International System of Units

• Measurement involves comparing a physical quantity to a standard unit.
• The result is expressed as a number with the unit.
• Physical quantities are interrelated.
• A few base units can express manyquantities.
• Fundamental/base units describe base quantities.
• Derived units are combinations of base units.
• A units system includes both base and derived units.

International System of Units

• Scientists used different unit systems earlier.
• The CGS, FPS, and MKS systems were once common.
• CGS uses centimetre, gram, and second for length, mass, and time.
• FPS uses foot, pound, and second for length, mass, and time.
• MKS uses metre, kilogram, and second for length, mass, and time.
• The current internationally accepted system is Système Internationale d' Unités (SI).
• It establishes standard symbols, units, and abbreviations.
• The Bureau International des Poids et measures (BIPM) developed the SI standards.
• The system was revised by the General Conference on Weights and Measures in 2018.
• SI units are valuable for scientific, technical, industrial, and commercial applications.
• Decimal-based SI units make conversions straightforward.
• The SI has seven base units.
• Plane angle (dθ) is the ratio of arc length (ds) to radius (r), measured in radians (rad).
• Solid angle (dΩ) is the ratio of area (dA) to radius squared (r²), measured in steradians (sr).
• Both radians and steradians are dimensionless.

SI Base Quantities and Units

• Length is measured in metres (m).
• A metre is defined by the speed of light in vacuum (c), which is 299,792,458 m/s.
• Mass is measured in kilograms (kg).
• A kilogram is defined by the Planck constant (h), which is 6.62607015 × 10⁻³⁴ J s (kg m² s⁻¹).
• Time is measured in seconds (s).
• A second is defined by the caesium frequency (ΔνCs), which is 9,192,631,770 Hz (s⁻¹).
• Electric current is measured in amperes (A).
• An ampere is defined by the elementary charge (e), which is 1.602176634 × 10⁻¹⁹ C (A s).
• Thermodynamic temperature is measured in kelvin (K).
• A kelvin is defined by the Boltzmann constant (k), which is 1.380649 × 10⁻²³ J K⁻¹ (kg m² s⁻² K⁻¹).
• The amount of substance is measured in moles (mol).
• A mole contains 6.02214076 × 10²³ elementary entities defined by the Avogadro constant (NA ).
• Luminous intensity is measured in candelas (cd).
• A candela is defined by the luminous efficacy (Kcd) of monochromatic radiation, which is 683 lm/W (cd sr kg⁻¹ m⁻² s³).
• With technology improvements, measuring techniques evolve for greater precision; definitions of base units are revised accordingly.

Other Retained Units

• minute (min) equals 60 s
• hour (h) equals 60 min or 3600 s
• day (d) equals 24 h or 86400 s
• year (y) equals 365.25 d or 3.156 × 10⁷ s
• degree (°) equals (π/180) rad
• litre (L) equals 1 dm³ or 10⁻³ m³
• tonne (t) equals 10³ kg
• carat (c) equals 200 mg
• bar equals 0.1 MPa or 10⁵ Pa
• curie (Ci) equals 3.7 × 10¹⁰ s⁻¹
• roentgen (R) equals 2.58 × 10⁻⁴ C/kg
• quintal (q) equals 100 kg
• barn (b) equals 100 fm² or 10⁻²⁸ m²
• are (a) equals 1 dam² or 10² m²
• hectare (ha) equals 1 hm² or 10⁴ m²
• standard atmospheric pressure (atm) equals 101325 Pa or 1.013 × 10⁵ Pa

Significant Figures

• Measurement results should indicate precision, including all reliably known digits plus the first uncertain digit.

• These reliable digits, including the first uncertain one, are significant figures.

• Reporting more digits than significant figures would wrongly imply greater precision.

• Changing units does not alter the number of significant figures.

• Digits, such as 2, 3, 0, and 8, in numbers like 2.308 cm, 0.02308 m, 23.08 mm, or 23080 μm, remain four.

• The location of the decimal point is irrelevant in determining significant figures.

  • all non-zero digits are significant
  • zeros between non-zero digits are significant regardless of the decimal point's location
  • in numbers less than 1, zeros to the right of the decimal point but to the left of the first non-zero digit are not significant
  • terminal or trailing zeros in a number without a decimal point are not significant
  • trailing zeros in a number with a decimal point are significant
  • using scientific notation removes ambiguity and accurately represents a measurement.

• Scientific notation format requires expressing every number as a × 10b.

• Where 'a' is between 1 and 10, and 'b' is any exponent of 10.

• When rounding quantities to estimate, 'a' is rounded to 1 if a < 5, and 10 if 5 < a < 10.

• The exponent 'b' of 10 indicates the order of magnitude of the physical quantity.

• When only an estimate is needed, report the quantity as being on the order of 10b.

• In scientific notation, all zeros in the base number are significant.

  • for numbers greater than 1, without any decimal, trailing zeros are not significant
  • for numbers with a decimal, the trailing zeros are significant.

• The zero to the left of a decimal in numbers less than 1 (e.g., 0.1250) is insignificant, but trailing zeros are significant in measurements.

• Multiplying or dividing factors that aren't rounded or measured are considered exact numbers with infinite significant digits.

Arithmetic Operations with Sig Figs

• Calculations with approximate values must reflect the uncertainties of the original data and should not be more precise.
• The final result should not have more significant figures than the original data with the least significant figures.
• Density Example if mass = 4.237 g (4 sig figs) volume = 2.51 cm³ (3 sig figs), then the density = 1.69 g cm⁻³ (3 sig figs)
• In multiplication or division, the result retains as many significant figures as the original number with the least significant figures.
• In addition or subtraction, the result retains as many decimal places as are present in the number with the least decimal places.

Rounding Uncertain Digits

• Computation results with approximate numbers should be rounded off appropriately to have the correct number of significant figures.
• convention states that the preceding digit is increased by 1 if the insignificant digit to be dropped is more than 5 or is otherwise left unchanged if the latter is less than 5
• If the number is equal to 5, the convention is that if the preceding digit is even, the insignificant digit is simply dropped, and, if it is odd, the preceding digit is raised by 1.
• In multi-step calculations, retain one extra digit in intermediate steps.
• Round off to the appropriate significant figures at the end.
• Exact numbers in formulae have infinite significant figures.

Rules for Determining Uncertainty

• The error for combined quantities is calculated using a combination of errors rule.
• If experimental data is specified to n significant figures, the result is valid to n significant figures unless data is subtracted and significant figures are reduced.
• Relative error depends on the number's value and the number of significant figures.
• Intermediate results in complex calculations should maintain one more significant figure than the least precise measurement to justify the data.

Dimensions of Physical Quantities

• The nature of a physical quantity is described by its dimensions
• Derived units can be expressed in terms of fundamental/base quantities
• Base quantities are considered the seven dimensions of the physical world.
• Length has a dimension of [L].
• Mass has a dimension of [M].
• Time has a dimension of [T].
• Electric Current has a dimension of [A].
• Thermodynamic Temperature has a dimension of [K].
• Luminous Intensity has a dimension of [cd].
• Amount of Substance has a dimension of [mol].
• The dimensions of a physical quantity are defined by the powers of base quantities needed to represent that quantity.
• Using square brackets [ ] around a quantity indicates dimensions are being discussed.

Dimensional Formulae and Equations

• A dimensional formula shows how basic quantities represent a physical quantity's dimensions.
  • the dimensional formula of volume is [M° L³ T°]
  • the dimensional formula of speed or velocity is [M° LT⁻¹]
  • the dimensional formula of acceleration is [M° LT⁻²]
  • the dimensional formula of mass density is [M° L⁻³ T°]
• A dimensional equation equates a physical quantity with its dimensional formula.
• Dimensional equations represent physical quantity dimensions in terms of base quantities.

Dimensional Analysis and Applications

• The concepts of dimensions are essential for guiding physical behavior descriptions
• Only quantities with the same dimensions can be added or subtracted.
• Dimensional analysis helps deduce relations among different physical quantities.
• It also helps check derivation accuracy and dimensional consistency of mathematical expressions.
• Dimensions of physical quantities must be the same on both sides of a mathematical equation.
• Magnitudes of physical quantities can only be added or subtracted if they share the same dimensions.
• Dimensional homogeneity checks equation correctness by ensuring all terms have the same dimensions.
• If dimensions do not match, the equation is incorrect.
• Dimensions test is equivalent to a units consistency test but does not ensure equations are entirely correct.
• A dimensionally correct equation may not be exact, but a dimensionally wrong equation must be incorrect.