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Overview

(Unit One→Physical Quantities and Units of Measurement — Second Secondary, Term 1) 🌳 Concept tree Chapter 1: Physical Measurement

├── What you will learn

├── 1.1 Why we measure

├── 1.2 Definition of measurement

├── 1.3 The three elements of any measurement

├── 1.4 Physical quantities

├── 2.1 Fundamental physical quantities

├── 2.2 Derived physical quantities

├── 2.3 Physics and mathematics

├── 2.4 Systems of units

├── 2.5 The International System of Units (SI)

├── 2.6 Scientists who benefited humanity

├── 3.1 From the body to the instrument

├── 3.2 Examples of instruments, old and modern

├── 3.3 The vernier caliper — measuring length precisely
Standard Units (Length, Mass, Time)

├── 4.1 The standard of length — the metre

├── 4.2 The standard of mass — the kilogram

├── 4.3 The standard of time — the second

├── 5.1 The idea

├── 5.2 Getting the unit from the dimensional formula

├── 5.4 Table of dimensional formulas

├── 5.5 Rules to remember ("Pay attention")

├── 6.1 Dimensional homogeneity

├── 6.3 Limits of the method (important!)

├── Lesson 7→Multiples and Submultiples of Units; Scientific Notation

├── 8.1 No measurement is perfect

├── 8.2 Sources of error

├── 8.3 Precautions when using a metre ruler

├── 8.4 Direct and indirect measurement

├── Extra worked problems

├── Summary

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What you will learn

The meaning of measurement and its three elements.
Fundamental and derived physical quantities, and the systems of units.
The International System of Units (SI) its seven base units and two supplementary units.
Measuring instruments for length, mass and time, and how to read a vernier caliper.
The standards of length (metre), mass (kilogram) and time (second).
Dimensional formulas, derived units, and using dimensional analysis to test equations.
Multiples and submultiples of units and scientific notation.
Sources of error, and the difference between direct and indirect measurement.

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1.1 Why we measure

Natural science studies the phenomena that happen around us it describes them, tries to explain them, and then tests the explanation by experiment.
A description such as "the patient has a high temperature" is scientifically weak, because "high" means different things to different people.
Saying "the patient’s temperature is 40 °C" is scientific, because it gives a number and a unit that anyone, anywhere, can understand and check.
Measurement therefore turns our observations into quantitative values that can be written as numbers.

Without measurement we could not design a bridge, dose a medicine, schedule a train, or test any law of physics.
Every experiment you will perform this year ends with a measurement.

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1.2 Definition of measurement

Measurement→is the process of comparing an unknown quantity with another quantity of the same kind (called the unit of measurement) in order to find how many times the unit is contained in the unknown quantity.

Example when you say a table is 2 m long, you mean that the standard length "one metre" fits twice into the length of the table.
The number 2 tells how many times, and the metre tells which kind of quantity was used for the comparison.

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1.3 The three elements of any measurement

Every act of measurement involves three things:

The physical quantityto be measured (for example the length of a desk, the mass of a stone, the time of a swing).

The measuring instrument(ruler, balance, stopwatch, ammeter …).

The unit of measurementthe standard in which the result is expressed.

If any one of the three is missing, the measurement is meaningless.
A result such as "mass = 5" is incomplete: is it 5 grams, 5 kilograms or 5 tonnes?
Only "5 kg" fully defines the quantity.
This is why a number without a unit has no physical meaning.

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1.4 Physical quantities

Quantities such as mass, time, length, volume, speed and force are called physical quantities: they can be measured and used in the mathematical laws of physics.
Physical quantities are divided into two groups.
Fundamental quantities cannot be defined in terms of other quantities; length, mass and time are the main examples.
Derived quantities are defined using the fundamental ones: volume is derived from length, since the volume of a box is found by multiplying its three lengths; speed is derived from length and time; and kinetic energy depends on mass, length and time.

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2.1 Fundamental physical quantities

A fundamental (base) physical quantity is a quantity that is not defined in terms of other physical quantities.
You cannot break it down into simpler ones.
Examples length, mass, time (and also electric current, temperature, amount of substance and luminous intensity, as we will see in the SI table).

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2.2 Derived physical quantities

A derived physical quantity is a quantity that is defined in terms of fundamental quantities (by multiplying or dividing them).
Examples volume, speed, acceleration, force, density, pressure.
Worked idea the volume of a rectangular block is
Vol = length × width × height = L₁ × L₂ × L₃.
Volume is built from three lengths, so it is derived from length. In the same way:

area = length × length,

speed = length ÷ time,

density = mass ÷ volume = mass ÷ (length)³.

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2.3 Physics and mathematics

Physical quantities are linked by equations.
An equation is a short, exact way of describing a physical relationship, and every equation carries a physical meaning.
For example, v = d/t does not only say "divide d by t": it says that speed is the rate at which distance is covered.
Always ask yourself what an equation means, not just how to substitute in it.

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2.4 Systems of units

Different parts of the world have used different systems to define the three mechanical fundamental quantities:

Fundamental quantity	C.G.S (French / Gauss system)	F.P.S (Imperial system)	M.K.S (Metric system)
Length	centimetre (cm)	foot (ft)	metre (m)
Mass	gram (g)	pound (lb)	kilogram (kg)
Time	second (s)	second (s)	second (s)

Notice that time is measured in seconds in every system; the systems differ in length and mass.
Having several systems caused confusion in trade and science, so scientists agreed on one system for the whole world.

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2.5 The International System of Units (SI)

The SI is the modern metric system.
At the 11th General Conference on Weights and Measures (1960) it was agreed to add four more base units to the old metric system (M.K.S), giving seven base quantities:

No.	Fundamental quantity	Symbol	SI unit	Unit symbol
1	Length	L	metre	m
2	Mass	M	kilogram	kg
3	Time	t (T)	second	s
4	Electric current intensity	I	ampere	A
5	Absolute (thermodynamic) temperature	T	kelvin	K
6	Amount of substance	n	mole	mol
7	Luminous intensity	I_v	candela	cd

Two supplementary units are also included:

the radian (rad) — the unit of plane angle,→the steradian (sr) — the unit of solid angle (the angle of a cone-like region in space, as seen from its apex).

The SI is used consistently in all scientific fields all over the world.
In this book, unless told otherwise, you should convert every given value to SI units before substituting in an equation.

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2.6 Scientists who benefited humanity

Comparison aspect	Main idea	Keywords
William Thomson (Lord Kelvin)	a British scientist contributed greatly to the development of the metric system. He fixed the absolute zero of temperature precisely: it equals 0 kelvin which corresponds to −273 °C. (You will meet absolute zero again in Chapter 6.)	Britishscientistcontributedgreatly
Ahmed Zewail	an Egyptian scientist won the Nobel Prize in Chemistry in 1999 for using laser technology to observe chemical reactions between molecules. These reactions happen in times measured in femtoseconds (10⁻¹⁵ s) which opened the new field of femtochemistry. This is a good example of how very precise time measurement opens new science	EgyptianscientistNobelPrize

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3.1 From the body to the instrument

Early humans used parts of their bodies and natural events as measuring tools: the arm, hand and foot for length, and the sunrise, sunset and cycles of the moon for time.
Because every person’s arm is a different length, these units differed from one country (even one town) to another.
With the growth of industry people needed agreed standards and better instruments, so instruments developed greatly and helped us describe phenomena more accurately.

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3.2 Examples of instruments, old and modern

Quantity	Instruments (from older to more modern)
Length	measuring tape, ruler, vernier caliper, micrometer
Mass	steelyard balance two-pan balance single-pan balance digital balance (the sensitive ones are used after calibration)
Time	hourglass pendulum clock stopwatch digital clock

A sensible choice of instrument is the first step to an accurate result: you would not use a metre tape to measure the thickness of a coin, nor a kitchen balance to weigh a gold ring (see sources of error, Lesson 7).

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3.3 The vernier caliper — measuring length precisely

Why do we need it?

A ruler graduated in millimetres cannot give a reading better than about 1 mm.
The vernier caliper allows a reading to 0.1 mm.
Parts. A vernier caliper has

a fixed main scale (graduated in mm), and

a sliding vernier scale that moves along it,

jaws between which the object is placed.
The principle. The main scale is divided into millimetres: one main-scale division = 1 mm. The vernier scale has 10 divisions that together are exactly as long as 9 main divisions (9 mm). So each vernier division is 9 mm ÷ 10 = 0.9 mm, which is 0.1 mm shorter than a main-scale division. Therefore:
Vernier reading = (number of the vernier line that coincides with a main-scale line) × 0.1 mm
and the smallest length the instrument can measure (its precision, or least count) is 0.1 mm.

Procedure for measuring a length

Place the object between the jaws and close them gently (do not squeeze hard — you would bend the object or the jaws).
Read the main scale at the point just before the zero of the vernier scale. (Example: 29 mm.)
Look along the vernier scale for the line that lines up exactly with some line of the main scale. Multiply its number by 0.1 mm. (Example: the 6th line coincides → 6 × 0.1 = 0.6 mm.)
29 mm + 0.6 mm = 29.6 mm.
Repeat the measurement several times and take the average to reduce random error.

Worked example 1 (new numbers). The zero of the vernier lies just after the 41 mm mark of the main scale, and the 3rd vernier line coincides with a main-scale line.
Reading = 41 + 3 × 0.1 = 41.3 mm = 4.13 cm.

Worked example 2. Four repeated readings of the diameter of a steel ball were 12.7 mm, 12.8 mm, 12.7 mm and 12.6 mm. The best value is the mean:
(12.7 + 12.8 + 12.7 + 12.6) ÷ 4 = 50.8 ÷ 4 = 12.7 mm.

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Lesson 4: Standard Units (Length, Mass, Time)

Without a standard, "5" means nothing.
Scientists have always searched for the most accurate and most stable definitions of the standard units.

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4.1 The standard of length — the metre

The French were the first to use the metre as a standard.
First definition (the international prototype metre) the distance between two fine marks engraved near the ends of a bar made of a platinum–iridium alloy, kept at 0 °C at the International Bureau of Weights and Measures near Paris.
Enrichment (1960) the metre was redefined by an atomic constant: it equals 1 650 763.73 wavelengths of the orange-red light emitted in a vacuum by atoms of the krypton-86 isotope in an electric discharge tube.
Why was the atomic definition preferred over the metal bar?

A metal bar can expand or contract with temperature, can be scratched or damaged, and only one original exists in Paris.
The wavelength of light from an atom is the same everywhere and at all times; anyone with a suitable apparatus can reproduce the metre in their own laboratory, with higher accuracy.
Why not a glass rod of the same length as the standard?

Glass is brittle and can break or be damaged; its length changes with temperature and humidity; and it ages.
A standard must be unchanging, reproducible and available everywhere.
Idea to think about how can the Earth–Moon distance or the Earth’s circumference be measured? (Hint: the Moon distance can be found by timing a laser pulse reflected from mirrors left on the Moon, using distance = speed × time ÷ 2; the Earth’s circumference was first estimated from the difference in the Sun’s shadow angle at two cities a known distance apart.)

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4.2 The standard of mass — the kilogram

The standard kilogram is the mass of a cylinder of platinum–iridium alloy of specified dimensions, kept at 0 °C at the International Bureau of Weights and Measures near Paris.
(The alloy is chosen because it is very hard and does not corrode.)

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4.3 The standard of time — the second

Ancient peoples used the day and night cycle as a natural, simple and stable reference for time:

60 s = 1 min, 60 min = 1 h, 24 h = 1 day

1 day = 24 × 60 × 60 = 86 400 s

So the second was defined as 1/86 400 of the mean solar day.

Later, scientists proposed atomic clocks, such as the caesium clock, because of their extreme accuracy.
(Enrichment) The standard second is the time needed by a caesium-133 atom to emit 9 192 631 700 waves (periods) of a particular radiation.
Uses of atomic clocks studying changes in the Earth’s rotation period (the length of the day), improving air and ground navigation systems (GPS depends on very exact timing), and refining space missions.
Why do scientists seek ever more accurate standards? Because the accuracy of every measurement in science and technology depends on the accuracy of the standards; better standards allow better instruments, better experiments and new discoveries.

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5.1 The idea

Every physical quantity has a definition that is agreed upon worldwide. For example, speed = distance ÷ time.
We use letters for the three mechanical fundamental quantities:

L for length, Mfor mass, T for time.

(Carefulhere T means time dimension, not temperature.)

When the definition of a quantity is written using these symbols we get its dimensional formula.
Square brackets [ ] mean "the dimensions of":

[speed] = [distance] ÷ [time] = L ÷ T = LT⁻¹

In general
most derived quantities can be written as

[A] = La Mb Tc

where a, b, c are the dimensional exponents (positive, negative or zero).
Dimensional formula of a physical quantity an expression showing how the quantity depends on the fundamental quantities L, M, T and the power of each.

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5.2 Getting the unit from the dimensional formula

Replace L
M
T by their SI units (m
kg
s).

Speed LT⁻¹ → m s⁻¹ = m/s.
Acceleration is the time rate of change of velocity:
[a] = [velocity] ÷ [time] = LT⁻¹ ÷ T = LT⁻²; unit m/s².
The method has four steps.
First, write the relation that defines the quantity.
Second, replace each quantity in it by its dimension (M, L or T).
Third, give each dimension its correct power.
Fourth, leave out any dimension whose power is zero, because X0 equals 1.
The unit is then read from the final formula, and the process can be reversed to find the dimensional formula from a unit.
Dimensional formulae cannot be combined by addition or subtraction, though they can be multiplied or divided: for example, multiplying LT-1 by T gives L, the dimension of distance.

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5.4 Table of dimensional formulas

Quantity	Relation	Dimensional formula	SI unit
Area	length × width	L²	m²
Volume	length × width × height	L³	m³
Density ρ	mass ÷ volume	ML⁻³	kg/m³
Velocity v	displacement ÷ time	LT⁻¹	m/s
Acceleration a	velocity ÷ time	LT⁻²	m/s²
Force F	mass × acceleration	MLT⁻²	newton (N) = kg·m/s²
Pressure P	force ÷ area	MLT⁻² ÷ L² = ML⁻¹T⁻²	pascal (Pa) = kg·m⁻¹·s⁻²
Work (energy) W	force × displacement	MLT⁻² × L = ML²T⁻²	joule (J) = kg·m²·s⁻²

More derived examples (your own practice):

Comparison aspect	Main idea	Keywords
Momentum	p = mv: M × LT⁻¹ = MLT⁻¹ unit kg·m/s	unit
Power	P = W/t: ML²T⁻² ÷ T = ML²T⁻³ unit watt (W) = kg·m²·s⁻³	ML²Tunitwatt
Weight density / specific gravity	relative density = ρ/ρ_water → dimensionless, no unit.	relativeρwaterdimensionlessunit
Frequency	f = 1/T: T⁻¹ unit hertz (Hz) = s⁻¹	unithertz
Viscosity coefficient	η = Fd/(Av): [MLT⁻² · L] ÷ [L² · LT⁻¹] = ML⁻¹T⁻¹ unit kg/(m·s) = N·s/m² (used in Chapter 4)	unitusedChapter

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5.5 Rules to remember ("Pay attention")

those having the same dimensional formula. You cannot add a mass (kg) to a distance (m).
If they are of the same kind but in different units, convert one first: 1 m + 170 cm = 100 cm + 170 cm = 270 cm.
Quantities with different dimensional formulas can be multiplied or divided, and a new quantity results (distance ÷ time → speed; force × distance → work).
Pure numbers, fractions and constants like ½, 2, π, are dimensionless.

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6.1 Dimensional homogeneity

In any correct physical equation, the dimensions of the left-hand side must equal those of the right-hand side (and of every separate term that is added or subtracted).
This is called dimensional consistency or dimensional homogeneity.
We use it to check an equation quickly.
Example A — kinetic energy. Is E_k = ½mv² possible?

Right sideM × (LT⁻¹)² = ML²T⁻²; the ½ has no dimensions.

EnergyML²T⁻². They match, so the equation may be valid.

Example B — a wrong volume formula.A student suggests that the volume of a cylinder is Vol = πrh. Test it.

LeftL³. Right: L × L = L². Not equal → the equation is incorrect.

Example C — pressure in a liquid.Test P = P_a + ρgh.

[P] = ML⁻¹T⁻². [P_a] is also pressure, ML⁻¹T⁻².
[ρgh] = ML⁻³ × LT⁻² × L = ML⁻¹T⁻². All three terms agree → consistent.
Example D — a pendulum. Test T = 2π√(l/g).

l/g = L ÷ LT⁻² = T²; √(T²) = T. The left side is a time, T. Consistent.
Example E — a wrong kinematic equation. Test v = u + at².

[v] = LT⁻¹; [u] = LT⁻¹; [at²] = LT⁻² × T² = L. A velocity and a length cannot be added: the equation is wrong. (Correct: v = u + at, since at = LT⁻².T = LT⁻¹.)
Example F — Einstein’s relation E = mc². [mc²] = M(LT⁻¹)² = ML²T⁻², the same as energy, so its SI unit is kg·m²/s² = joule.

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6.3 Limits of the method (important!)

If both sides have the same dimensions, the equation is not guaranteed correct — it only may be valid. For example, E = 3mv² has the right dimensions but the wrong number; numerical factors (½, 2, π …) cannot be checked by dimensions.
But if the dimensions of the two sides are different, the equation is definitely wrong.
So dimensional analysis can reject but cannot prove an equation.

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Lesson 7: Multiples and Submultiples of Units; Scientific Notation

Physical quantities can be gigantic or tiny.
The distance between stars is about 100 000 000 000 000 000 m, while the spacing of atoms in a solid is about 0.000 000 001 m.
Such numbers are hard to read, so we use powers of ten (scientific notation): a × 10ⁿ with 1 ≤ a < 10.
The two distances become 1 × 10¹⁷ m and 1 × 10⁻⁹ m.
The powers of ten have agreed names (prefixes):

Factor	10⁻⁹	10⁻⁶	10⁻³	10⁻²	10³	10⁶	10⁹
Prefix	nano	micro	milli	centi	kilo	mega	giga
Symbol	n	µ	m	c	k	M	G

Comparison aspect	Main idea	Keywords
Method of conversion.	Replace the prefix by its power of ten and simplify	Replaceprefixpowersimplify
Worked example 1.	Express 7 mA in µA	Express
•	1 mA = 10⁻³ A and 1 µA = 10⁻⁶ A. Dividing: 1 mA/1 µA = 10³ so 1 mA = 10³ µA	Dividing
•	7 mA = 7 × 10³ µA = 7000 µA	7000
Worked example 2.	A wire is 0.35 mm in diameter. In metres: 0.35 × 10⁻³ m = 3.5 × 10⁻⁴ m	wirediametermetres

✍️ Worked examples

Worked example 3.A hard disc stores 2.5 GB (gigabytes). In bytes: 2.5 × 10⁹ B.

Worked example 4.Convert 108 km/h to m/s. 1 km/h = 1000 m ÷ 3600 s = 5/18 m/s. So 108 × 5/18 = 30 m/s.

Writing numbers in scientific notation (practice).

Mass of a car ≈ 1 200 kg = 1.2 × 10³ kg.
Thickness of a leaf 0.000 12 m = 1.2 × 10⁻⁴ m.
Mass of the Earth ≈ 6 000 000 000 000 000 000 000 000 kg = 6 × 10²⁴ kg.

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8.1 No measurement is perfect

Accuracy of measurement is tied directly to scientific and technological progress, so humans keep improving methods and instruments.
But it is impossible to obtain a 100 % exact measurement; there is always a margin of error, however small.
When we measure the length of a room we usually find a small difference between the measured and the true value; the difference may be small or large depending on how carefully we measure.
Discussion exercise. Five students measured the same pencil: 10.1, 10.0, 9.8, 10.0, 10.0 cm.

The readings are close but not identical → measurement involves small errors, and the best value is the mean: (10.1+10.0+9.8+10.0+10.0)/5 = 49.9/5 ≈ 10.0 cm.
Possible reasons reading the scale from the wrong angle, not starting at the zero mark, different judgment of where the end falls, a worn ruler end.
A ruler graduated in 1 mm is more accurate than one graduated in 1 cm, because its smaller division allows a reading closer to the true length.

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8.2 Sources of error

Comparison aspect	Main idea	Example	Keywords
Choosing an inappropriate measuring instrument.	For instance weighing a gold ring on an ordinary kitchen balance instead of a sensitive balance gives a large error	—	instanceweighinggoldordinary
Defects in the instrument.	An old ammeter whose magnet has become weak or one whose pointer does not return to zero when the current is cut off (zero error)	—	ammeterwhosemagnetbecome
Incorrect measuring technique	by a beginner. Examples: not knowing how to use the ranges of a multi-range instrument	the multimeter viewing the pointer or scale at a slant instead of keeping the line of sight perpendicular to the scale at the reading (this is called a parallax error)	beginnerExamplesknowingranges
Environmental factors	such as temperature humidity and air currents. A sensitive balance is placed inside a glass case so that air currents do not move the pan	—	temperaturehumiditycurrentssensitive

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8.3 Precautions when using a metre ruler

Choose a ruler with suitable graduation and check that its end is not worn.
Place the zero mark exactly at one end of the object (or use another mark and subtract).
Keep the ruler along the length being measured, with its graduated edge in contact with the object.
Keep your eye vertically above the reading mark (line of sight perpendicular).
Repeat several times and take the average.

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8.4 Direct and indirect measurement

Comparison aspect	Main idea	Keywords
Direct measurement	uses one instrument and gives the quantity immediately with no substitution in an equation. Example: density of a liquid with a hydrometer volume of liquid with a graduated cylinder length with a ruler	usesinstrumentgivesquantity
Indirect measurement	uses more than one instrument or measurement and then substitution in an equation. Example: density = mass ÷ volume with the mass found by a balance and the volume by a graduated cylinder volume of a block by measuring L W and H with a ruler and multiplying	usesinstrumentthensubstitution

Point of comparison	Direct measurement	Indirect measurement
Number of measurements	one	more than one
Mathematical relation	not needed	needed to calculate the quantity
Errors	a single error	several errors may add up — accumulation of errors
Examples	volume with a graduated cylinder	volume = L × W × H

Comparison aspect	Main idea	Keywords
Worked example (indirect measurement).	A stone has a mass of 135 g (balance) and displaces water in a graduated cylinder from 40 mL to 90 mL. Volume = 50 mL = 50 cm³. Density = 135/50 = 2.7 g/cm³ = 2700 kg/m³ (probably aluminium)	stonemassbalancedisplaces
Worked example (error accumulation).	If a length is measured as 10.0 cm ± 0.1 cm and a width as 5.0 cm ± 0.1 cm the area 50 cm² could in the worst case be as large as 10.1 × 5.1 = 51.5 cm² or as small as 9.9 × 4.9 = 48.5 cm²: the errors of both measurements pass into the result	lengthmeasuredwidtharea

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Extra worked problems

Comparison aspect	Main idea	Keywords
Problem 1.	Find the dimensional formula and SI unit of the impulse J = F × t	Finddimensionalformulaunit
•	[F] = MLT⁻² [t] = T → [J] = MLT⁻¹ unit N·s = kg·m/s (same as momentum)	unitsamemomentum
Problem 2.	The speed v of a wave on a stretched string might depend on the tension F and the mass per unit length μ as v = √(F/μ). Check it	speedwavestretchedstring
•	[F] = MLT⁻² [μ] = ML⁻¹ → F/μ = MLT⁻² ÷ ML⁻¹ = L²T⁻² √ = LT⁻¹ = velocity. Consistent	velocityConsistent
Problem 3.	Convert a density of 13 600 kg/m³ (mercury) to g/cm³	Convertdensitymercury
•	1 kg = 10³ g 1 m³ = 10⁶ cm³ → 13 600 × 10³/10⁶ = 13.6 g/cm³
Problem 4.	A cube has an edge of 4.0 cm and mass 512 g. Density = 512/4³ = 512/64 = 8.0 g/cm³ = 8000 kg/m³	cubeedgemassDensity
Problem 5.	How many seconds in a year of 365 days? 365 × 86 400 = 3.15 × 10⁷ s	manysecondsyeardays

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Summary

Measurement = comparing an unknown quantity with a unit of the same kind; its elements are quantity, instrument, unit.
Fundamental quantities are not defined by others (L, M, t, I, T, n, I_v); derived quantities are defined from them.
SI m, kg, s, A, K, mol, cd, plus rad and sr.
Vernier caliper precision 0.1 mm: reading = main-scale value + (coinciding vernier line × 0.1 mm).
Standards metre (Pt–Ir bar, then krypton-86 wavelengths), kilogram (Pt–Ir cylinder), second (1/86 400 of the mean solar day, then caesium-133 clock).
Dimensional formula [A] = La Mb Tc; F MLT⁻², P: ML⁻¹T⁻², W: ML²T⁻², ρ: ML⁻³.
Both sides of a correct equation must have the same dimensions; agreement does not guarantee correctness, but disagreement proves an error.
Prefixes nano 10⁻⁹, micro 10⁻⁶, milli 10⁻³, centi 10⁻², kilo 10³, mega 10⁶, giga 10⁹.
Error sources wrong instrument, faulty instrument, wrong technique, environment. Direct = one measurement; indirect = several + equation (accumulated errors).

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Test yourself on this lesson

The lesson’s question bank is split across the points above; each point has its own questions underneath.
To work through the whole bank on one page:

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Part 33

📖

📌 Comprehensive Scientific Terminology & Definitions:

Comparison aspect	Main idea	Keywords
Fundamental vs Derived Quantities	Fundamental quantities cannot be defined in terms of other quantities (Length Mass Time Current Temp Amount Luminosity). Derived quantities are defined mathematically via fundamentals (Velocity Acceleration Force Work)	cannotdefinedtermsLength
Dimensional Formula	Mathematical representation expressing a derived quantity in terms of the fundamental dimensions of Mass [M] Length [L] and Time [T] raised to powers: [M^a L^b T^c]	Mathematicalrepresentationexpressingderived
Vernier Caliper and Micrometer	Precision measurement instruments Vernier caliper measures internal/external diameters to 0.1 mm Micrometer measures extremely fine thickness down to 0.01 mm precision	Precisionmeasurementinstrumentsmeasures
Absolute and Relative Error	Absolute error (Δx = \|x₀ – x\|) is the magnitude of the difference between true and measured values. Relative error (r = Δx / x₀) is dimensionless and provides the true indicator of measurement precision	magnitudedifferencebetweentrue

⚖️ Governing Scientific Laws, Principles & Mathematical Formulas:

Comparison aspect	Main idea	Keywords
Absolute Error	Δx = \|x₀ – x\| (carries measurement units)	carriesmeasurementunits
Relative Error	r = Δx / x₀ \| Percentage Error = r × 100%	Percentage
Addition & Subtraction Error	Δxtotal = Δx₁ + Δx₂	Δxtotal
Multiplication & Division Error	rtotal = r₁ + r₂ \| Δxtotal = rtotal × x₀	rtotalΔxtotal
Key Dimensions	Velocity [LT⁻¹] \| Acceleration [LT⁻²] \| Force [MLT⁻²] \| Work/Energy [ML²T⁻²]	VelocityAccelerationForceWork

🔍 In-Depth Explanations, Deductions & Core Concepts:

Comparison aspect	Main idea	Keywords
Why is zero measurement error impossible?	Due to inherent error sources: (1) Instrumental defects (e.g zero error) (2) Inappropriate tool selection (3) Environmental factors (temperature air currents) (4) Parallax observation angle errors	inherentsourcesInstrumentaldefects
Why is relative error more meaningful than absolute error?	Because it normalizes error against the magnitude of the measured quantity: an error of 1 cm over 100 m represents far superior accuracy than 1 cm over a 10 cm pencil	Becausenormalizesagainstmagnitude

Physical Quantity	Dimensional Formula	SI Base Units
Velocity (v)	[M⁰ L T⁻¹]	m/s
Acceleration (a)	[M⁰ L T⁻²]	m/s²
Force (F)	[M L T⁻²]	Newton (N = kg·m/s²)
Work / Kinetic Energy	[M L² T⁻²]	Joule (J = kg·m²/s²)
Pressure (P)	[M L⁻¹ T⁻²]	Pascal (Pa = N/m²)

💡 High-Yield Exam Traps & Key Takeaways:

Quantities can only be added or subtracted if they possess identical dimensional formulas and units.
Pure numbers, constants (π, e), and trigonometric functions are completely dimensionless.

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