UST FOE Bridging Program – Physics Module 0 Comprehensive Notes

What Is Physics?

  • Science that studies nature & properties of matter and energy

  • Subject matter: heat, light, sound, electricity, magnetism, atomic structure

  • Also called the “science of measurement” – every law is formulated & verified experimentally through precise measurement

  • Applied Physics = interface/bridge between physics & engineering; focuses on practical problem-solving

Branches by Historical Era

  • Classical Physics (pre-1900)

    • Mechanics (a.k.a. Newtonian Mechanics)

    • Electromagnetism, Thermodynamics, Optics (19ᵗʰ cent.)

  • Modern Physics (≈1900-present)

    • Special & General Relativity – high-speed, gravitation-related phenomena

    • Quantum Mechanics – atomic/subatomic domain → modern electronics, lasers, MRI, etc.

Special Relativity Highlights

  • Correctly describes motion near light speed

  • Redefines space, time, energy; c is ultimate speed limit

  • Shows equivalence E = mc^2 (mass-energy relation)

Objectives of Physics

  1. Identify a limited set of fundamental laws governing natural phenomena

  2. Use these laws to predict outcomes of future experiments (theory building)

  3. Express laws mathematically – mathematics forms the bridge between theory & experiment

  • Theory and experiment must complement; discrepancies lead to theory refinement (e.g., Newtonian mechanics → relativistic mechanics)

Units, Dimensions & Measurements (Lecture 1)

Measurement & Standards

  • Measurement = comparison of an unknown physical quantity with a unit (known fixed quantity)

  • Desirable properties of a standard unit:

    • Well-defined, invariant with time/place/conditions, reproducible, convenient size, internationally accepted, easy access

Fundamental vs Derived Quantities

  • Fundamental (mechanics): Length L, Mass M, Time T

  • Derived: formed from fundamental (Area = L^2, Speed = L/T, etc.)

  • Physical quantity expressed as: Q = n\,u (numerical value × unit)

International System of Units (SI)

  • Adopted 1960; seven base units (relevant 3 for mechanics plus ampere, kelvin, mole, candela)

  • Fundamental units used here:

    • Length: meter (m) – defined via distance light travels in 1/299{,}792{,}458\ \text{s}

    • Mass: kilogram (kg) – defined via Planck constant (historically by platinum-iridium cylinder)

    • Time: second (s) – 9 192 631 770 periods of Cs-133 radiation

Other Systems of Units

  • CGS: cm-g-s

  • FPS (British): foot-pound-second (mass sometimes slug)

  • MKS: meter-kilogram-second → predecessor of SI

  • US Customary still used domestically (ft, slug, s)

Supplementary / Angular Units

  • Plane angle: radian (rad) – arc length = radius

  • Solid angle: steradian (sr) – area = r^2 on sphere

Handy Length & Mass Conversions

  • 1\,\mu\text{m} = 10^{-6}\,\text{m} 1\,\text{Å} = 10^{-10}\,\text{m} 1\,\text{fm} = 10^{-15}\,\text{m}

  • 1\,\text{ly} = 9.46\times10^{15}\,\text{m} 1\,\text{pc} = 3.26\,\text{ly}

  • 1\,\text{kg} = 2.2046\,\text{lb}; 1\,\text{lb} = 453.6\,\text{g}

  • Metric prefixes for smaller sub-grams (cg, dg, mg); larger (quintal = 100 kg; metric ton = 1000 kg)

Scientific Notation

  • Compactly expresses very large/small numbers as a \times 10^{b}, with one non-zero digit left of decimal

  • Conversion steps:

    1. Locate decimal (implied if absent)

    2. Shift decimal until one non-zero digit left

    3. Count shifts → exponent b (+ if left, − if right)

    4. Write a \times 10^{b}

  • Example: 1 500 000 mi → 1.5 \times 10^{6}\ \text{mi}

Uncertainties & Deviations in Measurements (Lecture 2)

Concept of Uncertainty

  • Every measurement has doubt; need two numbers:

    1. Width of margin/interval

    2. Confidence level (probability true value lies inside)

  • Importance: Calibration certificates, pass/fail tests, tolerance checks

Standard Deviation (σ)

  • Statistical dispersion measure for repeated readings around mean \bar{x}

  • ~68 % of readings within ±1σ, ~95 % within ±2σ (for normal-like distributions)

Errors vs Uncertainties

  • Error = Measured − True (difference)

  • Uncertainty = doubt due to unknown errors

  • Known errors are corrected; unknown contribute to uncertainty

  • Error cannot be eliminated but can be minimized

Sources of Error (4)

  1. Instrumental – faulty, broken, mis-calibrated equipment

  2. Procedural – inappropriate methods, inconsistent procedures

  3. Environmental – temperature, humidity, vibrations, etc.

  4. Human – negligence, estimation, transcription mistakes

    • Sub-types: Transcriptional vs Estimation errors

Types of Error (3 major)

  • Random Error – unpredictable scatter around mean; reduce by multiple trials and averaging
    \bar{X} = \tfrac{\sum X}{n}

  • Systematic Error – consistent bias affecting all readings equally; reduce via comparison/calibration & applying correction factor

  • Gross Error – blunders & oversight; reduce by careful work, multiple experimenters, averaging

Error Calculations

  • Absolute Error: |\text{Measured} - \text{True}|

  • Percentage Error: \dfrac{|\text{Measured} - \text{True}|}{\text{True}} \times 100\%

  • Relative Error: \dfrac{|\text{Measured} - \text{True}|}{\text{True}} (dimensionless)

Accuracy vs Precision

  • Accuracy – closeness to true value

    • Types: Point Accuracy, % of Scale Range, % of True Value (typically ±0.5 %)

  • Precision – repeatability / reproducibility of readings; independent of accuracy

    • Repeatability (same operator, short time) vs Reproducibility (different operators/instruments, longer time)

  • Dartboard metaphor:

    • (A) accurate & precise, (B) precise not accurate, (C) accurate not precise, (D) neither

Uncertainties of Derived Quantities (Lecture 3)

Derived Quantities & SI Derived Units

  • Examples with units:

    • Area m^2, Volume m^3, Speed m\,s^{-1}, Acceleration m\,s^{-2}

    • Density kg\,m^{-3}, Force Newton kg\,m\,s^{-2}

    • Pressure Pascal kg\,m^{-1}\,s^{-2}, Energy Joule kg\,m^{2}\,s^{-2}

    • Electrical: Volt kg\,m^{2}\,s^{-3}\,A^{-1}, Ohm kg\,m^{2}\,s^{-3}\,A^{-2}, Coulomb A\,s

Best Estimate ± Uncertainty Format

  • Report measurements as:
    \text{Measurement} = \text{Best Estimate} \pm \text{Uncertainty}

  • Rules:

    • Round uncertainty to 1 significant figure (2 if leading digit = 1)

    • Round best estimate to same decimal place as uncertainty

    • Example: WRONG 52.3 cm ± 4.1 cm ➔ CORRECT 52.3 cm ± 4 cm

Propagating Uncertainties (Four Operations)

Given X = 5 \pm 0.2\,\text{cm}, Y = 4.5 \pm 0.3\,\text{cm}

  1. Addition: X + Y = 9.5 \pm 0.5\,\text{cm} (max–avg or avg–min)

  2. Subtraction: X - Y = 0.5 \pm 0.1\,\text{cm}

  3. Multiplication: XY = 22.5 \pm 4.8\,\text{cm}^2 (larger of two deviation calculations)

  4. Division: X/Y = 1.11 \pm 0.12 (use extreme quotients then half-range)

Percentage Method for Derived Units

Rules:

  1. Convert absolute uncertainties to percentage uncertainties:
    \%\,u = \dfrac{\Delta q}{q} \times 100\%

  2. Add percentage uncertainties to get total; apply to calculated value
    Example – bench top area:

  • Length = 2.25 ± 0.01 m (0.44 %)

  • Width = 0.75 ± 0.01 m (1.33 %)

  • Area = 1.7 m² ± (0.44 + 1.33 ≈ 2 %) → 1.7 m² ± 2 %

Practice Problem (density of crystal)

  • Mass = 3.52 ± 0.01 g (0.28 %)

  • Volume = 1.34 ± 0.02 mL (1.49 %)

  • Density ≈ \dfrac{3.52}{1.34} = 2.63\,\text{g/mL}

  • Total % uncertainty ≈ 1.77 % → absolute uncertainty ≈ 0.05 g/mL
    \rho = 2.63 \pm 0.05\,\text{g/mL}

  • Quartz range 2.635–2.660 g/mL → Possible but at lower bound; additional identification needed

Graphical Analysis (brief preview – Module 1 continued)

  • Linear fitting & transforming functional dependencies to a straight-line form for easier parameter extraction (details in upcoming sessions)

Ethical, Practical & Real-World Relevance

  • Precise measurement underpins engineering safety (bridges, buildings, circuits)

  • Calibration & uncertainty reporting are ethical obligations in professional practice to avoid catastrophic failures

  • Understanding accuracy vs precision guides quality control and instrumentation design

Connections to Later Modules

  • Vector analysis, kinematics, free-fall, etc., will rely heavily on correct unit handling, vector notation, and uncertainty propagation techniques learned here.

End-of-Module Checklist

  • Able to:

    • State definitions of physics, classical vs modern branches

    • Identify fundamental SI units & convert between unit systems

    • Express and manipulate numbers in scientific notation

    • Distinguish error types & compute standard deviation

    • Propagate uncertainties through algebraic operations

    • Report results using correct significant figures and uncertainty conventions