Chapter One: Units and Measurement

CHAPTER ONE UNITS AND MEASUREMENT

1.1 Introduction

• Measurement: Involves comparing a physical quantity with an internationally accepted reference standard called a unit.

• Result of Measurement: Expressed as a number (numerical measure) with a unit.

• Despite the large variety of physical quantities, a limited set of base units covers all.

• Base Units: Fundamental units for basic quantities.

• Derived Units: Combinations of base units used for other physical quantities.

• System of Units: Complete set including both base and derived units.

1.2 The International System of Units (SI)

• Historically, varying systems existed (e.g., CGS, FPS, MKS).

• Base Units of major systems:

  • CGS: centimetre (cm), gram (g), second (s)

  • FPS: foot (ft), pound (lb), second (s)

  • MKS: metre (m), kilogram (kg), second (s)

• SI Units: Currently accepted system, standardized by the BIPM. The latest revision was in November 2018.

• Base Units in SI (Seven units):

  • Length: Metre (m), defined via the speed of light.

  • Mass: Kilogram (kg), defined via the Planck constant.

  • Time: Second (s), defined via caesium frequency.

  • Electric Current: Ampere (A).

  • Thermodynamic Temperature: Kelvin (K).

  • Amount of Substance: Mole (mol).

  • Luminous Intensity: Candela (cd).

• Additional Units: Radian (rad) for angles and steradian (sr) for solid angles are dimensionless quantities.

1.3 Significant Figures

• Definition: Reflect the precision of a measurement, indicating certain digits and the first uncertain digit.

• Examples:

  • Period of a pendulum: 1.62 s (3 significant figures)

  • Length: 287.5 cm (4 significant figures)

• Rules for Significant Figures:

  1. Non-zero digits are significant.

  2. Zeros between non-zero digits are significant.

  3. Leading zeros (left of non-zero) are not significant.

  4. Trailing zeros without a decimal point are not significant, with the exception of trailing zeros in decimals (significant).

  5. Scientific notation always conveys significant figures reliably (e.g., 4.700 × 10^2 has 4 significant figures).

• Arithmetic Operations:

  • In multiplication/division, the total should have as many significant figures as the measurement with the least significant figures.

  • In addition/subtraction, the sum should have as many decimal places as the measurement with the least decimal places.

1.4 Dimensions of Physical Quantities

• Definition: The nature of a quantity expressed as a combination of base quantities (dimensions).

  • Length: [L], Mass: [M], Time: [T], Electric Current: [A], Temperature: [K], Luminous Intensity: [cd], Amount of Substance: [mol].

• Examples of Dimensional Analysis:

  • Volume: [L]^3.

  • Force: [M L T^-2].

1.5 Dimensional Formulae and Dimensional Equations

• Dimensional Formula: Represents the dimensions of a physical quantity.

• Dimensional Equation: Equates a physical quantity to its dimensional formula.

  • Example: Volume (V) in terms of dimensions: [V] = [M^0 L^3 T^0].

1.6 Dimensional Analysis and Its Applications

• Applications:

  • Ensure dimensional consistency of equations.

  • Can determine relations among physical quantities.

• Homogeneity Principle: Quantities can only be added/subtracted if they share the same dimensions.

• Verification of equations for dimensional correctness without specific unit choice.

• Example of Dimensional Analysis: Examining equations like kinetic energy to ensure consistency in dimensions.

Summary

• Key Points:

  • Physics relies on the measurement of physical quantities using a system of units (SI).

  • Significant figures provide the necessary precision for measurements in calculations.

  • Dimensional analysis offers a method to derive relations and confirm the correctness of equations.