Units and Measurement: SI Units

• Units and Measurement involves comparing physical quantities to standards.
• A unit is a basic, arbitrarily chosen reference standard.
• Results are expressed as a number and a unit.
• A limited number of units are needed because physical quantities are interrelated.
• Fundamental or base units are for fundamental quantities.
• Derived units are combinations of base units.
• System of units are complete sets of base and derived units.

The International System of Units

• Scientists used different unit systems (CGS, FPS, MKS) in the past.
• CGS base units: centimetre, gram, second.
• FPS base units: foot, pound, second.
• MKS base units: metre, kilogram, second.
• Système Internationale d' Unités (SI) is internationally accepted today.
• SI has standard symbols, units, and abbreviations.
• SI was developed in 1971 and revised in 2018 by BIPM .
• SI is used in scientific, technical, industrial, and commercial work.
• Conversions are simple due to the decimal system.

SI Base Quantities and Units

• Seven base units in SI include Length-metre(m); Mass-kilogram(kg); Time-second(s); Electric current-ampere(A); Thermodynamic Temperature-kelvin(K); Amount of substance-mole(mol); and Luminous intensity-candela(cd).
• Two additional units define plane angles (radian, rad) and solid angles (steradian, sr).
• Radian (rad) is the ratio of arc length to radius.
• Steradian (sr) is the ratio of intercepted area to squared radius.
• Both radian and steradian are dimensionless.
• The metre (m) is defined by the speed of light in a vacuum (299792458 m/s).
• The kilogram (kg) is defined by the Planck constant (6.62607015 × 10⁻³⁴ J s).
• The second (s) is defined by the caesium-133 frequency (9192631770 Hz).
• The ampere (A) is defined by the elementary charge (1.602176634 × 10⁻¹⁹ C).
• The kelvin (K) is defined by the Boltzmann constant (1.380649 × 10⁻²³ J K⁻¹).
• The mole (mol) contains exactly 6.02214076 × 10²³ elementary entities (Avogadro constant).
• The candela (cd) is defined by the luminous efficacy of monochromatic radiation at 540 × 10¹² Hz (683 lm/W).

Other Units

• Minute (min) is 60 s.
• Hour (h) is 3600 s.
• Day (d) is 86400 s.
• Year (y) is 3.156 × 10's.
• Degree (°) is (π/180) rad.
• Liter (L) is 10⁻³ m³.
• Tonne (t) is 10³ kg.
• Carat (c) is 200 mg.
• Bar is 10⁵ Pa.
• Curie (Ci) is 3.7 × 10¹⁰ s⁻¹.
• Roentgen (R) is 2.58 × 10⁻⁴ C/kg.
• Quintal (q) is 100 kg.
• Barn (b) is 10⁻²⁸ m².
• Are (a) is 10² m².
• Hectare (ha) is 10⁴ m².
• Standard atmosphere (atm) is 101325 Pa.

Significant Figures

• Significant figures are reliable digits plus the first uncertain digit.
• A measurement's precision is indicated by significant figures.
• Units do not impact the number of significant figures.

Rules for Determining Significant Figures

• Non-zero digits are always significant.
• Zeros between non-zero digits are significant.
• For numbers less than 1, zeros to the left of the first non-zero digit are not significant (e.g., 0.002308).
• Trailing zeros in numbers without a decimal point are not significant (e.g., 123 m = 12300 cm).
• Trailing zeros in numbers with a decimal point are significant (e.g., 3.500).

Scientific Notation

• Report measurements in scientific notation (a × 10ᵇ) to avoid ambiguity.
• The term 'a' should be a number between 1 and 10.
• The power of 10 is irrelevant which determining significant figures, so all zeroes in the number are significant.
• For numbers greater than 1 (without a decimal), trailing zeros are insignificant.
• For numbers with a decimal, trailing zeros are significant.
• The digit 0 to the left of a decimal for numbers less than 1 is never significant.
• Multiplying or dividing factors are exact and have infinite significant digits.

Arithmetic Operations

• Calculations must reflect original measurement uncertainties.
• Results should not have more significant figures than the original data.

Multiplication/Division

• The final result should have as many significant figures as the original number with the least significant figures.

Addition/Subtraction

• The final result should have as many decimal places as the number with the least decimal places.

Rounding off Uncertain Digits

• Round off results of computation with approximate numbers.
• The preceding digit is raised by 1 if the insignificant digit is more than 5.
• It is left unchanged if the insignificant digit is less than 5.
• For insignificant digit of 5: If the preceding digit is even, drop the 5; if odd, raise the preceding digit by 1.
• Retain one extra digit during intermediate steps and round off at the end.
• Round numbers known to many significant figures, like the speed of light, for computations.
• Exact numbers have an infinite number of significant figures.

Rules for Determining Uncertainty in Arithmetic Calculations

• These rules include uncertainty or error in arithmetic operations.
• For length (l) and breadth (b) measurements with 3 significant figures:
• l = 16.2 ± 0.1 cm = 16.2 cm ± 0.6% and b = 10.1 ± 0.1 cm = 10.1 cm ± 1%.
• Error in the product is calculated using the combination of errors rule: lb = 163.62 cm² ± 1.6% = 163.62 ± 2.6 cm².
• Final result: lb = 164 ± 3 cm².
• Uncertainty indicated occurs in the area of rectangular sheet.
• 'n' significant figures for data results in combination valid to 'n' significant figures, unless data is subtracted
• Relative error depends on 'n' and the number itself.
• Computation needs one more significant figure than the least precise measurement, justified by data.

Dimensions of Physical Quantities

• Dimensions describe the nature of physical quantities.
• Base quantities: length [L], mass [M], time [T], electric current [A], thermodynamic temperature [K], luminous intensity [cd], amount of substance [mol].
• Dimensions are the powers to which base quantities are raised.
• Mechanics uses [L], [M], [T].
• Volume is length × breadth × height, with dimensions [L]³.
• Volume has zero dimension in mass [M⁰] and time [T⁰].
• Force = mass × acceleration, with dimensions [MLT⁻²].
• Velocity quantities are expressed as length/time, or [LT⁻¹].

Dimensional Formulae and Equations

• Dimensional formula expresses dimensions of a physical quantity.
• Volume: [M⁰ L³ T⁰], speed: [M⁰ LT⁻¹], acceleration: [M⁰ LT⁻²], mass density: [M L⁻³ T⁰].
• Dimensional equation equates a physical quantity with its dimensional formula.
• Equations represent dimensions of a physical quantity in terms of base quantities.

Dimensional Analysis and its Applications

• Recognition of dimension concepts guides description of physical behavior.
• Quantities with the same dimensions can be added or subtracted.
• Dimensional analysis helps deduce relations, check derivation, accuracy, and consistency of equations.
• Units are treated as algebraic symbols during multiplication.
• Physical quantities on both sides of an equation must have the same dimensions.
• Principle of Homogeneity of Dimensions states that only quantities with the same dimensions can be added or subtracted.
• Dimensions test the consistency of equations.
• Dimensional consistency does not guarantee correct equations.
• Arguments of special functions must be dimensionless.
• Pure numbers and ratios of similar quantities are dimensionless.

Learn about units and measurement, including the International System of Units (SI). Understand base units, derived units, and the importance of standard units in scientific and technical fields. Explore the evolution from CGS, FPS, and MKS systems to the universally accepted SI system.