Engineering Calculations: Units and Dimensions

Units and Dimensions

Dimensions

• Physical quantities that can be measured (length, time, mass, temperature) or calculated by multiplying/dividing other dimensions (velocity, volume, density).

Units

• Arbitrary names correlating to dimensions, defined by conventions, laws, or customs.
  • Examples: meter (length), second (time), kilogram (mass), Kelvin (temperature).

Classification of Units and Dimensions

Fundamental Units/Dimensions

• Measured independently; sufficient to describe a physical quantity.

Multiple Units/Dimensions

• Defined as multiples or fractions of base units (minutes, hours, years).

Derived or Compound Dimensions/Units

• Obtained through multiplication/division of base or multiple units.
  • Examples: cm², ft³/h, kg⋅m/s².
• Defined equivalents of compound units:
  • erg = 1 g⋅cm²/s²
  • P (Poise) = 1 g/cm⋅s

Systems of Units of Measurement

English Units

• Used in England until 1826.

Imperial Units

• Also known as the imperial system.
• Used across the British Empire from 1826.

Metric System

• Officially adopted in France in 1799.
• MKS (meter-kilogram-second) system: Introduced in 1889 by the Conférence Générale des Poids et Mesures (CGPM).
• CGS (centimeter-gram-second) system: Introduced in 1874 by the British Association for the Advancement of Science (BAAS).

American Engineering System (AES)

• Also known as English Engineering Units; commonly used in the United States.
• Based on:
  • Foot (ft) for length.
  • Pound-mass (lb) for mass.
  • Second (s) for time.
  • Degree Fahrenheit (°F) for temperature.
  • Rankine (R) for absolute temperature.

Système International d'Unités (SI)

• Also known as the International System of Units.
• Adopted by the CGPM in 1960; revised periodically (latest revision: 2018, implemented May 20, 2019).
• Revision changed the definition of base units except for meter, second, and candela; base units defined by relations of defining constants.

SI Base Units and Defining Constants

• Revision of SI units approved in November 2018, redefining kilogram, ampere, Kelvin, and mole.
• All SI units defined in terms of constants describing the natural world.
• The Republic of the Philippines became an Associate of the CGPM on June 1, 2002.

SI Base Units

Defining Constants of the Base Units

• Notes:
  • 1 Hz (Hertz) = 1 s⁻¹
  • 1 C (Coulomb) = 1 A⋅s
  • 1 lm (lumen) = 1 cd⋅sr

Equation for Quantity Values:

$|Q| = {Q} [Q]$
where:

• Q = quantity
• {Q} = numerical value
• [Q] = associated unit

Twenty-Two Named Derived Units

SI Base Unit - METER (m)

• Defined in 2019 using the fixed numerical value of the speed of light in vacuum (c) and the hyperfine transition frequency of Cesium-133 ($Δν_{Cs}$).

• $c = 299 792 458 m/s$ and $Δν_{Cs} = 9 192 631 770 Hz$

• One meter is obtained as follows:
  $1 m = C<em>m$ where $C</em>m = \frac{c}{Δν_{Cs}} = \frac{299 792 458}{9 192 631 770} ≈ 0.032612258$ Hz

• The hyperfine transition frequency of Cs-133 is the frequency of microwave radiation causing the valence electron to jump between two closely spaced low-energy states.

Historical Standards:

• 1889: The International Bureau of Weights and Measures (BIPM) provided the National Prototype Meter Bar No. 27 to the United States, serving as the standard for length (1893-1960).
  • Prototype made of 90% platinum and 10% iridium alloy with an X-shaped (Tresca) cross-section.
• 1960: Krypton-86 lamp defined one meter as 1,650,763.73 wavelengths of light emitted by electrically excited Kr-86 gas.
• 1983: 633 nm Iodine Stabilized He-Ne laser defined one meter as approximately 1,579,800.762 wavelengths.
  • Established an exact relationship between time and length.

SI Base Unit - KILOGRAM (kg)

• Defined in 2019 using fixed numerical values of:

  • Speed of light in vacuum: $c = 299 792 458 m/s$
  • Hyperfine transition frequency of Cesium-133: $Δν_{Cs} = 9 192 631 770 Hz$
  • Planck's constant: $h = 6.626 070 15 × 10^{-34} Js$

• One kilogram is obtained as follows:
  $1 kg = C<em>{kg}$ where: $C</em>{kg} = \frac{h Δν_{Cs}}{c^2} = approx$ a very small number

• Planck's constant (h): A fundamental constant in quantum mechanics, defining the size of quanta (packets of energy) exchanged by matter.

Historical Standards:

• 1793: The grave (gv) was defined as the mass of one liter (dm³) of water at the ice point (0°C).
• 1795: The gram was defined as the mass of one cubic centimeter (cm³) of water at the ice point (0°C).
• 1799: Kilogramme des Archives: A platinum standard equal to the mass of one cubic decimeter of water at 4°C.
• 1875: The Treaty of the Meter established new international prototypes for mass and length.
  • International Prototype Kilogram (IPK): An alloy of 90% platinum and 10% iridium, machined into a right circular cylinder (height and diameter of 39 mm).

SI Base Unit - SECOND (s)

• Defined in 2019 by taking the fixed numerical value of the hyperfine transition frequency of Cs-133: $Δν_{Cs} = 9 192 631 770 Hz$

• One second is obtained as follows:
  $1 s = C<em>s$ where: $C</em>s = \frac{1}{Δν_{Cs}} = \frac{1}{9 192 631 770} Hz$

Historical Timekeeping:

• Ancient Egypt: Sundials divided day and night into 12 hours each using a duodecimal system.
• 1657: Christiaan Huygens built the first pendulum clock.
• 1730: John Harrison invented the marine chronometer accurate to within one second in 100 days.
• 1889: Sigmund Riefler invented the Riefler clock, a highly stable pendulum clock.
• 1904: The National Bureau of Standards (NBS) purchased the Riefler clock.
• 1929: NBS purchased the Shortt clock with a precision of about one millisecond per day.
• 1947: Harold Lyons constructed a clock using ammonia molecules bombarded with microwave radiation.
• 1955: Louis Essen at the National Physical Laboratory in UK constructed the first atomic clock based on Isidor Isaac Rabi's idea.
  • The first atomic clock used a beam of Cesium atoms rather than ammonia molecules.
  • 1956: Commercial Cesium clocks became available costing $20,000 each.

National Bureau of Standards' (NBS) Atomic Clocks:

• 1952: NBS completed NBS-1, the first accurate measurement device for the frequency of the Cesium clock resonance.
• 1960: NBS-2 was inaugurated in Colorado, used to calibrate secondary standards.
• 1963: NBS-3 was created with improved accuracy and stability.
• 1968: NBS-4 was completed and considered the world's most stable Cesium clock.
• 1960: NBS-5 was completed and served as the primary standard.
• 1975: NBS-6 commenced operation, considered an outgrowth of NBS-5.

Nobel Prize:

• 1989: The Nobel Prize in Physics was awarded to Norman F. Ramsey, Hans G. Dehmelt, and Wolfgang Paul for their work in the development of atomic clocks.

National Institute of Standards and Technology's' (NIST) Atomic Clocks:

• 1993: The NIST-7 atomic clock was developed, being more accurate and would not gain or lose a second in 6 million years, with an uncertainty of 5 × 10⁻¹⁵.
• 1999: The NIST-F1 began operation with an uncertainty of 1.7 x 10⁻¹⁵ and an accuracy of one second in 20 million years.
• NIST launched the NIST-F2 to serve as a new US civilian time and frequency standard with an accuracy of one second in about 300 million years. It was run with liquid nitrogen to around - 200°C and was retired in 2015.
• 2023: The NIST-F3, a Cesium fountain frequency reference was presented at the 54th Annual Precise Time and Time Interval Systems and Applications Meeting.
• On April 2025, the NIST-F4 was described in a publication by Gerginov et al and has an accuracy of one second in 100 million years.

SI Base Unit - KELVIN (K)

• Defined in November 2018 using fixed numerical values of:

  • Planck's constant: $h = 6.626 070 15 × 10^{-34} Js$
  • Hyperfine transition frequency of Cs-133: $Δν_{Cs} = 9 192 631 770 Hz$
  • Boltzmann constant: $k_B = 1.380 649 × 10^{-23} J K^{-1}$

• One Kelvin is obtained as follows:
  $1 K = \frac{h \cdot Δν<em>{Cs}}{k</em>B}$

Calculation:
$C_K = \frac{6.62607015 × 10^{-34} × 9192631770}{1.380649 × 10^{-23}} \approx 44065937647987x10^{-11}$

• The Boltzmann constant relates the energy of an object to its temperature.
• Thermodynamic temperature: The absolute measure of the average total internal energy of an object.

Historical Context:

• 1948: The 9th CGPM defined the zero of the centesimal thermodynamic scale as 0.0100 degree below the triple point of water.
• 1954: The 10th CGPM decided that the triple point of water would serve as the fundamental fixed point with an exact value of 273.16 K.
• 1967: The 13th CGPM defined the Kelvin as the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

Temperature Scales:

• 1968: The National Bureau of Standards used the triple-point cell to determine the unique temperature at which water exists in solid, liquid, and vapor phases in equilibrium.
• 2005: The International Committee for Weights and Measures (CIPM) specified the isotopic composition of water influencing the definition of the Kelvin.
  • Composition of Vienna Standard Mean Ocean Water (VSMOW).

SI Base Unit - MOLE (mol)

• Defined in 2019 as exactly $6.022 14076 × 10^{23}$ elementary entities (Avogadro's constant).
• An elementary entity can be an atom, molecule, ion, electron, or any specified group of particles.

Historical Context:

• 1894: Wilhelm Friedrich Ostwald coined the term