Physical Quantities and Measurements - Comprehensive Notes

Physical Quantities and Measurements

Purpose: introduce what physical quantities are, how we measure them, and how to handle units, notation, and errors in measurements.
Foundational idea: measurements quantify properties of materials or systems (e.g., length, mass, time, pressure, temperature, current).
Key outcomes: convert units, express in scientific notation, distinguish accuracy vs precision, and distinguish random vs systematic errors.

SI Units, Fundamental vs Derived Quantities, and Unit Definitions

What is a physical quantity?
- A property of a material or system that can be quantified.
- Examples: length, mass, time, pressure, temperature, electric current.
- Typical statements: e.g., the mass of a person is 65 kg; the length of a table is 3 m; the area of a hall is in m^2; the room temperature is 300 K.
Types of physical quantities
- Fundamental quantities (base quantities): do not depend on other quantities for their measurements.
- Examples: Mass, Time, Length, Temperature (and others depending on the system; the slide lists these as fundamental quantities).
- Derived quantities: depend on one or more fundamental quantities for their measurements.
- Examples: Area (L^2), Volume (L^3), Speed (L T^{-1}), Force (M L T^{-2}).
SI base system (SI units)
- SI is based on the metric system with seven fundamental units.
- Fundamental units (base units):
- Length: metre, symbol $ext{m}$ , dimension $L$
- Mass: kilogram, symbol $ext{kg}$ , dimension $M$
- Time: second, symbol $ext{s}$ , dimension $T$
- Electric current: ampere, symbol $ext{A}$ , dimension $I$
- Thermodynamic temperature: kelvin, symbol $ext{K}$ , dimension $heta$ (often written as $ext{T}$ in some tables but here linked to temperature)
- Amount of substance: mole, symbol $ext{mol}$ , dimension $N$ or amount of substance
- Luminous intensity: candela, symbol $ext{cd}$ , dimension corresponds to luminous intensity
- Derived quantities use these base units (e.g., speed is m s^{-1}).
Units and symbols
- A unit is a standard for measurement (e.g., metre, foot, inch; kilogram, pound; second, minute, hour; Fahrenheit, Kelvin).
- Some notes:
- Symbols for quantities are not mandatory (you may write the quantity name with units).
Characteristics of a good unit
- Well-defined, suitable size, reproducible, invariable, indestructible, internationally acceptable.

Fundamental and Derived Quantities (Three core quantities noted)

Three fundamental quantities highlighted:
- Mass
- Length
- Time
The kilogram (kg) is the fundamental unit of mass.
- Conversion examples:
- 1 kg = 1000 g
- 1 g = 1000 mg
- 1 mg = 1000 µg
The second (s) is the fundamental unit of time.
- Time conversions:
- 1 day = 24 h
- 1 h = 60 min
- 1 min = 60 s

SI Prefixes and Units for Measurement

Common prefixes (power-of-ten multipliers):
- tera (T) = $10^{12}$
- giga (G) = $10^{9}$
- mega (M) = $10^{6}$
- kilo (k) = $10^{3}$
- deci (d) = $10^{-1}$
- centi (c) = $10^{-2}$
- milli (m) = $10^{-3}$
- micro (µ) = $10^{-6}$
- nano (n) = $10^{-9}$
- pico (p) = $10^{-12}$

Unit Abbreviations, Conversions, and Multipliers

The standard unit system (SI) uses seven base units as listed above; other units are derived.
Examples of measurement units across domains:
- Length: metre (m)
- Mass: kilogram (kg)
- Time: second (s)
- Temperature: kelvin (K)
- Electric current: ampere (A)
- Luminous intensity: candela (cd)
- Amount of substance: mole (mol)
Common non-SI units exist (e.g., foot, inch, pound), but SI base and derived units are preferred for consistency in science.

Scientific Notation

Purpose: express very large or very small numbers concisely.
Form: a × 10^n, where a is a decimal number with typically three significant figures and n is an integer.
How to form:
- Move the decimal point to place it after the first nonzero digit.
- The number of places the decimal is moved becomes the exponent n (positive if moved left, negative if moved right).
Examples from notes:
- 0.0000000005385 → 5.385 × 10^{-10}
- 5,125,487,000,000 → 5.125487 × 10^{12}
Three significant figures convention:
- In most cases, scientific notation is expressed with three significant figures, e.g., 5.39 × 10^{-10} or 5.13 × 10^{12}.
Addition/Subtraction vs Multiplication/Division in scientific notation
- Addition/Subtraction: align exponents; convert to the same exponent before adding/subtracting.
- Multiplication/Division: multiply/divide the decimal parts, and add/subtract exponents accordingly.
Formatting rules for decimal placement and exponent adjustments when the result’s decimal part is not properly placed are described through examples in the material.

Significant Figures (Sig Figs)

Definition: The digits in a number that carry meaning contributing to its precision; exclude certain zeros.
Guidelines:
- All non-zero digits (1–9) are significant.
- Zeros between nonzero digits are significant.
- Zeros to the right of a decimal point and to the left of a nonzero digit are significant (trailing zeros after a decimal point are significant).
- Zeros to the left of a number (leading zeros) are not significant.
- Trailing zeros without a decimal point are not necessarily significant; adding a decimal point (e.g., 300.) makes them significant.
- Trailing zeros to the right of a decimal point and to the right of a nonzero digit are significant (e.g., 0.00250 has 3 sig figs).
Sample counts (from notes):
- 0.0678 m → 3 sig figs
- 30.7 °C → 3 sig figs
- 300,000,000 m/s → 1 sig fig
- 20.00 cm → 4 sig figs
- 1.860 × 10^5 mi/s → 4 sig figs
1. mm → 2 sig figs
- 200 m → 1 sig fig
- 400 km → 1 sig fig
- 0.005430 m → 4 sig figs
- 250.0 feet → 4 sig figs
Practical takeaway: use sig figs to reflect measurement precision; when performing calculations, carry extra figures and round only at the end to the correct precision.

Practical Notes on Measurement Practice

Accuracy vs. precision
- Accuracy: closeness of a measurement to the true value.
- Precision: reproducibility of measurements (how closely they agree with each other).
- A measurement can be accurate without being precise, precise without being accurate, both, or neither.
- Random errors affect precision; systematic errors affect accuracy.
Visual representations (conceptual, not shown here):
- A results plot can show accuracy (closeness to true value) and precision (spread of measurements).
Measurement culture and implications
- Understanding errors improves reliability, reproducibility, and interpretability of data.
- Ethical and practical implications: reporting uncertainty is essential for scientific integrity; comparisons across experiments rely on consistent uncertainties and units.

Connections to Foundational Principles and Real-World Relevance

Consistent units and dimensional analysis are essential for comparing measurements, performing calculations, and communicating results in science and engineering.
Scientific notation is fundamental for handling extremely large or small numbers typical in physics and chemistry (e.g., atomic scales, astrophysical distances).
Significant figures reflect measurement precision and inform how much confidence to assign to reported numbers.
Understanding different error sources (systematic vs random) guides the design of experiments and interpretation of data.
Variance and SEM provide quantitative frameworks for estimating the precision of a mean value from repeated measurements.
In real-world practice, precise and accurate measurements underpin quality control, safety, and scientific conclusions.

Quick Reference: Key Formulas

Base quantities and units
- Length: $L = \text{m}$
- Mass: $M = \text{kg}$
- Time: $T = \text{s}$
Derived quantities (examples)
- Area: $A = L \times W$
- Volume: $V = L \times W \times H$
- Speed: $v = \frac{\text{distance}}{\text{time}}$
- Force: $F = m a$
Variance and dispersion