SI units and prefixes Knowledge-Based Study

Study notes

1. The Foundation of Scientific Measurement

   In physics, quantitative observation relies on measurement. To ensure consistency, reproducibility, and clear communication across the global scientific community, physicists use the International System of Units, abbreviated as SI (from the French Système international d'unités).

   The SI system is built upon seven fundamental base quantities, each defined by an invariant standard:

   • 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)

   Every other physical quantity in mechanics, thermodynamics, electromagnetism, and optics can be expressed as a mathematical combination of these base units.

   Combining Base Units

   Quantities that are defined algebraically in terms of two or more base quantities are known as derived quantities. Their corresponding units are called derived units.

   Common Mechanical Derived Units

   • Speed or Velocity: Defined as displacement divided by time elapsed. The SI unit is meters per second ($\text{m/s}$ or $\text{m}\cdot\text{s}^{-1}$).
   • Acceleration: Defined as the rate of change of velocity. The SI unit is meters per second squared ($\text{m/s}^2$ or $\text{m}\cdot\text{s}^{-2}$).
   • Force: Defined through Newton's second law ($F = ma$). The derived unit is the newton (N), where:
     $1\text{ N} = 1\text{ kg}\cdot\text{m/s}^2$.
   • Energy and Work: Defined as force acting over a distance ($W = F\cdot d$). The derived unit is the joule (J), where:
     $1\text{ J} = 1\text{ N}\cdot\text{m} = 1\text{ kg}\cdot\text{m}^2/\text{s}^2$.
   • Power: Defined as work done per unit time ($P = W/t$). The derived unit is the watt (W), where:
     $1\text{ W} = 1\text{ J/s} = 1\text{ kg}\cdot\text{m}^2/\text{s}^3$.
   • Pressure: Defined as force applied perpendicular to a surface per unit area ($P = F/A$). The derived unit is the pascal (Pa), where:
     $1\text{ Pa} = 1\text{ N/m}^2 = 1\text{ kg}/(\text{m}\cdot\text{s}^2)$.

   Scaling Physical Quantities

   Physical phenomena occur over an immense range of scales, from the subatomic radius of a proton (around $10^{-15}\text{ m}$) to the diameter of observable space (around $10^{26}\text{ m}$). To express extremely large or small quantities cleanly without cumbersome strings of zeros, standard metric prefixes representing decimal powers of ten are attached to SI units.

   Common Prefixes for Large Quantities (Multiples)

   • giga- (G): $10^9$ (e.g., $1\text{ GHz} = 10^9\text{ Hz}$)
   • mega- (M): $10^6$ (e.g., $1\text{ MJ} = 10^6\text{ J}$)
   • kilo- (k): $10^3$ (e.g., $1\text{ km} = 10^3\text{ m}$)

   Common Prefixes for Small Quantities (Submultiples)

   • centi- (c): $10^{-2}$ (e.g., $1\text{ cm} = 10^{-2}\text{ m}$)
   • milli- (m): $10^{-3}$ (e.g., $1\text{ ms} = 10^{-3}\text{ s}$)
   • micro- ($\mu$): $10^{-6}$ (e.g., $1\ \mu\text{m} = 10^{-6}\text{ m}$)
   • nano- (n): $10^{-9}$ (e.g., $1\text{ nm} = 10^{-9}\text{ m}$)
   • pico- (p): $10^{-12}$ (e.g., $1\text{ pF} = 10^{-12}\text{ F}$)

   Note: The base unit for mass, the kilogram, already includes the prefix kilo-. Prefix symbols are attached directly to the root unit symbol without spaces (for instance, $\text{nm}$ for nanometer, not $\text{n m}$).

   Systematic Unit Conversion

   Converting between different prefixes or derived forms requires the factor-label method (chain-link conversion), where a quantity is multiplied by conversion factors equal to unity (1).

   Handling Squared and Cubic Units

   A frequent source of error occurs when converting area or volume units. Because the prefix modifies the unit itself, any exponent applies to both the prefix and the unit:

   • $1\text{ cm} = 10^{-2}\text{ m}$
   • $1\text{ cm}^2 = (10^{-2}\text{ m})^2 = 10^{-4}\text{ m}^2$
   • $1\text{ cm}^3 = (10^{-2}\text{ m})^3 = 10^{-6}\text{ m}^3$

   Worked Example

   Convert a density of $2.7\text{ g/cm}^3$ into the standard SI unit ($\text{kg/m}^3$):

   1. Convert grams to kilograms: $1\text{ kg} = 10^3\text{ g}$, so multiply by $\frac{1\text{ kg}}{10^3\text{ g}}$.
   2. Convert cubic centimeters to cubic meters: $1\text{ cm}^3 = 10^{-6}\text{ m}^3$, so multiply by $\frac{1\text{ cm}^3}{10^{-6}\text{ m}^3}$.
   3. Combine the factors: $$2.7\text{ g/cm}^3 \times \left(\frac{1\text{ kg}}{10^3\text{ g}}\right) \times \left(\frac{1\text{ cm}^3}{10^{-6}\text{ m}^3}\right) = 2.7 \times 10^3\text{ kg/m}^3 = 2700\text{ kg/m}^3$$

   Always verify that units cancel algebraically to leave the desired target unit.