Chapter 1: Measurement - Physics Daily

Introduction to Measurement

Contextual Reflection

Consider the challenge of cooking without measuring ingredients or constructing a building without measuring materials. Scientific investigation likewise depends upon reliable, standardised measurement. Every accepted physical law has emerged from reproducible, quantitative observation.

Physics is fundamentally an experimental science. Progress arises from precise observation, controlled comparison, and quantitative record. From Galileo’s kinematics studies to precision timing in modern particle accelerators, refined measurement enables theoretical development and technological application.

Modern navigation illustrates this dependence: global positioning systems function only because relativistic time dilation, gravitational frequency shifts, and atomic clock behaviour are measured and corrected with sub‑nanosecond resolution.

Power of High-Resolution Measurement

The 2017 gravitational‑wave detection (LIGO) required resolving differential length changes smaller than 10⁻¹⁸ m—orders of magnitude below atomic dimensions—demonstrating the role of instrumentation fidelity in contemporary physics.

Chapter Learning Objectives

Physical Quantities: Distinguish scalar and vector forms; interpret operational definitions.
SI Units: Apply the base units and construct derived units coherently.
Dimensional Analysis: Validate relations and obtain structural dependence among variables.
Precision & Errors: Characterise, combine, and report measurement uncertainty rigorously.

Advanced Perspective

Measurement at quantum and relativistic scales introduces intrinsic limits (e.g., Heisenberg uncertainty) distinct from instrumental limitations. These fundamental bounds inform experimental design and data interpretation in modern research domains.

Representative Application Thresholds

Satellite Navigation: Timing stability ~10⁻⁸ s
Magnetic Resonance Imaging: Spatial resolution ~10⁻⁴ m
Semiconductor Fabrication: Feature control ~10⁻⁹ m
Gravitational Wave Interferometry: Displacement sensitivity ~10⁻¹⁸ m
Precision Spectroscopy: Frequency determination at parts per 10⁹ or better

1.1 Physical Quantities, Standards, and Units

Definition of a Physical Quantity

A physical quantity is any measurable property of matter or energy that can be used to explain physical phenomena in a quantitative way.

Components of a Physical Quantity

For a physical quantity to exist, it must have two essential parts:

Numerical part (n): The numeric value (pure number)
Unit part (u): An established standard reference part

Physical Quantity = Numerical part × Unit part

Fundamental Relationship: n ∝ 1/u

The numerical part is inversely proportional to the unit part:

n₁u₁ = n₂u₂ = constant

This formula is fundamental for unit conversions!

Example: Representation of Length

Numerical part (n)	Unit part (u)
5	km
5 × 10³ = 5000	m
5 × 10⁵ = 500,000	cm
5 × 10⁶ = 5,000,000	mm

Notice: As the unit gets smaller, the numerical value gets larger!

Principal Categories

Scalar Quantities: Have magnitude only
Examples: mass (5 kg), time (10 s), temperature (25°C), energy (100 J)
Vector Quantities: Have both magnitude and direction
Examples: displacement (5 m north), velocity (10 m/s east), acceleration (9.8 m/s² downward)

Operational Definitions

Each physical quantity must have an operational definition - a precise description of how to measure it. This ensures anyone can reproduce the measurement anywhere in the world.

Example: Operational Definition of Velocity

Operational definition: "Change in position divided by time interval"

v = Δx/Δt

This tells us exactly how to measure velocity: measure the change in position, measure the time it took, then divide.

Characteristics of Good Physical Standards

For measurements to be meaningful worldwide, our standards must be:

                        Accessible: Can be reproduced in laboratories worldwide
Invariable: Unchanging over time and location
Precise: Allows measurement to desired accuracy
Reproducible: Independent measurements yield consistent results

                    

1.2 The International System of Units (SI)

The International System of Units (SI) provides a standardized framework for measurement that enables scientists worldwide to communicate effectively. This system is built on seven fundamental base units from which all other units are derived.

Base Units of SI

Physical Quantity	SI Base Unit	Symbol	Definition
Length	meter	m	Distance light travels in vacuum in 1/299,792,458 second
Mass	kilogram	kg	Mass of the international prototype kilogram
Time	second	s	Duration of 9,192,631,770 periods of cesium-133 radiation
Electric Current	ampere	A	Current producing force of 2×10⁻⁷ N/m between parallel wires
Temperature	kelvin	K	1/273.16 of thermodynamic temperature of water's triple point
Amount of Substance	mole	mol	Amount containing as many particles as atoms in 0.012 kg carbon-12
Luminous Intensity	candela	cd	Luminous intensity of 1/683 watt per steradian at 540×10¹² Hz

Derived Units

Many physical quantities are expressed using derived units, which are combinations of base units. Some common derived units have special names:

Quantity	Unit Name	Symbol	Base Unit Expression
Force	newton	N	kg⋅m⋅s⁻²
Energy	joule	J	kg⋅m²⋅s⁻²
Power	watt	W	kg⋅m²⋅s⁻³
Pressure	pascal	Pa	kg⋅m⁻¹⋅s⁻²
Electric Charge	coulomb	C	A⋅s
Voltage	volt	V	kg⋅m²⋅s⁻³⋅A⁻¹

SI Prefixes

SI prefixes allow concise representation of very large and very small numerical values. The most frequently used ranges in typical physics problems are highlighted.

Negative Powers (Sub-multiples)
Prefix	Symbol	Multiplier	Power	Typical Example
yocto	y	0.000000000000000000000001	10⁻²⁴	Quantum events
zepto	z	0.000000000000000000001	10⁻²¹	High-energy collisions
atto	a	0.000000000000000001	10⁻¹⁸	Attosecond lasers
femto	f	0.000000000000001	10⁻¹⁵	Nuclear radius (1 fm)
pico	p	0.000000000001	10⁻¹²	Charge (pC), atomic spacings
nano	n	0.000000001	10⁻⁹	Wavelength (nm)
micro	μ	0.000001	10⁻⁶	Cell size (μm)
milli	m	0.001	10⁻³	Length (mm), current (mA)
centi	c	0.01	10⁻²	Everyday length (cm)
deci	d	0.1	10⁻¹	Rarely used

Positive Powers (Multiples)
Prefix	Symbol	Multiplier	Power	Typical Example
deca	da	10	10¹	Very rare
hecto	h	100	10²	Hectare (area)
kilo	k	1,000	10³	Mass (kg), distance (km)
mega	M	1,000,000	10⁶	Frequency (MHz)
giga	G	1,000,000,000	10⁹	Data (GB), energy
tera	T	1,000,000,000,000	10¹²	Power (TW)
peta	P	1,000,000,000,000,000	10¹⁵	Astro data
exa	E	10¹⁸	10¹⁸	Big data scale
zetta	Z	10²¹	10²¹	Cosmic energy
yotta	Y	10²⁴	10²⁴	Stellar mass est.

Memory Aids

Small (10⁻²⁴ → 10⁻¹): Young Zebras Are Fearfully Pale, Notably Miniature Creatures Drink
Large (10¹ → 10²⁴): Do Hungry Kids Meticulously Gobble Tremendous Portions Every Zestful Year

Usage emphasis: nano (light, semiconductor physics), micro (microscopy), milli (laboratory scales), kilo (macroscopic distances), mega/giga (frequency, large data throughput).

Order of Magnitude

Definition: The order of magnitude of a number N is the power of 10, say x, such that 0.5 ≤ N/10ˣ < 5. This gives a fast approximate scale of size.

Examples

0.6 = 0.6 × 10⁰ → Order 0
49 = 4.9 × 10¹ → Order 1
51 = 0.51 × 10² → Order 2
2040 = 2.04 × 10³ → Order 3
9163 = 0.9163 × 10⁴ → Order 4

Frequently Used Non-SI Units

🌌 Astronomical Distances

Light year (ly): 9.46 × 10¹⁵ m (distance light travels in 1 year)
Astronomical Unit (AU): 1.496 × 10¹¹ m ≈ 1.5 × 10¹¹ m (mean Earth–Sun distance)
Parsec (pc): 3.08 × 10¹⁶ m = 3.26 ly (parallax of 1 arc‑second)
Hierarchy: 1 pc > 1 ly > 1 AU

🧪 Small Length Scales

Micron / micrometre (μm): 10⁻⁶ m
Nanometre (nm): 10⁻⁹ m
Angstrom (Å): 10⁻¹⁰ m (atomic spacing)
X‑ray unit (XU): 10⁻¹³ m (X‑ray wavelength scale)
Fermi (fm): 10⁻¹⁵ m (nuclear radius)

⚖️ Mass Units

Atomic mass unit (u): 1.67 × 10⁻²⁷ kg
Slug: 14.57 kg (imperial dynamics)
Quintal: 100 kg
Metric ton: 10³ kg
Solar mass (M☉): 2.0 × 10³⁰ kg

⏱️ Time Units

Mean solar day: 24 h
Sidereal month: 27.3 days
Lunar month: ≈ 29.5 days (phase cycle)
Mean solar year: 365.25 days
Shake: 10⁻⁸ s (nuclear physics)

Do not confuse: AU (astronomical unit) with Å (angstrom).

Exercises

Express 0.000035 meters using appropriate SI prefix notation.

Solution:

0.000035 m = 3.5 × 10⁻⁵ m

Since 10⁻⁵ = 10⁻⁶ × 10¹ = μ × 10, we can write:

3.5 × 10⁻⁵ m = 35 × 10⁻⁶ m = 35 μm

Alternatively: 3.5 × 10⁻⁵ m = 0.035 × 10⁻³ m = 0.035 mm

1.3 Dimensional Analysis

Dimensional analysis is a powerful tool for checking equations, converting units, and solving physics problems. It uses the principle that physical equations must be dimensionally consistent - both sides must have the same dimensions.

Definition of Dimensions

Definition: The dimension of any physical quantity is defined as the powers of the fundamental quantities in the product form.

[Physical Quantity] = [M]ˣ[L]ʸ[T]ᶻ...

Where x, y, z are called the dimensions (powers) of that quantity.

Example: Force

Force = mass × acceleration = (mass) × (length)/(time)²

[F] = [M][L][T⁻²] = [MLT⁻²]

This is called the dimensional equation of force.

[MLT⁻²] is called the dimensional formula of force.

Fundamental Dimensions

All physical quantities can be expressed in terms of fundamental dimensions:

Physical Quantity	Dimension Symbol	SI Base Unit
Length	[L]	meter (m)
Mass	[M]	kilogram (kg)
Time	[T]	second (s)
Electric Current	[I]	ampere (A)
Temperature	[Θ]	kelvin (K)
Amount of Substance	[N]	mole (mol)
Luminous Intensity	[J]	candela (cd)

Derived Dimensions

Common derived quantities and their dimensional expressions:

Physical Quantity	Dimensional Formula	SI Unit
Area	[L²]	m²
Volume	[L³]	m³
Velocity	[LT⁻¹]	m/s
Acceleration	[LT⁻²]	m/s²
Force	[MLT⁻²]	N (kg⋅m/s²)
Energy	[ML²T⁻²]	J (kg⋅m²/s²)
Power	[ML²T⁻³]	W (kg⋅m²/s³)
Pressure	[ML⁻¹T⁻²]	Pa (kg/(m⋅s²))

Selected Additional Dimensional Forms
                            Speed/Velocity:

                            [v] = [L]/[T] = [LT⁻¹] = [M⁰LT⁻¹]
                        
                            Acceleration:

                            [a] = [v]/[T] = [LT⁻²] = [M⁰LT⁻²]
                        
                            Work/Energy:

                            [W] = [F][s] = [MLT⁻²][L] = [ML²T⁻²]
                        
                            Power:

                            [P] = [W]/[T] = [ML²T⁻²]/[T] = [ML²T⁻³]
                        
                            Pressure:

                            [P] = [F]/[A] = [MLT⁻²]/[L²] = [ML⁻¹T⁻²]
                        
                            Potential Difference:

                            [V] = [W]/[Q] = [ML²T⁻²]/[AT] = [ML²T⁻³A⁻¹]
                        
                            Resistance:

                            [R] = [V]/[I] = [ML²T⁻³A⁻¹]/[A] = [ML²T⁻³A⁻²]
                        
                            Gravitational Constant:

                            From F = Gm₁m₂/r²

                            [G] = [M⁻¹L³T⁻²]

Comprehensive Dimensional Reference

The following consolidated tables summarise high‑frequency physical quantities. Users should prioritise foundational mechanical quantities (force, energy, power, pressure) before extending to electromagnetic and thermal constants.

Mechanics Core Quantities

Quantity	Symbol	Relation	Dimensions	Unit
Displacement / Length	x, l	—	[L]	m
Area	A	l·b	[L²]	m²
Volume	V	l·b·h	[L³]	m³
Density	ρ	m/V	[ML⁻³]	kg·m⁻³
Velocity / Speed	v	dx/dt	[LT⁻¹]	m·s⁻¹
Acceleration	a	dv/dt	[LT⁻²]	m·s⁻²
Momentum	p	mv	[MLT⁻¹]	kg·m·s⁻¹
Force / Weight	F, W	ma	[MLT⁻²]	N
Work / Energy	W, E	F·s	[ML²T⁻²]	J
Power	P	W/t	[ML²T⁻³]	W
Pressure / Stress	p, σ	F/A	[ML⁻¹T⁻²]	Pa
Surface tension	γ	F/l	[MT⁻²]	N·m⁻¹
Gravitational constant	G	Fr²/m²	[M⁻¹L³T⁻²]	N·m²·kg⁻²
Coefficient of friction	μ	F_f/N	[1]	—
Spring constant	k	F/x	[MT⁻²]	N·m⁻¹
Angular velocity	ω	dθ/dt	[T⁻¹]	rad·s⁻¹
Moment of inertia	I	Σmr²	[ML²]	kg·m²
Planck's constant	h	E/ν	[ML²T⁻¹]	J·s

Heat / Thermal

Quantity	Symbol	Relation	Dimensions	Unit
Temperature	T	—	[Θ]	K
Specific heat	c	Q=m c ΔT	[L²T⁻²Θ⁻¹]	J·kg⁻¹·K⁻¹
Latent heat	L	Q=mL	[L²T⁻²]	J·kg⁻¹
Entropy	S	Q/T	[ML²T⁻²Θ⁻¹]	J·K⁻¹
Thermal conductivity	k	Q= k A ΔT t / d	[MLT⁻³Θ⁻¹]	W·m⁻¹·K⁻¹
Stefan constant	σ	E=σT⁴	[MT⁻³Θ⁻⁴]	W·m⁻²·K⁻⁴

Light / Waves

Quantity	Symbol	Relation	Dimensions	Unit
Frequency	ν	1/T	[T⁻¹]	Hz
Wavelength	λ	v/ν	[L]	m
Wave speed	v	λν	[LT⁻¹]	m·s⁻¹
Intensity (radiative)	I	Power/Area	[MT⁻³]	W·m⁻²
Planck's constant	h	E=hν	[ML²T⁻¹]	J·s

Electricity & Magnetism

Quantity	Symbol	Relation	Dimensions	Unit
Charge	q	I t	[IT]	C
Potential (Voltage)	V	W/q	[ML²T⁻³I⁻¹]	V
Capacitance	C	q/V	[M⁻¹L⁻²T⁴I²]	F
Resistance	R	V/I	[ML²T⁻³I⁻²]	Ω
Resistivity	ρ	RA/l	[ML³T⁻³I⁻²]	Ω·m
Conductivity	σ	1/ρ	[M⁻¹L⁻³T³I²]	S·m⁻¹
Permittivity	ε₀	q²/ (F r²)	[M⁻¹L⁻³T⁴I²]	F·m⁻¹
Permeability	μ₀	F r /(I² l)	[MLT⁻²I⁻²]	H·m⁻¹
Magnetic flux	Φ	B A	[ML²T⁻²I⁻¹]	Wb
Inductance	L	Φ/I	[ML²T⁻²I⁻²]	H

Dimensionless Quantities

Some quantities have no dimensions:

Angle: [θ] = [arc length]/[radius] = [L]/[L] = [M⁰L⁰T⁰]
Trigonometric functions: [sin θ] = [cos θ] = [tan θ] = [M⁰L⁰T⁰]
Relative density: [density of substance]/[density of water] = 1
Strain: [ΔL/L] = [ΔA/A] = [ΔV/V] = [M⁰L⁰T⁰]

📌 Dimensional Clarifications Addendum

Density: ρ = m/V → [ρ] = [ML⁻³]
Gravitational Constant: From F = G m₁ m₂ / r² → [G] = [M⁻¹L³T⁻²]
General Unit Conversion: For quantity with dimensions M^αL^βT^γ: n₂ = n₁ (M₁/M₂)^α (L₁/L₂)^β (T₁/T₂)^γ
All Strain Types: Linear, area, and volume strain are ratios → dimensionless (unit 1)
Use Scientific Notation to make the intended significant figures explicit (e.g., 3.400 × 10³ vs 3400)

Unit Conversion Using Dimensional Analysis

The conversion factor method uses the principle that multiplying by a ratio equal to 1 doesn't change the value:

n₂ = n₁ × [M₁/M₂]ˣ × [L₁/L₂]ʸ × [T₁/T₂]ᶻ

Example: Converting 1 Newton to Dyne

Force has dimensions [MLT⁻²], so x=1, y=1, z=-2

SI system: n₁ = 1, M₁ = 1 kg, L₁ = 1 m, T₁ = 1 s

CGS system: n₂ = ?, M₂ = 1 g, L₂ = 1 cm, T₂ = 1 s

n₂ = 1 × [1 kg/1 g]¹ × [1 m/1 cm]¹ × [1 s/1 s]⁻²

n₂ = 1 × 10³ × 10² × 1 = 10⁵

Answer: 1 N = 10⁵ dyne

Example: Convert 75 mph to m/s

75 mph × (1609 m/1 mile) × (1 hr/3600 s) = 33.5 m/s

Checking Equation Validity - Homogeneity Principle

🎯 Homogeneity Principle

Definition: Two or more physical quantities can be added or subtracted only if they have the same dimensions.

This principle is used to check if physical equations are dimensionally correct.

Example: Verification of s = ut + ½at²

1st term: [s] = [L]

2nd term: [ut] = [LT⁻¹][T] = [L]

3rd term: [½at²] = [LT⁻²][T²] = [L]

Conclusion: All terms have dimension [L], so the equation is ✓ dimensionally correct

Counter-example: Invalid Relation s = ut + at³

1st term: [s] = [L]

2nd term: [ut] = [LT⁻¹][T] = [L]

3rd term: [at³] = [LT⁻²][T³] = [LT]

Conclusion: The third term has dimension [LT], not [L], so this equation is ✗ dimensionally incorrect

Important Limitation

Dimensionally correct equations need not be physically correct, but dimensionally incorrect equations are never physically correct.

Checking Equation Validity

Dimensional analysis can verify if an equation is physically meaningful by ensuring both sides have identical dimensions.

Representative Applications of Dimensional Analysis

🚀 Rocket Science
Verify thrust equations before multi-million dollar launches

💊 Drug Dosage
Ensure proper concentration units to prevent medical errors

🏗️ Engineering
Check structural calculations before construction

🌡️ Climate Models
Validate complex atmospheric equations

                    Guidelines for Effective Dimensional Analysis
                    Always check your work: If dimensions don't match, there's an error!
Use it for estimation: Get order-of-magnitude answers quickly
Derive new relationships: Combine quantities to find unknown formulas
Convert between systems: Imperial to metric, CGS to SI, etc.

                

Exercises

Find the dimensional formula for kinetic energy (KE = ½mv²).

Solution:

KE = ½mv²

Dimensions of mass: [M]

Dimensions of velocity: [LT⁻¹]

Dimensions of v²: [LT⁻¹]² = [L²T⁻²]

Therefore: [KE] = [M][L²T⁻²] = [ML²T⁻²]

This matches the dimensional formula for energy.

⚠️ Limitations of Dimensional Analysis

Dimensionally correct equations need not be physically correct, but dimensionally incorrect equations are never physically correct.
Pure numbers or numerical constants in physical equations cannot be determined by dimensional analysis (like π, ½, 2, etc.).
Only equations in product form can be derived by dimensional analysis.
Equations containing trigonometric, logarithmic, or exponential functions cannot be derived by dimensional analysis.
Dimensional analysis doesn't indicate whether a quantity is vector or scalar.

Important Points to Remember (Quick Scan)

Frequency, angular frequency, angular velocity, velocity gradient → all [T⁻¹].
Dimensionless: angle, solid angle, trigonometric ratios, strain (all), Poisson ratio, relative density, relative permittivity, refractive index, relative permeability, coefficient of friction.
[L²T⁻²] group: (speed)², gravitational potential, latent heat.
[ML²T⁻²] group: work, energy, torque, heat.
[MLT⁻¹] group: momentum, impulse.
Acceleration and gravitational field intensity share [LT⁻²].
Force, weight, thrust → [MLT⁻²].
Entropy, gas constant R, Boltzmann constant k, heat capacity share same M,L,T dimensions (differ by Θ power).
Light year, radius of gyration, wavelength all [L].
Surface tension and spring constant share [MT⁻²].
Planck's constant and angular momentum share [ML²T⁻¹].
Rydberg constant & propagation constant: [L⁻¹].
Pressure, stress, elastic moduli, energy density → [ML⁻¹T⁻²].

1.4 Precision, Accuracy, and Error Analysis

Understanding measurement uncertainty is crucial in physics. We distinguish between precision (repeatability) and accuracy (closeness to truth), identify sources of error, and apply rules for combining uncertainties.

Instrument Precision Limits

The precision (least count) of a measuring instrument determines the smallest change it can reliably detect.

Instrument	Least Count
Linear Scale (ruler)	1 mm = 0.1 cm = 0.001 m
Vernier Caliper	0.1 mm = 0.01 cm = 0.0001 m
Micrometer Screw Gauge	0.01 mm = 0.001 cm = 0.00001 m

Interpretation: The micrometer provides the most precise measurement; the ruler the least. A smaller least count means finer resolution.

Precision vs. Accuracy

Precision

How closely repeated measurements agree with each other.

Example: Readings: 5.21, 5.22, 5.21, 5.22 cm → High precision.

Accuracy

How close a measurement is to the true value.

Example: True length = 5.80 cm but repeated readings cluster near 5.22 cm → Precise but inaccurate (systematic error).

Key Point: Precision (repeatability) and accuracy (closeness to true value) are distinct; systematic errors impair accuracy while random fluctuations limit precision.

Precision vs. Accuracy

Concept	Definition	Analogy
Precision	How close repeated measurements are to each other	Tight grouping of arrows on target
Accuracy	How close a measurement is to the true value	Arrows hitting the bullseye

Types of Errors

1. Systematic Errors

Definition: Consistent, repeatable errors that shift all measurements in the same direction
Causes: Faulty instruments, environmental factors, procedural mistakes
Effect: Reduces accuracy but not precision
Examples: Uncalibrated scale, parallax error, thermal expansion

2. Random Errors

Definition: Unpredictable variations that fluctuate randomly around the true value
Causes: Human limitations, electrical noise, environmental fluctuations
Effect: Reduces precision
Examples: Reading variations, timing uncertainties, atmospheric turbulence

Significant Figures

Significant figures (sig figs) are the reliably known digits in a measured quantity, plus the first uncertain (estimated) digit.

Rules & Examples

All non-zero digits are significant.
Example: 2345 → 4 sig figs
Leading zeros are never significant.
Example: 0.0234 → 3 sig figs
Zeros between non-zero digits are significant.
Example: 2003 → 4 sig figs
In numbers < 1, zeros after decimal but before first non-zero are not significant.
Example: 0.00250 → 3 sig figs
Any zero between non-zero digits (even after decimal) is significant.
Example: 0.2035 → 4 sig figs
In numbers ≥ 1 with a decimal, trailing zeros are significant.
Example: 1.200 → 4 sig figs
Trailing zeros in whole numbers without a decimal may be ambiguous.
Example: 4500 → 2–4 sig figs (write 4.500×10³ to show 4)
Digits in the power of 10 (scientific notation) are not counted.
Examples: 3 × 10⁸ → 1 sig fig; 6.67 × 10⁻¹¹ → 3 sig figs

Best Practice: Use scientific notation to make significant figures explicit; e.g., 2.300 × 10⁴ conveys four significant figures whereas 23000 is ambiguous.

Operations with Significant Figures:

Addition/Subtraction: Round to the least precise decimal place
Multiplication/Division: Round to the fewest significant figures

Fundamental Error (Uncertainty) Definitions

Error (or uncertainty) is the difference between the measured value and the accepted (true) value. We characterize how errors combine using absolute, relative (fractional), and percentage forms.

Notation: A measurement written as (7.52 ± 0.03) cm means 7.52 cm is the best estimate and 0.03 cm is the absolute uncertainty.

Absolute Error (Δy)

Represents the uncertainty in the same units as the quantity.
For sums/differences: Δy = Δa + Δb (worst-case combination).
For a power: y = aⁿ → Δy = n aⁿ⁻¹ Δa (approx) or fractional form below.

Relative (Fractional) Error

Fractional uncertainty: Δy / y

For products/quotients: Δy / y = (Δa / a) + (Δb / b) (worst-case)
For powers: y = aⁿ → Δy / y = n (Δa / a)
General: y = aᵐ bⁿ → Δy / y = m (Δa / a) + n (Δb / b)

Percentage Error

(Δy / y) × 100%

Example: If radius r has 0.3% error, area A = 4πr² has 2 × 0.3% = 0.6% error.

Error Propagation (Summary Table)

When combining measurements, uncertainties propagate according to these common patterns:

Operation	Formula	Error Propagation
Addition	z = x + y	δz = √(δx² + δy²)
Subtraction	z = x - y	δz = √(δx² + δy²)
Multiplication	z = xy	δz/z = √((δx/x)² + (δy/y)²)
Division	z = x/y	δz/z = √((δx/x)² + (δy/y)²)

Exercises

Calculate: 12.34 + 5.6 + 0.789. Express with proper significant figures.

Solution:

12.34 (2 decimal places)

+ 5.6 (1 decimal place) ← least precise

+ 0.789 (3 decimal places)

= 18.729

Round to 1 decimal place: 18.7

Calculate: 2.5 × 3.14159. Express with proper significant figures.

Solution:

2.5 has 2 significant figures ← fewest

3.14159 has 6 significant figures

2.5 × 3.14159 = 7.853975

Round to 2 significant figures: 7.9

A measurement gives x = 15.2 ± 0.3 cm and y = 8.7 ± 0.2 cm. Find z = x + y with uncertainty.

Solution:

z = x + y = 15.2 + 8.7 = 23.9 cm

For addition: δz = √(δx² + δy²)

δz = √(0.3² + 0.2²) = √(0.09 + 0.04) = √0.13 = 0.36 cm

Therefore: z = 23.9 ± 0.4 cm

How many significant figures are in: (a) 0.00234 (b) 1.200 (c) 3400?

Solutions:

(a) 0.00234: Leading zeros don't count → 3 significant figures

(b) 1.200: Trailing zeros after decimal count → 4 significant figures

To clarify (c), write as 3.4 × 10³ (2 sig figs) or 3.400 × 10³ (4 sig figs)

A student measured the radius of a sphere with a percentage error of 0.3%. Find the percentage error in its surface area.

Solution:

Surface area: A = 4πr²

Percentage error propagates with powers: ΔA/A × 100% = 2 (Δr/r × 100%)

= 2 × 0.3% = 0.6%

A block has dimensions (8 ± 0.4) cm by (6 ± 0.2) cm. Find the area with its absolute uncertainty.

Solution:

A = l × b = 8 × 6 = 48 cm²

Fractional uncertainty (worst-case): ΔA/A = Δl/l + Δb/b = 0.4/8 + 0.2/6

= 0.05 + 0.0333 = 0.0833

ΔA = 0.0833 × 48 ≈ 4 cm²

A = (48 ± 4) cm² (Range: 44 cm² to 52 cm²)

Mass and velocity of a body have percentage errors of 3% and 2% respectively. Find the percentage error in kinetic energy.

Solution:

K.E. = ½ m v²

Δ(K.E.)/K.E. × 100% = (Δm/m × 100%) + 2(Δv/v × 100%)

= 3% + 2 × 2% = 7%

Percentage error in kinetic energy = 7%

⚡ Tricks & Exam Strategies (Consolidated)

This section gathers only time-saving techniques, pattern recognition, and exam‑oriented heuristics. All conceptual theory has already been integrated into the core sections above to avoid duplication.

🚀 Lightning-Fast Dimensional Shortcuts

Energy Relationships (Direct Forms)

Kinetic: E = p²/(2m) → p²/m has energy dimensions.
Inductance: E = ½ L I² → [L] = [ML²T⁻²A⁻²].
Capacitance: E = ½ C V² → [C] = [Q²/E] = [M⁻¹L⁻²T⁴A²].

Force-Derived Quantity Patterns

Surface Tension = F/Length → [MT⁻²]
Pressure (or Energy Density) = F/Area → [ML⁻¹T⁻²]
Spring Constant = F/Length → [MT⁻²]

If two unfamiliar options share the same base form F divided by some geometry, reduce power of L quickly to decide.

🏃 Rapid Speed Recognition

√(Pressure / Density) → Acoustic speed
√(Tension / Linear density) → Wave on string
√(GM / r) → Orbital / circular speed
1 / √(μ₀ ε₀) → Speed of light

🎯 Unit Conversion Efficiency

Method 1: Power Counting Template

n₂ = n₁ (M₁/M₂)^a (L₁/L₂)^b (T₁/T₂)^c for dimensions MᵃLᵇTᶜ (extend with I, Θ if present).

Example: 1 N to dyne → [MLT⁻²]; (kg→g)=10³, (m→cm)=10² ⇒ 10⁵ dyne.

Method 2: High-Value Memorised Conversions

Energy:
1 J = 10⁷ erg
1 eV = 1.6×10⁻¹⁹ J
1 cal = 4.18 J

Force:
1 N = 10⁵ dyne
1 kgf ≈ 9.8 N

Pressure:
1 atm ≈ 10⁵ Pa
1 bar = 10⁵ Pa
1 Pa = 10 dyne/cm²

🧠 Dimension Families (Pattern Recognition)

[MT⁻²]

Surface tension
Spring constant
Force/length forms

[ML²T⁻²]

Energy / Work
Torque
Angular momentum ÷ time

[T⁻¹]

Frequency
Angular frequency
Velocity gradient

⚠️ Classic Traps

Prefix Case Sensitivity

M (mega, 10⁶) vs m (milli, 10⁻³). One capital letter changes factor by 10⁹!

Dimension ≠ Unit

Always separate: Energy has dimension [ML²T⁻²], unit Joule.

Angles Are Dimensionless

θ, sinθ, tanθ, solid angle Ω → all [1]. Never introduce L.

🏆 Pro Techniques

1. Eliminate by Dimensions First

Check each option’s M-L-T powers before algebra. Often leaves 1–2 survivors fast.

2. Limiting Cases

If l → 0 in pendulum, T must → 0 (√(l/g)). Reject forms that diverge incorrectly.

3. Symmetry & Independence

Pendulum period independent of mass ⇒ discard any option containing m.

📝 Rapid Revision Checklist

[Force] [MLT⁻²], [Energy] [ML²T⁻²], [Power] [ML²T⁻³]
[Pressure] & Energy density both [ML⁻¹T⁻²]
All strains & angles dimensionless
n₁u₁ = n₂u₂ for conversions
1 N = 10⁵ dyne, 1 J = 10⁷ erg

⚡ Speed Checks Framework

Dimensions consistent?
Units plausible?
Limits sensible?
Order of magnitude OK?

🏅 3-2-1 Rule

3 seconds: identify target quantity; 2 methods: have a backup; 1 final physical sense check.

Chapter 1 Assessment

Test your understanding of measurement principles with this comprehensive assessment. These problems integrate concepts from all sections of this chapter.

🎯 Competitive Exam Style Questions

SI unit of surface tension is: (a) N·s·m⁻¹ (b) J·s⁻¹ (c) N·m⁻² (d) N·m⁻¹

Quick Solution:

Surface tension = Force/Length

[Surface tension] = [MLT⁻²]/[L] = [MT⁻²]

SI unit = N/m = N·m⁻¹

Answer: (d) N·m⁻¹

Trick: Remember the formula immediately - don't derive from stress!

Dimensional analysis of (speed)ⁿ = [pressure/density]^(1/2) gives n = ?

Solution:

[LT⁻¹]ⁿ = ([ML⁻¹T⁻²]/[ML⁻³])^(1/2)

[LT⁻¹]ⁿ = ([L²T⁻²])^(1/2) = [LT⁻¹]

Comparing: n = 1

Physics insight: This is the formula for sound speed in a fluid!

The self-inductance L has dimensions: (a) [ML²T⁻¹A⁻¹] (b) [ML²T⁻¹A⁻²] (c) [ML²T⁻²A⁻¹] (d) [ML²T⁻²A⁻²]

Time-saving method:

Energy = ½LI²

[L] = [Energy]/[I²] = [ML²T⁻²]/[A²] = [ML²T⁻²A⁻²]

Answer: (d)

Pro tip: Use energy formulas for quick dimensional analysis!

60 J/min in a system with units 100g, 100cm, 1min equals how many new units?

Solution:

Power: [ML²T⁻³], so a=1, b=2, c=-3

60 J/min = 1 J/s = 1 W

n₂ = 1 × (1kg/100g)¹ × (1m/100cm)² × (1s/1min)⁻³

= 10 × 1 × (60)³ = 2.16 × 10⁶ new units

Answer: 2.16 × 10⁶

If p is momentum, then p²/m has dimensions of: (a) power (b) impulse (c) force (d) acceleration (e) energy

Super quick method:

Remember: E = p²/(2m) (kinetic energy formula)

So p²/m has dimensions of energy

Answer: (e) energy

Time saved: 2 seconds vs 30 seconds of calculation!

Comprehensive Problems

A student measures the length of a table five times and gets: 1.52 m, 1.49 m, 1.53 m, 1.51 m, 1.50 m. The true length is 1.65 m. Comment on the precision and accuracy of these measurements.

Solution:

Analysis of Precision:

Range = 1.53 - 1.49 = 0.04 m

The measurements are clustered closely together (small range), indicating high precision.

Analysis of Accuracy:

Average = (1.52 + 1.49 + 1.53 + 1.51 + 1.50)/5 = 1.51 m

Error = |1.51 - 1.65| = 0.14 m = 14 cm

This is a significant deviation from the true value, indicating low accuracy.

Conclusion: The measurements show high precision but low accuracy, suggesting a systematic error.

Express the following in proper scientific notation with appropriate significant figures: The speed of light is approximately 299,800,000 m/s.

Solution:

299,800,000 m/s = 2.998 × 10⁸ m/s

Since we're given "approximately" and the number ends in zeros, we should consider how many figures are truly known.

The most precise commonly used value is: 2.998 × 10⁸ m/s (4 significant figures)

For many calculations, 3.00 × 10⁸ m/s (3 significant figures) is sufficient.

A rectangular field has dimensions 127.3 ± 0.5 m by 89.6 ± 0.3 m. Calculate the area with proper uncertainty.

Solution:

Area = length × width = 127.3 × 89.6 = 11,406.08 m²

Uncertainty calculation for multiplication:

Relative uncertainty in length: δl/l = 0.5/127.3 = 0.00393

Relative uncertainty in width: δw/w = 0.3/89.6 = 0.00335

Combined relative uncertainty: √(0.00393² + 0.00335²) = 0.00517

Absolute uncertainty: δA = A × 0.00517 = 11,406 × 0.00517 = 59 m²

Final answer: A = 11,400 ± 60 m²

Convert 55 mph to km/h using dimensional analysis, and express your answer with appropriate significant figures.

Solution:

55 mph × (1.609 km/1 mile) = 55 × 1.609 km/h = 88.495 km/h

Significant figures analysis:

• 55 mph has 2 significant figures

• 1.609 km/mile is a conversion factor with 4 significant figures

Result should have 2 significant figures: 88 km/h

Note: For exact conversion factors (like 12 inches = 1 foot), the number of significant figures is unlimited.

Check whether the equation P = ½ρv²A is dimensionally consistent, where P is power, ρ is density, v is velocity, and A is area.

Solution:

Left side dimensions:

[P] = [ML²T⁻³] (power)

Right side dimensions:

[ρ] = [ML⁻³] (density)

[v²] = [L²T⁻²] (velocity squared)

[A] = [L²] (area)

[½ρv²A] = [ML⁻³][L²T⁻²][L²] = [ML⁻³⁺²⁺²T⁻²] = [MLT⁻²]

Comparison:

Left: [ML²T⁻³], Right: [MLT⁻²]

No, the equation is dimensionally inconsistent.

The equation might be for force (F = ½ρv²A) rather than power.

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