Measurement

Overview

Measurement is the foundation of experimental physics. Every physical law is tested through observations and measured data. Good measurement requires:

• correct units
• sensible precision
• awareness of uncertainty
• clear presentation of data
• fair experimental design

This topic supports all later chapters such as Kinematics, Forces, and Current Electricity Fundamentals.

A strong understanding of measurement helps students avoid common practical and examination mistakes.

Core Ideas

Physical Quantities

A physical quantity is a measurable property that can be expressed by:

• a numerical value
• a unit

Examples:

• length = $2.35\text{m}$
• time = $4.2\text{s}$
• mass = $0.150\text{kg}$

Physical quantities are divided into:

• base quantities
• derived quantities

SI Base Quantities and Units

Base Quantity	SI Unit	Symbol
length	metre	m
mass	kilogram	kg
time	second	s
electric current	ampere	A
thermodynamic temperature	kelvin	K
amount of substance	mole	mol
luminous intensity	candela	cd

For H2 Physics, the most frequently used are $m,kg,s,A,K$.

Derived Quantities and Units

Derived quantities are formed from base quantities.

Examples:

Quantity	Expression	Unit
velocity	displacement / time	m s${}^{-1}$
acceleration	velocity / time	m s${}^{-2}$
force	mass × acceleration	N
energy	force × distance	J
pressure	force / area	Pa
charge	current × time	C

Named units:

• $1\text{N}=1{\text{kg m s}}^{-2}$
• $1\text{J}=1{\text{kg m}}^2{\text{s}}^{-2}$
• $1\text{Pa}=1{\text{N m}}^{-2}$

See Measurement Units and Dimensions.

Dimensions and Homogeneity

Dimensions describe the physical nature of quantities using symbols such as:

• Mass: $M$
• Length: $L$
• Time: $T$
• Current: $I$

Examples:

• velocity: $L{T}^{-1}$
• acceleration: $L{T}^{-2}$
• force: $ML{T}^{-2}$

Principle of Homogeneity

All terms in a valid physical equation must have the same dimensions.

Example:

$$s=ut+\frac{1}{2}a{t}^2$$

• $ut$ has dimension $L$
• $a{t}^2$ has dimension $L$

Hence the equation is dimensionally consistent.

Dimensional checks help detect mistakes, but do not prove an equation is physically correct.

Prefixes and Orders of Magnitude

Common SI Prefixes

Prefix	Symbol	Value
pico	p	$10^{-12}$
nano	n	$10^{-9}$
micro	$\mu$	$10^{-6}$
milli	m	$10^{-3}$
centi	c	$10^{-2}$
kilo	k	$10^3$
mega	M	$10^6$
giga	G	$10^9$

Examples:

• $3.0\text{mm}=3.0\times 10^{-3}\text{m}$
• $5.0\text{kV}=5.0\times 10^3\text{V}$

Order of Magnitude

Approximate power of ten estimate.

Examples:

• diameter of atom $\sim 10^{-10}\text{m}$
• human height $\sim 10^0\text{m}$
• Earth radius $\sim 10^6\text{m}$

See Measurement Estimation and Experimental Design.

Precision vs Accuracy

Precision

How close repeated readings are to one another.

• small spread of values
• related to random error

Accuracy

How close a reading is to the true or accepted value.

• related to systematic error

Example

Readings of $9.81,9.82,9.80$

• high precision

If true value is $9.81$:

• also high accuracy

A set of readings can be:

• precise but inaccurate
• accurate on average but imprecise

Figure: Accuracy and precision are different. Measurements may cluster tightly without being close to the true value, or scatter around the true value without being precise.

Systematic Errors vs Random Errors

Systematic Errors

Errors that shift readings consistently in one direction.

Examples:

• zero error
• poor calibration
• heat loss in calorimetry
• friction ignored

Effects:

• reduces accuracy
• repeating readings does not remove it

Random Errors

Unpredictable fluctuations between readings.

Examples:

• reaction time
• changing surroundings
• reading scale by eye

Effects:

• reduces precision
• reduced by repeated readings and averaging

See Measurement Uncertainty and Errors.

Uncertainty Overview

Every measured value has uncertainty.

Example:

$$x=12.4\pm 0.1\text{cm}$$

This means likely range:

$$12.3\text{ cm to }12.5\text{ cm}$$

Fractional Uncertainty

$$\frac{\Delta x}{x}$$

Percentage Uncertainty

$$\frac{\Delta x}{x}\times 100\%$$

Example:

$$x=20.0\pm 0.2$$

Percentage uncertainty:

$$\frac{0.2}{20.0}\times 100\%=1.0\%$$

Propagation

When values are combined, uncertainties combine too.

See Uncertainty Propagation Methods.

Data and Graph Overview

Experimental data must be presented clearly.

Tables

Use headings with quantity and unit.

Example:

$t/\text{s}$	$x/\text{m}$
0.0	0.00
1.0	2.10
2.0	4.20

Graphs

Include:

• title if needed
• labelled axes
• units
• sensible scales
• best-fit line/curve

Useful Interpretation

• gradient often represents physical quantity
• intercept may have meaning

Example:

Velocity-time graph gradient = acceleration.

See Measurement Data Presentation.

Estimation and Experimental Design Overview

Physics students should be able to estimate sensible values and plan fair tests.

Good Experiment Design Includes:

• suitable instrument choice
• control of variables
• repeated measurements
• reduction of uncertainty
• clear method
• safe procedure

Example

To measure pendulum period:

• time 20 oscillations instead of 1
• divide by 20

This reduces reaction-time uncertainty.

See Measurement Estimation and Experimental Design.

Vector Bridge Note

Some older syllabuses introduced vectors within measurement. In this wiki, vectors are developed separately in:

Vectors

This keeps the Measurement chapter focused on experimental skills and data handling.

Formula Summary

Density

$$\rho =\frac{m}{V}$$

Speed

$$v=\frac{s}{t}$$

Fractional Uncertainty

$$\frac{\Delta x}{x}$$

Percentage Uncertainty

$$\frac{\Delta x}{x}\times 100\%$$

Gradient

$$\text{gradient}=\frac{\Delta y}{\Delta x}$$

Common Exam Mistakes

• omitting units
• wrong unit prefixes
• confusing precision with accuracy
• confusing systematic and random errors
• too many significant figures
• graph axes not labelled
• using poor scale
• drawing line through every point instead of best-fit line
• forgetting uncertainty in final answer

Fast Revision Checklist

You should be able to:

• state SI base quantities and units
• convert prefixes
• determine derived units
• check homogeneity
• explain precision vs accuracy
• distinguish systematic vs random errors
• calculate percentage uncertainty
• present tables and graphs properly
• describe methods to reduce uncertainty

Related Links

• Measurement Units and Dimensions
• Measurement Uncertainty and Errors
• Uncertainty Propagation Methods
• Measurement Data Presentation
• Measurement Estimation and Experimental Design
• Vectors
• Kinematics
• Forces
• Current Electricity Fundamentals
• Vectors

Exam Relevance

Measurement concepts are fundamental across H2 Physics. Students are frequently tested on dimensional consistency, experimental uncertainty, systematic and random errors, graph interpretation, significant figures, and vector operations. These skills also support later topics such as vectors, kinematics, and dynamics.

Common exam traps include:

• using the gram instead of kilogram as the SI base unit of mass;
• treating a dimensionally valid equation as automatically correct;
• confusing systematic error with random error;
• claiming too many significant figures from a calculator;
• adding percentage uncertainties for addition or subtraction;
• adding absolute uncertainties for multiplication or division;
• ignoring direction when working with vector quantities.

Links

• Related: vectors
• Related: kinematics
• Related: Thermal Physics A
• Related: current electricity fundamentals
• Related: measurement units and dimensions
• Related: measurement data presentation
• Related: measurement uncertainty and errors
• Related: uncertainty propagation methods
• Related: measurement estimation and experimental design
• Related: scalars vectors and vector operations

Provenance

• source file: 1_PDFsam_01_Measurement.pdf
• generated by: bridging_tools/ingest_JC_phy_wiki.py
• manifest entry: inbox/lecture_notes/1_PDFsam_01_Measurement.pdf
• source hash: f6acbd161d261405eabedc8d61ebe91cc9b2e187cc1345bbe06c7233d22d2be3