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