Units and Measurement Notes

Introduction

• Measurement is the process of comparing a physical quantity to a standard unit to determine its magnitude.

• The result of a measurement is expressed as a numerical value accompanied by a unit.

• Due to the interrelation of physical quantities, a limited number of units can be used to express all of them.

• Fundamental quantities are those that are directly measurable and do not depend on other quantities. They have fundamental units assigned to them.

• Derived quantities are expressed in terms of fundamental quantities through mathematical relationships. They have derived units.

• A complete and consistent set of units used for measuring all kinds of physical quantities is called the system of units.

The International System of Units (SI)

• Historically, various systems of units were used, including:

  • CGS (centimetre, gram, second):

  • Primarily used in some fields of physics and chemistry.

  • FPS (foot, pound, second):

  • Commonly used in engineering in the United States.

  • MKS (metre, kilogram, second):

  • A precursor to the SI system.

• The Système Internationale d’ Unites (SI) is the internationally accepted system of units, ensuring uniformity and accuracy in measurements worldwide.

• Developed and maintained by the Bureau International des Poids et Mesures (BIPM), the SI system was revised in 2018 to enhance accuracy and reliability.

• SI uses the decimal system, which simplifies conversions between units, making it convenient for scientific and engineering calculations.

• The SI system includes seven base units (Table 1.1), each representing a fundamental physical quantity:

  • Length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity.

• Two supplementary units are defined for angular measurements:

  • Radian (rad) for plane angle, defined as $dθ = \frac{ds}{r}$.

  • Steradian (sr) for solid angle, defined as $dΩ = \frac{dA}{r^2}$.

SI Base Quantities

• Length: metre (m)

  • Defined by the speed of light in vacuum $c = 299792458 \frac{m}{s}$. This definition provides an invariant standard for length based on a fundamental constant of nature.

• Mass: kilogram (kg)

  • Defined by the Planck constant $h = 6.62607015 × 10^{-34} Js = kg m^2 s^{-1}$. This definition links the unit of mass to a fundamental constant, ensuring accuracy and stability.

• Time: second (s)

  • Defined by the caesium frequency $∆ν_{Cs} = 9192631770 Hz = s^{-1}$. The second is defined based on an atomic property, providing a highly precise and reproducible standard.

• Electric Current: ampere (A)

  • Defined by the elementary charge $e = 1.602176634 × 10^{-19} C = A s$. This definition connects the unit of electric current to a fundamental constant, ensuring consistency and accuracy.

• Thermodynamic Temperature: kelvin (K)

  • Defined by the Boltzmann constant $k = 1.380649 × 10^{-23} J K^{-1} = kg m^2 s^{-2} K^{-1}$. This definition links temperature to energy at the atomic level, providing a fundamental and precise standard.

• Amount of Substance: mole (mol)

  • Contains exactly $6.02214076 × 10^{23}$ elementary entities (Avogadro constant $N_A$). The mole is a unit that facilitates counting atoms and molecules in chemical reactions.

• Luminous Intensity: candela (cd)

  • Defined by the luminous efficacy of monochromatic radiation of frequency $540 × 10^{12} Hz$, $K_{cd} = 683 \frac{lm}{W} = cd sr W^{-1}$. This definition provides a standard for measuring the brightness of light sources based on human perception.

Notes on Base Units

• The definitions of base units are periodically revised to achieve greater precision and stability, incorporating the latest advancements in measurement science.

• When using the mole, it is crucial to specify the elementary entities being counted, such as atoms, molecules, ions, or other particles.

• Derived units can be expressed using the seven base units through mathematical formulas, allowing for a coherent and interconnected system of measurement (Appendix A 6).

• Some derived units have special names and symbols for convenience in specific fields of science and engineering (Appendix A 6.2).

• SI prefixes and symbols for multiples and sub-multiples are listed in Appendix A2, providing a convenient way to express very large or very small quantities.

• Guidelines for using symbols and writing unit names are provided in Appendices A7 and A8 to ensure consistency and clarity in scientific communication.

Significant Figures

• Measurements always involve some degree of error, so it is essential to indicate the precision of the results.

• Reported results should include all digits that are known reliably, plus the first digit that is uncertain.

• These digits, known as significant figures, indicate the degree of confidence in a measurement.

• Reporting more digits than significant figures is misleading and overstates the precision of the measurement.

• The number of significant figures indicates the precision of a measurement, which depends on the least count of the measuring instrument.

• Changing units of a measurement does not change the number of significant figures.

Rules for Determining Significant Figures:

• All non-zero digits are significant.

• Zeros between two non-zero digits are significant.

• Zeros to the right of the decimal point but to the left of the first non-zero digit are not significant (for numbers less than 1).

• Trailing zeros in a number without a decimal point are not significant.

• Trailing zeros in a number with a decimal point are significant.

Ambiguity with Trailing Zeros

• If a length is reported as 4.700 m, the zeros indicate that the measurement was made to the nearest millimetre, and they are significant.

• Changing units (e.g., to 470.0 cm) should not change the number of significant figures, as the precision of the measurement remains the same.

Scientific Notation

• To avoid ambiguity in the number of significant figures, use scientific notation: $a × 10^b$, where 1 ≤ a < 10.

• 'b' is the order of magnitude, representing the power of 10 by which the base number is multiplied.

• Example: Diameter of Earth ($1.28 × 10^7 m$) is of the order of $10^7 m$, with magnitude 7.

• In scientific notation, all zeros in the base number are significant, clearly indicating the precision of the measurement.

Additional Rules:

• For numbers greater than 1 without a decimal, trailing zeros are not significant.

• For numbers with a decimal, trailing zeros are significant.

• The zero to the left of a decimal for numbers less than 1 is not significant.

• Multiplying or dividing factors that are exact numbers have an infinite number of significant digits.

• For example, in $d = 2r$, the factor 2 is exact and does not limit the number of significant figures in the result.

Rules for Arithmetic Operations with Significant Figures

• Calculations should reflect the uncertainties in the original measured values to provide meaningful results.

• The final result of a calculation should not have more significant figures than the original data with the least number of significant figures.

Multiplication and Division:

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

• Example: Density = $\frac{4.237 g}{2.51 cm^3} = 1.69 \frac{g}{cm^3}$

Addition and Subtraction:

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

• Example: 436.32 g + 227.2 g + 0.301 g = 663.8 g

Rounding off Uncertain Digits

• If the insignificant digit to be dropped is more than 5, the preceding digit is raised by 1.

• If the insignificant digit is less than 5, the preceding digit is unchanged.

• If the insignificant digit is 5:

  • If the preceding digit is even, the digit is dropped.

  • If the preceding digit is odd, the preceding digit is raised by 1.

• Retain one extra digit in intermediate steps to avoid accumulation of rounding errors and round off at the end.

• Exact numbers in formulas (like $2π$) have infinite significant figures and do not limit the number of significant figures in the result.

Example 1.1

• Side of a cube = 7.203 m (4 significant figures)

• Surface area = $6(7.203)^2 m^2 = 311.3 m^2$

• Volume = $(7.203)^3 m^3 = 373.7 m^3$

Example 1.2

• Mass = 5.74 g (3 significant figures)

• Volume = 1.2 cm3 (2 significant figures)

• Density = $\frac{5.74 g}{1.2 cm^3} = 4.8 \frac{g}{cm^3}$

Rules for Determining Uncertainty in Arithmetic Calculations

Example 1

• Length = 16.2 ± 0.1 cm = 16.2 cm ± 0.6%

• Breadth = 10.1 ± 0.1 cm = 10.1 cm ± 1%

• Area = $163.62 cm^2 ± 1.6$

Example 2

• If data is subtracted, the number of significant figures can be reduced, affecting the precision of the result.

• 12.9 g – 7.06 g = 5.8 g

Example 3

• Relative error depends on the number itself and is crucial in assessing the reliability of measurements.

• Error in 1.02 g = ±0.01 g, relative error = ±1%

• Error in 9.89 g = ±0.01 g, relative error = ±0.1%

• Retain one extra digit in intermediate steps to avoid rounding errors that can accumulate and affect the accuracy of the final result.

Dimensions of Physical Quantities

• The nature of a physical quantity is described by its dimensions, which represent the fundamental physical concepts underlying the quantity.

• Base quantities are the seven dimensions: [L], [M], [T], [A], [K], [cd], [mol].

• Dimensions are the powers to which base quantities are raised to express the quantity.

• Volume = [L] × [L] × [L] = [L3]

• Force = mass × acceleration = [M][L]/[T]2 = [M L T–2]

• Change in velocity, initial velocity, average velocity, final velocity, and speed are all equivalent and have dimensions [L T–1].

Dimensional Formulae and Dimensional Equations

• Dimensional formula: Expression showing how base quantities represent the dimensions of a physical quantity.

• Volume: [M0 L3 T0]

• Speed: [M0 L T-1]

• Acceleration: [M0 L T–2]

• Mass density: [M L–3 T0]

• Dimensional equation: Equation equating a physical quantity with its dimensional formula.

• [V] = [M0 L3 T0]

• [v] = [M0 L T–1]

• [F] = [M L T–2]

• [ρ] = [M L–3 T0]

Dimensional Analysis and its Applications

• Only quantities with the same dimensions can be added or subtracted, ensuring dimensional consistency in calculations.

• Dimensional analysis helps deduce relations among physical quantities and check the correctness of derivations.

Checking Dimensional Consistency of Equations

• The principle of homogeneity states that the dimensions of all terms in a physical equation must be the same.

• Example:- $x = x<em>0 + v</em>0t + \frac{1}{2}at^2$

  • [x] = [L]

  • [x0] = [L]

  • [v0t] = [L T–1][T] = [L]

  • [$\frac{1}{2}at^2$] = [L T–2][T2] = [L]

• This equation is dimensionally correct, as all terms have the same dimensions.

• Dimensional consistency does not guarantee that the equation is correct, as it does not account for dimensionless constants or other factors.

• Arguments of trigonometric, logarithmic, and exponential functions must be dimensionless to ensure mathematical validity.

• Pure numbers and ratios of similar quantities are dimensionless, allowing for meaningful comparisons and calculations.

Example 1.3

• $\frac{1}{2}mv^2 = mgh$

• LHS: [M][L T–1]2 = [M L2 T–2]

• RHS: [M][L T–2][L] = [M L2 T–2]

• The equation is dimensionally correct.

Example 1.4

• K = m2 v3 : [M2 L3 T–3] - Ruled out.

• K = (1/2)mv2 : [M L2 T–2] - Possible.

• K = ma : [M L T–2] - Ruled out.

• K = (3/16)mv2 : [M L2 T–2] - Possible.

• K = (1/2)mv2 + ma : No proper dimensions - Ruled out.

Deducing Relation among the Physical Quantities

• The method of dimensions can be used to deduce relations among physical quantities by analyzing their dimensions.

• Assume dependence as a product type, where the physical quantity is expressed as a product of powers of other quantities.

Example 1.5

• Consider a simple pendulum: T = k lx gy mz

• [L0 M0 T1]=[Lx][L yT–2y][Mz]

• [L0 M0 T1] = Lx+y T–2y Mz

• x + y = 0, –2y = 1, z = 0

• x = 1/2, y = -1/2, z = 0

• $$T = k \sqrt{\frac{l}{