\text{m}}{\text{s}} $$ Common Physical Constants Constant Symbol Value (approx.) Unit Speed of light in vacuum $c$ $2.998 \times 10^8$ $\text{m/s}$ Elementary charge $e$ $1.602 \times 10^{-19}$ C Planck constant $h$ $6.626 \times 10^{-34}$ $\text{J} \cdot \text{s}$ Gravitational constant $G$ $6.674 \times 10^{-11}$ $\text{N} \cdot \text{m}^2 \cdot \text{kg}^{-2}$ Avogadro constant $N_A$ $6.022 \times 10^{23}$ $\text{mol}^{-1}$ Boltzmann constant $k_B$ $1.381 \times 10^{-23}$ $\text{J/K}$ Electron mass $m_e$ $9.109 \times 10^{-31}$ kg Proton mass $m_p$ $1.673 \times 10^{-27}$ kg