Comprehensive Notes on SI Units, Prefixes, and Unit Conversions

Measurements and the Need for Standards

• Measurements are observed daily, from speed limits on highways to the mass of grocery items.
• The widespread use of measurement requires a reference standard to ensure consistency across people, places, and time.
• In science and engineering, standard systems of measurement enable comparisons, reproducibility, and safety (e.g., in construction, manufacturing, and research).
• Visual elements in the transcript emphasize this theme with reference to general physics, chemistry, and everyday items (e.g., speed limits, cake ingredients, and energy concepts like E=mc^2).

What is the SI System and Why Do We Use It?

• International System of Units (SI) is the modern standard system for fundamental quantities.
• SI is the standard system of measurement used today and is often referred to as the metric system.
• Establishment: SI was established in 1960.
• SI base units provide a universal foundation for all measurements; they are defined by fixed constants and reproducible phenomena.
• In scientific notation and documentation, the quantities are typically written in italic (e.g., $t$ for time), while the unit symbols are upright (e.g., s for second).

SI Base Units: The Seven Fundamental Quantities

• SI base units and their typical symbols:
  • Time: Second, symbol $t\,$, unit $\mathrm{s}$, quantity: time; Unit symbol: $\mathrm{s}$
  • Length: Meter, symbols $l, x, r, \dots$, unit $\mathrm{m}$, quantity: length/space
  • Mass: Kilogram, symbol $m$, unit $\mathrm{kg}$, quantity: mass
  • Electric current: Ampere, symbol $I, i$, unit $\mathrm{A}$, quantity: electric current
  • Thermodynamic temperature: Kelvin, symbol $T$, unit $\mathrm{K}$, quantity: temperature
  • Amount of substance: Mole, symbol $n$, unit $\mathrm{mol}$, quantity: amount of substance
  • Luminous intensity: Candela, symbol $Iv$, unit $\mathrm{cd}$, quantity: luminous intensity
• Base units are defined to be invariant and universal, with their symbols written upright and quantity symbols in italics.

Base Units: Historical Definitions and Modern Redefinitions

• Early definitions of base units relied on artefacts or human body parts, which led to variability and disputes.

• Today, base units are defined by fixing the values of fundamental constants, ensuring stability and universality across time and space.

• Example: the table of redefining constants used to define each base unit (values given in the transcript are representative):

  • Second (s): Defined by the cesium-133 atom hyperfine transition. The duration of 9,192,631,770 cycles of microwave radiation corresponding to this transition defines the second.
  • Meter (m): Defined by the distance light travels in vacuum in the time interval of $\frac{1}{299{,}792{,}458}\text{ s}$, i.e., the speed of light $c = 299{,}792{,}458\ \mathrm{m\,s^{-1}}$.
  • Kilogram (kg): Redefined by fixing the Planck constant $h = 6.626\,070\,15\times 10^{-34} \ \mathrm{J\,s}$, which is equivalent to defining the kilogram via $h\, (=:)\ \mathrm{kg\,m^2\,s^{-1}}$.
  • Ampere (A): Redefined by fixing the elementary charge $e = 1.602\,176\,634\times 10^{-19} \ \mathrm{C}$; the ampere is then tied to a fixed amount of charge per unit time.
  • Kelvin (K): Redefined by fixing the Boltzmann constant $k = 1.380\,649\times 10^{-23} \ \mathrm{J\,K^{-1}}$, linking temperature to energy per kelvin.
  • Mole (mol): Defined by fixing the Avogadro constant $N_A = 6.022\,140\,76\times 10^{23}\ \mathrm{mol^{-1}}$, so one mole contains exactly this number of entities.
  • Candela (cd): Defined by fixing the luminous efficacy of monochromatic radiation with frequency $f = 540\times 10^{12}\ \mathrm{Hz}$ to be $K_{cd} = 683\ \mathrm{lm\,W^{-1}}$, corresponding to the luminous intensity of a light source in a given direction.

• Notes on constants and definitions:

  • The shift to fixed constants eliminates artefacts and abstract or time-varying references.
  • The fixed constants provide stability under real-world measurement changes and improvements in technology.

Seven SI Base Units: Detailed Definitions (Concisely)

• Second (s): Time required for 9,192,631,770 cycles of the microwave radiation of cesium-133 atoms.

• Meter (m): Distance light travels in vacuum in $1/299{,}792{,}458\ \text{s}$; equivalently, speed of light $c = 299{,}792{,}458\ \mathrm{m\,s^{-1}}.$

• Kilogram (kg): Defined by fixed value of Planck constant $h = 6.626\,070\,15\times 10^{-34}\ \mathrm{J\,s}$; unit expressed as $\mathrm{kg\,m^2\,s^{-1}}$.

• Ampere (A): Defined by fixed value of elementary charge $e = 1.602\,176\,634\times 10^{-19}\ \mathrm{C}.$ Current relates to charge per unit time.

• Kelvin (K): Defined by fixed value of Boltzmann constant $k = 1.380\,649\times 10^{-23}\ \mathrm{J\,K^{-1}}.$

• Mole (mol): One mole contains exactly $N_A = 6.022\,140\,76\times 10^{23}$ elementary entities.

• Candela (cd): Defined by luminous efficacy constant $K_{cd} = 683\ \mathrm{lm\,W^{-1}}.$

• Summary equation set (key relationships):

  • Speed of light: $c = 299{,}792{,}458\ \mathrm{m\,s^{-1}}.$
  • Planck relation (conceptual link between energy and frequency): $E = h\,\nu.$
  • 1 kg in base units: represented as $1\ \mathrm{kg} = 1\ \mathrm{J\,s}\, /\, h.$ (illustrative connection to Planck constant)

SI Prefixes: Powers of Ten Used with SI Units

• Prefixes are added to base units to scale values up or down by factors of ten. Common prefixes and their factors (from smallest to largest in the transcript and standard usage):
  • yocto (y): $10^{-24}$
  • zepto (z): $10^{-21}$
  • atto (a): $10^{-18}$
  • femto (f): $10^{-15}$
  • pico (p): $10^{-12}$
  • nano (n): $10^{-9}$
  • micro (\mu): $10^{-6}$
  • milli (m): $10^{-3}$
  • centi (c): $10^{-2}$
  • deci (d): $10^{-1}$
  • kilo (k): $10^{3}$
  • mega (M): $10^{6}$
  • giga (G): $10^{9}$
  • tera (T): $10^{12}$
  • peta (P): $10^{15}$
  • exa (E): $10^{18}$
  • zetta (Z): $10^{21}$
  • yotta (Y): $10^{24}$
• Practical tips:
  • Memorize common prefixes and their values (e.g., femto- to centi-, kilo- to giga-).
  • Prefixes appear frequently in physics and everyday life (e.g., microgram vs milligram, kilometer vs meter).

Derived Quantities and Derived Units

• Derived quantities are formed from the product (or quotient) of two or more base units.
• Examples of derived quantities:
  • Volume: $V = m^{3}$ (unit: $\mathrm{m^3}$)
  • Speed/velocity: $v = \frac{\mathrm{m}}{\mathrm{s}}$ (unit: $\mathrm{m\,s^{-1}}$)
  • Force: $F = \mathrm{N} = \mathrm{kg\,m\,s^{-2}}$
  • Energy/Work: $E = \mathrm{J} = \mathrm{kg\,m^{2}\,s^{-2}}$
  • Heat capacity: $C = \mathrm{J\,K^{-1}}$
  • Electric charge: $Q = \mathrm{C} = \mathrm{A\,s}$
• Derived units summarised:
  • Volume: $\text{Unit} = \mathrm{m^3}$
  • Speed/velocity: $\text{Unit} = \mathrm{m\,s^{-1}}$
  • Newton (force): $\mathrm{N} = \mathrm{kg\,m\,s^{-2}}$
  • Joule (energy): $\mathrm{J} = \mathrm{kg\,m^2\,s^{-2}}$
  • Coulomb (electric charge): $\mathrm{C} = \mathrm{A\,s}$

Other Systems of Measurement

• British Imperial system and U.S. customary units are still used in some contexts, but SI is preferred for consistency.
• Historical act: the Weights and Measures Act (1824) influenced imperial units; U.S. customary units were adapted from this act.
• Examples of imperial and U.S. customary units and their metric equivalents:
  • Pound (lb) → $4.448\ \mathrm{N}$
  • Slug → $14.59\ \mathrm{kg}$
  • Ounce (oz) → $28.350\ \mathrm{g}$
  • Mile (mi) → $1.609\ \mathrm{km}$
  • Foot (ft) → $0.3048\ \mathrm{m}$
  • Inch (in) → $2.54\ \mathrm{cm}$
• The transcript notes: most quantities in problems require SI units; imperial and U.S. customary units may be mentioned but SI should be used whenever possible.

Conversion of Units: Core Principles

• Key idea: an equation or expression must be consistent with units; units can act as algebraic quantities and cancel during calculations.
• Example demonstration: converting $5.0\ \,in$ to cm using the conversion factor $1\ \,\mathrm{in} = 2.54\ \mathrm{cm}$:
  • $5.0\ \mathrm{in} = (5.0\ \mathrm{in})\left(\frac{2.54\ \mathrm{cm}}{1\ \mathrm{in}}\right) = 12.7\ \mathrm{cm}$
  • The inch unit cancels, leaving the desired unit (cm).
• Another example: converting $55\ \mathrm{m}$ to km:
  • Using the prefix relation $1\ \mathrm{km} = 10^{3}\ \mathrm{m}$, we get $55\ \mathrm{m} = 0.055\ \mathrm{km}$.
• Rule of thumb: interchange numerator and denominator in the conversion factor to obtain the desired unit; the remaining unit is the target unit.

Practice Problems and Worked Solutions

• Quick conversions:
  • A common housefly is 5.0 mm long. In meters:
  • Answer: $5.0\times 10^{-3}\ \mathrm{m} = 0.005\ \mathrm{m}$
  • A largest white shark mass: 907{,}185 g to kilograms:
  • Answer: $907{,}185\ \mathrm{g} = 907.185\ \mathrm{kg}$
  • A three-story building height: 10 ft to meters:
  • Answer: $10\ \mathrm{ft} \approx 3.048\ \mathrm{m}$
  • A medium-sized truck mass: 544 slug to kilograms (1 slug = 14.59 kg):
  • Calculation: $544\ \text{slug} \times 14.59\ \mathrm{kg/slug} = 7936.96\ \mathrm{kg}$
  • A car on the NLEX at 35 km/h: is it exceeding a speed limit of 17 m/s?
  • Convert 35 km/h to m/s: $35\ \mathrm{km/h} = 35\times\frac{1000}{3600} \ \mathrm{m/s} \approx 9.72\ \mathrm{m/s}$
  • Since 9.72 < 17, it is not exceeding the limit.
  • If the speed limit is 80 km/h, what is the equivalent speed in m/s and in mph?
  • In m/s: $80\ \mathrm{km/h} = 80\times\frac{1000}{3600} \ \mathrm{m/s} \approx 22.22\ \mathrm{m/s}$
  • In mph (approximate): $80\ \mathrm{km/h} \approx 49.72\ \mathrm{mph}$
• Additional practice questions from the transcript:
  • Check movement between units and ensure consistent use of SI.

Check Your Understanding (Direct from Transcript)

• Identify the correct word(s) for each item:
  1) It is a process of assigning a quantity to describe the property of an object by comparing it with a standard.
  Answer: Measurement
  2) It is the standard system of measurement used today.
  Answer: SI
  3) It is expressed from the product of two or more base units.
  Answer: Derived quantities
  4) It is the standard unit of luminous intensity in a given direction.
  Answer: Candela (cd)
  5) It is the standard unit of amount of substance defined based on the fixed value of Avogadro’s number.
  Answer: Mole (mol)

Practice Conversions (From Transcript)

• Convert to SI units:
  1) The Philippines is 3{,}070 km away from Japan.
  Answer: $3{,}070\ \mathrm{km} = 3{,}070{,}000\ \mathrm{m}$
  2) An Asian elephant mass: 370 slug to kg.
  Answer: $370\ \mathrm{slug} \times 14.59\ \mathrm{kg/slug} = 5398.3\ \mathrm{kg} (\approx 5.40\times 10^3\ \mathrm{kg})$
  3) Osmium density: 22.59 g/cm³ to kg/m³.
  Note: $1\ \mathrm{g/cm^3} = 1000\ \mathrm{kg/m^3}$ so $22.59\ \mathrm{g/cm^3} = 22590\ \mathrm{kg/m^3}$
  4) The Eiffel Tower height: 1{,}064 ft to meters.
  Answer: $1064\ \mathrm{ft} \times 0.3048\ \mathrm{m/ft} = 324.0\ \mathrm{m}$
  5) Fastest land animal speed: cheetah at 109.4 km/h to m/s.
  Answer: $109.4\ \mathrm{km/h} \times \frac{1000\ \mathrm{m}}{3600\ \mathrm{s}} \approx 30.4\ \mathrm{m/s}$

The Challenge: Hands as a Unit of Horse Height

• A horse height is often given in hands; 1 hand equals 4 inches.
• Question: Is hands a good standard? Why or why not?
  • Pros: Easy to approximate horse height by a simple, common unit; familiar in equestrian contexts.
  • Cons: Not SI-based; includes variability due to horse stance, saddle, and measurement conventions. Less precise and not universally standardized outside the equestrian community.
  • Conclusion: For scientific and engineering measurements, SI units are preferred; hands are useful in veterinary and riding contexts but not as a universal standard for cross-discipline communication.

Final Notes and Practical Guidelines

• Always include units with numerical values; carry units through calculations to avoid mistakes.
• If a unit appears as a derived quantity, know the base-unit decomposition (e.g., N = kg m s^{-2}, J = kg m^2 s^{-2}).
• When converting, use consistent significant figures and round only at the end if required by the problem.
• Remember: the SI base units are defined by constants and fixed values, which makes SI stable and universally reproducible.
• The Apostrophized values and notations (italic quantity symbols, upright unit symbols) help distinguish quantity from unit in technical writing.

Bibliography (Referenced in Transcript)

• Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore: Brooks/Cole, 2006.
• International Committee for Weights and Measures. SI Brochure (9th ed). France: Bureau International des Poids et Mesures, 2019.
• Knight, Randall Dewey. Physics for Scientists and Engineers: a Strategic Approach with Modern Physics. Pearson, 2017.
• Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics (9th ed). USA: Brooks/Cole, 2014.
• Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky's University Physics with Modern Physics (13th ed). USA: Pearson Education, 2012.