Units, Systems and Dimensional Analysis

Absolute and Technical (or Gravitational) Systems of Units

Absolute Systems

In these systems, force is a derived quantity (F = m·a).

System Length (L) Mass (M) Time (T) Unit of Force
CGS centimeter (cm) gram (g) second (s) dyne (1 g·cm/s²)
MKS meter (m) kilogram (kg) second (s) newton (1 kg·m/s²)
FPS foot (ft) pound-mass (lbm) second (s) poundal (1 lbm·ft/s²)

Technical or Gravitational Systems

In these systems, force is a fundamental quantity and mass is derived (m = F/a).

System Length (L) Force (F) Time (T) Unit of Mass
Technical CGS cm gram-force (gf) s technical mass unit (TMU-CGS)
Technical MKS m kilogram-force (kgf) s Technical Mass Unit, TUE (9.8 kg)
Technical FPS ft pound-force (lbf) s slug (32.2 lbm)

International System of Units (SI)

SI Base Quantities

Quantity Dimension Unit Symbol Current Definition
Length L meter m Distance traveled by light in 1/299,792,458 s
Mass M kilogram kg Based on the Planck constant (2019)
Time T second s 9,192,631,770 periods of radiation of Cs-133
Thermodynamic temperature Θ kelvin K Based on the Boltzmann constant
Electric current I ampere A Based on the elementary charge (1.602×10⁻¹⁹ C/s)
Luminous intensity J candela cd Based on luminous efficacy
Amount of substance N mole mol Contains 6.022×10²³ elementary entities

SI Supplementary Quantities

Quantity Unit Symbol Definition
Plane angle radian rad m/m (dimensionless)
Solid angle steradian sr m²/m² (dimensionless)

Classification of Physical Quantities

Fundamental vs. Derived Quantities

  • Fundamental: Independent of each other, define the system
  • Derived: Expressed through dimensional equations

Scalar vs. Vector Quantities

Type Characteristics Examples
Scalars Magnitude only (numerical value + unit) Mass, time, temperature, energy
Vectors Magnitude, direction, and sense Displacement, velocity, force, acceleration

Examples of Important Derived Quantities

Quantity Formula Dimension SI Unit Name
Velocity v = Δx/Δt LT⁻¹ m/s —
Acceleration a = Δv/Δt LT⁻² m/s² —
Force F = m·a MLT⁻² kg·m/s² newton (N)
Work W = F·d ML²T⁻² N·m joule (J)
Power P = W/t ML²T⁻³ J/s watt (W)
Pressure p = F/A ML⁻¹T⁻² N/m² pascal (Pa)

Comparison of Systems of Units

Main Comparison Table

Quantity Symbol SI CGS Technical MKS Technical FPS
Length L meter (m) cm m foot (ft)
Mass M kg g TUE ≈ 9.8 kg slug ≈ 14.6 kg
Time T s s s s
Force F N (newton) dyne kgf (kilopond) lbf
Equivalence — 1 N = 1 kg·m/s² 1 dyne = 1 g·cm/s² 1 kgf = 9.80665 N 1 lbf = 4.448 N

Key relation: 1 N = 10⁵ dynes

Essential Conversions

Length

  • 1 m = 100 cm = 3.28084 ft
  • 1 ft = 30.48 cm = 12 in
  • 1 in = 2.54 cm
  • 1 mi = 1609.34 m = 5280 ft

Mass

  • 1 kg = 1000 g = 2.20462 lbm
  • 1 slug = 14.5939 kg = 32.174 lbm
  • 1 TUE = 9.80665 kg

Force

  • 1 N = 0.101972 kgf = 0.224809 lbf
  • 1 kgf = 9.80665 N = 2.20462 lbf
  • 1 lbf = 4.44822 N = 0.453592 kgf

Energy

  • 1 J = 10⁷ erg = 0.101972 kgf·m
  • 1 cal = 4.184 J (thermochemical calorie)
  • 1 eV = 1.602×10⁻¹⁹ J
  • 1 kWh = 3.6×10⁶ J

Power

  • 1 W = 1 J/s = 0.101972 kgf·m/s
  • 1 HP (English) = 745.7 W
  • 1 CV (French, metric horsepower) = 735.5 W

Dimensional Analysis

Fundamental Principle

Dimensional homogeneity: All terms in a physical equation must have the same dimensions.

Basic Dimensional Equations (Mechanics)

Quantity Dimension S.I. Unit C.G.S. Unit
ll (Length) LL m cm
mm (Mass) MM kg g
tt (Time) TT s s
SS (Area) L2L^2 m2\text{m}^2 cm2\text{cm}^2
VV (Volume) L3L^3 m3\text{m}^3 cm3\text{cm}^3
vv (Linear velocity) LT−1LT^{-1} m/s\text{m/s} cm/s\text{cm/s}
aa (Linear acceleration) LT−2LT^{-2} m/s2\text{m/s}^2 cm/s2\text{cm/s}^2
FF (Force) MLT−2MLT^{-2} kg⋅m/s2\text{kg} \cdot \text{m/s}^2 (Newton) g⋅cm/s2\text{g} \cdot \text{cm/s}^2 (dyne)
τ\tau (Torque) ML2T−2ML^2T^{-2} m⋅N\text{m} \cdot \text{N} cm⋅dyne\text{cm} \cdot \text{dyne}
WW (Energy/Work) ML2T−2ML^2T^{-2} N⋅m\text{N} \cdot \text{m} (Joule) dyne⋅cm\text{dyne} \cdot \text{cm} (erg)
PP (Power) ML2T−3ML^2T^{-3} J/s\text{J/s} (Watt) erg/s\text{erg/s}
pp (Momentum) MLT−1MLT^{-1} kg⋅m/s\text{kg} \cdot \text{m/s} g⋅cm/s\text{g} \cdot \text{cm/s}
II (Impulse) MLT−1MLT^{-1} N⋅s\text{N} \cdot \text{s} dyne⋅s\text{dyne} \cdot \text{s}
TT (Period) TT s s
ff (Frequency) T−1T^{-1} Hz\text{Hz} (Hertz) Hz\text{Hz} (Hertz)
ω\omega (Angular velocity) T−1T^{-1} rad/s\text{rad/s} rad/s\text{rad/s}
α\alpha (Angular acceleration) T−2T^{-2} rad/s2\text{rad/s}^2 rad/s2\text{rad/s}^2
MM (Moment of force) ML2T−2ML^2T^{-2} m⋅N\text{m} \cdot \text{N} cm⋅dyne\text{cm} \cdot \text{dyne}
LL (Angular momentum) ML2T−1ML^2T^{-1} kg⋅m2/s\text{kg} \cdot \text{m}^2/\text{s} g⋅cm2/s\text{g} \cdot \text{cm}^2/\text{s}
II (Moment of inertia) ML2ML^2 kg⋅m2\text{kg} \cdot \text{m}^2 g⋅cm2\text{g} \cdot \text{cm}^2

Basic Dimensional Equations (Thermodynamics)

Quantity Dimension S.I. Unit C.G.S. Unit
Temperature [θ][\theta] kelvin (K) kelvin (K)
Heat [ML2T−2][ML^2T^{-2}] joule (J) erg
Specific heat [L2T−2θ−1][L^2T^{-2}\theta^{-1}] J⋅kg−1⋅K−1\text{J} \cdot \text{kg}^{-1} \cdot \text{K}^{-1} erg⋅g−1⋅K−1\text{erg} \cdot \text{g}^{-1} \cdot \text{K}^{-1}
Latent heat [L2T−2][L^2T^{-2}] J⋅kg−1\text{J} \cdot \text{kg}^{-1} erg⋅g−1\text{erg} \cdot \text{g}^{-1}
Entropy [ML2T−2θ−1][ML^2T^{-2}\theta^{-1}] J⋅K−1\text{J} \cdot \text{K}^{-1} erg⋅K−1\text{erg} \cdot \text{K}^{-1}
Internal energy [ML2T−2][ML^2T^{-2}] joule (J) erg
Enthalpy [ML2T−2][ML^2T^{-2}] joule (J) erg
Thermodynamic work [ML2T−2][ML^2T^{-2}] joule (J) erg
Thermal power [ML2T−3][ML^2T^{-3}] watt (W) erg⋅s−1\text{erg} \cdot \text{s}^{-1}
Pressure [ML−1T−2][ML^{-1}T^{-2}] pascal (Pa) barye (dyn⋅cm−2\text{dyn} \cdot \text{cm}^{-2})
Density [ML−3][ML^{-3}] kg⋅m−3\text{kg} \cdot \text{m}^{-3} g⋅cm−3\text{g} \cdot \text{cm}^{-3}
Specific volume [L3M−1][L^3M^{-1}] m3⋅kg−1\text{m}^3 \cdot \text{kg}^{-1} cm3⋅g−1\text{cm}^3 \cdot \text{g}^{-1}
Heat capacity [ML2T−2θ−1][ML^2T^{-2}\theta^{-1}] J⋅K−1\text{J} \cdot \text{K}^{-1} erg⋅K−1\text{erg} \cdot \text{K}^{-1}
Thermal conductivity [MLT−3θ−1][MLT^{-3}\theta^{-1}] W⋅m−1⋅K−1\text{W} \cdot \text{m}^{-1} \cdot \text{K}^{-1} erg⋅cm−1⋅s−1⋅K−1\text{erg} \cdot \text{cm}^{-1} \cdot \text{s}^{-1} \cdot \text{K}^{-1}
Thermal expansion coefficient [θ−1][\theta^{-1}] K−1\text{K}^{-1} K−1\text{K}^{-1}
Dynamic viscosity [ML−1T−1][ML^{-1}T^{-1}] Pa⋅s\text{Pa} \cdot \text{s} poise (dyn⋅s⋅cm−2\text{dyn} \cdot \text{s} \cdot \text{cm}^{-2})
Kinematic viscosity [L2T−1][L^2T^{-1}] m2⋅s−1\text{m}^2 \cdot \text{s}^{-1} stokes (cm2⋅s−1\text{cm}^2 \cdot \text{s}^{-1})

Basic Dimensional Equations (Electromagnetism)

Quantity Dimension (SI) S.I. Unit C.G.S. Unit (esu/emu)
Electric charge [IT][IT] coulomb © statcoulomb (esu) / abcoulomb (emu)
Electric current [I][I] ampere (A) statampere (esu) / abampere (emu)
Electric potential [ML2T−3I−1][ML^2T^{-3}I^{-1}] volt (V) = J⋅C−1\text{J} \cdot \text{C}^{-1} statvolt (esu) / abvolt (emu)
Electric field [MLT−3I−1][MLT^{-3}I^{-1}] V⋅m−1\text{V} \cdot \text{m}^{-1} statvolt⋅cm−1\text{statvolt} \cdot \text{cm}^{-1} (esu)
Electrical resistance [ML2T−3I−2][ML^2T^{-3}I^{-2}] ohm (Ω\Omega) statohm (esu) / abohm (emu)
Electrical conductance [M−1L−2T3I2][M^{-1}L^{-2}T^3I^2] siemens (S) statsiemens (esu) / absiemens (emu)
Capacitance [M−1L−2T4I2][M^{-1}L^{-2}T^4I^2] farad (F) statfarad (esu) / abfarad (emu)
Magnetic flux [ML2T−2I−1][ML^2T^{-2}I^{-1}] weber (Wb) maxwell (Mx)
Magnetic flux density [MT−2I−1][MT^{-2}I^{-1}] tesla (T) = Wb⋅m−2\text{Wb} \cdot \text{m}^{-2} gauss (G)
Magnetic field strength (H) [IL−1][IL^{-1}] A⋅m−1\text{A} \cdot \text{m}^{-1} oersted (Oe)
Inductance [ML2T−2I−2][ML^2T^{-2}I^{-2}] henry (H) stathenry (esu) / abhenry (emu)
Electromotive force (emf) [ML2T−3I−1][ML^2T^{-3}I^{-1}] volt (V) statvolt (esu) / abvolt (emu)
Current density [IL−2][IL^{-2}] A⋅m−2\text{A} \cdot \text{m}^{-2} statampere⋅cm−2\text{statampere} \cdot \text{cm}^{-2}
Electric permittivity [M−1L−3T4I2][M^{-1}L^{-3}T^4I^2] F⋅m−1\text{F} \cdot \text{m}^{-1} dimensionless (esu: ε0=1\varepsilon_0 = 1)
Magnetic permeability [MLT−2I−2][MLT^{-2}I^{-2}] H⋅m−1\text{H} \cdot \text{m}^{-1} dimensionless (emu: μ0=1\mu_0 = 1)
Electric power [ML2T−3][ML^2T^{-3}] watt (W) erg⋅s−1\text{erg} \cdot \text{s}^{-1}
Electromagnetic energy [ML2T−2][ML^2T^{-2}] joule (J) erg
Radiant intensity [MT−3][MT^{-3}] W⋅m−2\text{W} \cdot \text{m}^{-2} erg⋅cm−2⋅s−1\text{erg} \cdot \text{cm}^{-2} \cdot \text{s}^{-1}
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Notes:
  • In the SI system, electric current (II) is a fundamental dimension.
  • In CGS there are variants: esu (electrostatic units) for electrical phenomena and emu (electromagnetic units) for magnetic phenomena. The Gaussian system combines both.
  • In CGS-esu, the vacuum permittivity ε0=1\varepsilon_0 = 1 (dimensionless); in CGS-emu, the vacuum permeability μ0=1\mu_0 = 1 (dimensionless).
  • 1 C=3×109 statcoulomb1\ \text{C} = 3 \times 10^9\ \text{statcoulomb}; 1 T=104 G1\ \text{T} = 10^4\ \text{G}; 1 Wb=108 Mx1\ \text{Wb} = 10^8\ \text{Mx}.

Basic Dimensional Equations (Optics)

Quantity Dimension S.I. Unit C.G.S. Unit
Wavelength [L][L] meter (m) centimeter (cm)
Frequency [T−1][T^{-1}] hertz (Hz) = s−1\text{s}^{-1} s−1\text{s}^{-1}
Speed of light [LT−1][LT^{-1}] m⋅s−1\text{m} \cdot \text{s}^{-1} cm⋅s−1\text{cm} \cdot \text{s}^{-1}
Refractive index dimensionless — —
Optical power (diopter) [L−1][L^{-1}] diopter (m−1\text{m}^{-1}) cm−1\text{cm}^{-1}
Plane angle dimensionless radian (rad) radian (rad)
Solid angle dimensionless steradian (sr) steradian (sr)
Luminous intensity [J][J] candela (cd) —
Luminous flux [J][J] lumen (lm) = cd⋅sr\text{cd} \cdot \text{sr} —
Illuminance [JL−2][JL^{-2}] lux (lx) = lm⋅m−2\text{lm} \cdot \text{m}^{-2} phot (ph) = lm⋅cm−2\text{lm} \cdot \text{cm}^{-2}
Luminance [JL−2][JL^{-2}] cd⋅m−2\text{cd} \cdot \text{m}^{-2} stilb (sb) = cd⋅cm−2\text{cd} \cdot \text{cm}^{-2}
Radiant energy [ML2T−2][ML^2T^{-2}] joule (J) erg
Radiant power (flux) [ML2T−3][ML^2T^{-3}] watt (W) erg⋅s−1\text{erg} \cdot \text{s}^{-1}
Radiant intensity [ML2T−3][ML^2T^{-3}] W⋅sr−1\text{W} \cdot \text{sr}^{-1} erg⋅s−1⋅sr−1\text{erg} \cdot \text{s}^{-1} \cdot \text{sr}^{-1}
Irradiance [MT−3][MT^{-3}] W⋅m−2\text{W} \cdot \text{m}^{-2} erg⋅s−1⋅cm−2\text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-2}
Radiance [MT−3][MT^{-3}] W⋅m−2⋅sr−1\text{W} \cdot \text{m}^{-2} \cdot \text{sr}^{-1} erg⋅s−1⋅cm−2⋅sr−1\text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-2} \cdot \text{sr}^{-1}
Radiant exitance [MT−3][MT^{-3}] W⋅m−2\text{W} \cdot \text{m}^{-2} erg⋅s−1⋅cm−2\text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-2}
Wavenumber [L−1][L^{-1}] m−1\text{m}^{-1} cm−1\text{cm}^{-1} (kayser)
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Notes:
  • In physical optics, radiometric quantities (energy, power, irradiance) have classical mechanical dimensions.
  • Photometric (luminous) quantities incorporate the luminous intensity dimension [J][J], which is fundamental in SI.
  • The CGS system does not define standard photometric units; luminous units are specific to SI.
  • 1 phot=104 lux1\ \text{phot} = 10^4\ \text{lux}; 1 stilb=104 cd⋅m−21\ \text{stilb} = 10^4\ \text{cd} \cdot \text{m}^{-2}.

Basic Dimensional Equations (Acoustics)

Quantity Dimension S.I. Unit C.G.S. Unit
Acoustic pressure [ML−1T−2][ML^{-1}T^{-2}] pascal (Pa) = N⋅m−2\text{N} \cdot \text{m}^{-2} barye = dyn⋅cm−2\text{dyn} \cdot \text{cm}^{-2}
Sound energy density [ML−1T−2][ML^{-1}T^{-2}] J⋅m−3\text{J} \cdot \text{m}^{-3} erg⋅cm−3\text{erg} \cdot \text{cm}^{-3}
Sound intensity [MT−3][MT^{-3}] W⋅m−2\text{W} \cdot \text{m}^{-2} erg⋅s−1⋅cm−2\text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-2}
Sound power [ML2T−3][ML^{2}T^{-3}] watt (W) erg⋅s−1\text{erg} \cdot \text{s}^{-1}
Specific acoustic impedance [ML−2T−1][ML^{-2}T^{-1}] Pa⋅s⋅m−1\text{Pa} \cdot \text{s} \cdot \text{m}^{-1} barye⋅s⋅cm−1\text{barye} \cdot \text{s} \cdot \text{cm}^{-1}
Characteristic acoustic impedance [ML−2T−1][ML^{-2}T^{-1}] rayl (Pa⋅s⋅m−1\text{Pa} \cdot \text{s} \cdot \text{m}^{-1}) barye⋅s⋅cm−1\text{barye} \cdot \text{s} \cdot \text{cm}^{-1}
Speed of sound [LT−1][LT^{-1}] m⋅s−1\text{m} \cdot \text{s}^{-1} cm⋅s−1\text{cm} \cdot \text{s}^{-1}
Frequency [T−1][T^{-1}] hertz (Hz) = s−1\text{s}^{-1} s−1\text{s}^{-1}
Wavelength [L][L] meter (m) centimeter (cm)
Sound pressure level dimensionless decibel (dB) decibel (dB)
Medium density [ML−3][ML^{-3}] kg⋅m−3\text{kg} \cdot \text{m}^{-3} g⋅cm−3\text{g} \cdot \text{cm}^{-3}
Particle displacement [L][L] meter (m) centimeter (cm)
Particle velocity [LT−1][LT^{-1}] m⋅s−1\text{m} \cdot \text{s}^{-1} cm⋅s−1\text{cm} \cdot \text{s}^{-1}
Particle acceleration [LT−2][LT^{-2}] m⋅s−2\text{m} \cdot \text{s}^{-2} cm⋅s−2\text{cm} \cdot \text{s}^{-2}
Acoustic volume flow rate [L3T−1][L^{3}T^{-1}] m3⋅s−1\text{m}^{3} \cdot \text{s}^{-1} cm3⋅s−1\text{cm}^{3} \cdot \text{s}^{-1}
Bulk modulus [ML−1T−2][ML^{-1}T^{-2}] pascal (Pa) barye
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Notes:
  • The sound pressure level in decibels is dimensionless and is defined as Lp=20log⁡10(p/p0)L_p = 20 \log_{10}(p/p_0), where p0=20 μPap_0 = 20\ \mu\text{Pa} is the reference pressure in air.
  • 1 rayl (SI)=10 rayl (CGS)1\ \text{rayl (SI)} = 10\ \text{rayl (CGS)}; 1 Pa=10 barye1\ \text{Pa} = 10\ \text{barye}; 1 W/m2=103 erg⋅s−1⋅cm−21\ \text{W/m}^2 = 10^3\ \text{erg} \cdot \text{s}^{-1} \cdot \text{cm}^{-2}.
  • The characteristic acoustic impedance of a medium is Z=ρcZ = \rho c, where ρ\rho is the density and cc is the speed of sound.

Applications of Dimensional Analysis

  1. Formula verification: Checking dimensional consistency
  2. Formula deduction: Rayleigh’s method
  3. Unit conversion: Changing between systems
  4. Phenomenon study: Similarity analysis and dimensionless numbers

SI Prefixes (International System)

Multiples

Prefix Symbol Factor Scale
Yotta Y 102410^{24} septillion
Zetta Z 102110^{21} sextillion
Exa E 101810^{18} quintillion
Peta P 101510^{15} quadrillion
Tera T 101210^{12} trillion
Giga G 10910^{9} billion
Mega M 10610^{6} million
Kilo k 10310^{3} thousand
hecto h 10210^{2} hundred
deca da 10110^{1} ten

Submultiples

Prefix Symbol Factor Scale
deci d 10−110^{-1} tenth
centi c 10−210^{-2} hundredth
milli m 10−310^{-3} thousandth
micro μ 10−610^{-6} millionth
nano n 10−910^{-9} billionth
pico p 10−1210^{-12} trillionth
femto f 10−1510^{-15} quadrillionth
atto a 10−1810^{-18} quintillionth
zepto z 10−2110^{-21} sextillionth
yocto y 10−2410^{-24} septillionth
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Note:

For areas and volumes, the prefixes apply to the base unit:

  • 1 cm³ = (0.01 m)³ = 10⁻⁶ m³
  • 1 km² = (1000 m)² = 10⁶ m²
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