Physics · Lesson 1

Measurements, units & dimensions

JAMBWAECNECO

What the syllabus asks

JAMB: Measurements and units: fundamental and derived quantities, SI units, dimensional analysis, measuring instruments, errors, scalars and vectors (introduction)

WAEC: Fundamental and derived quantities and units (SI), dimensions of physical quantities, measurement of length, mass, time, volume and density; use of metre rule, vernier calipers and micrometer screw gauge; accuracy, precision, significant figures and errors

NECO: Fundamental and derived quantities and units (SI), dimensions of physical quantities, measurement of length, mass, time, volume and density; use of metre rule, vernier calipers and micrometer screw gauge; accuracy, precision, significant figures and errors

What you will learn

Distinguish fundamental and derived quantities and state their SI units.
Express derived units in base units and convert between units and prefixes.
Write the dimensions of common quantities.
Use dimensional analysis to check equations and derive relationships.
Read a metre rule, vernier calipers and micrometer screw gauge correctly, including zero error.
Identify sources of error, use significant figures and calculate percentage error.

Key points

Seven SI base quantities; all others are derived.
Derived units reduce to base units: N = kg m s⁻², J = kg m² s⁻², Pa = kg m⁻¹ s⁻².
Prefixes: k 10³, M 10⁶, c 10⁻², m 10⁻³, μ 10⁻⁶; square or cube for area and volume.
Dimensions: force [MLT⁻²], energy [ML²T⁻²], pressure [ML⁻¹T⁻²].
Correct equations are dimensionally consistent; dimensions find powers but not pure numbers.
Metre rule 0.1 cm; vernier 0.01 cm; micrometer 0.01 mm; subtract positive zero errors.
Avoid parallax, repeat readings, time many oscillations.
Systematic vs random errors; percentage error; report to suitable significant figures.

Start here: Physical quantities and units

Physics is built on measurement. A physical quantity is anything that can be measured, and every measurement has two parts: a number and a unit. "The table is 1.5" means nothing; "the table is 1.5 m long" is a measurement.

Scientists across the world use the Système International (SI) so that results can be compared. A doctor in Ibadan, an engineer in Port Harcourt and a researcher in Japan all mean the same thing by 1 kilogram.

Fundamental (base) quantities are chosen and defined independently. There are seven: length (m), mass (kg), time (s), electric current (A), thermodynamic temperature (K), amount of substance (mol) and luminous intensity (cd).

Derived quantities are formed by multiplying or dividing fundamental quantities. Area = length × length (m²). Speed = length ÷ time (m s⁻¹). Density = mass ÷ volume (kg m⁻³).

Worked example 1.1

Classify: (a) volume; (b) temperature; (c) force; (d) electric current; (e) pressure.

(a) Volume = length³: derived.
(b) Temperature: fundamental (kelvin).
(c) Force = mass × acceleration: derived.
(d) Current: fundamental (ampere).
(e) Pressure = force/area: derived.

Derived, fundamental, derived, fundamental, derived

⚠️ Common mistake: calling weight a fundamental quantity. Weight is a force (mg), so it is derived; mass is fundamental.

Try it yourself. Is "charge" fundamental or derived in SI?

Show solution

Derived: charge = current × time (coulomb = A s).

Why we need standard units

In many Nigerian markets, grain is sold by the "mudu" or by a paint tin, and cloth is sold by the "yard". These measures work locally, but a mudu in one market may hold slightly more than in another. Science cannot work like that: an experiment in Zaria must give the same number as one in Enugu. Standard units are kept consistent worldwide by international agreement, and every laboratory instrument is ultimately calibrated against them. When you write a unit after every answer, you are taking part in this global system.

Notice also that units can be multiplied and divided just like numbers. If a car travels 150 m in 6 s, the speed is 150 m ÷ 6 s = 25 m/s; the unit m/s comes straight from the calculation. Keeping units in your working is one of the best ways to catch mistakes.

Continue this lesson in the free app →

Practice questions

1. Express the unit of power in base SI units.

Show worked answer

kg m² s⁻³.

2. Find the dimensions of momentum.

Show worked answer

[MLT⁻¹].

3. Convert 90 km/h to m/s.

Show worked answer

90/3.6 = 25 m/s.

4. Vernier: main scale 1.7 cm, 4th division coincides. Zero error −0.01 cm. True reading?

Show worked answer

1.74 − (−0.01) = 1.75 cm.

5. Micrometer: sleeve 3.5 mm, thimble 17. Reading?

Show worked answer

3.67 mm.

Key words

Physical quantity: A property that can be measured and expressed as a number with a unit.
Fundamental quantity: A quantity not defined in terms of others, e.g. mass.
Derived quantity: A quantity obtained by combining fundamental quantities, e.g. velocity.
SI unit: A unit of the International System of Units.
Dimension: The way a quantity depends on mass [M], length [L] and time [T].
Dimensionless: Having no dimensions, e.g. strain, angle in radians, refractive index.
Least count: The smallest reading an instrument can measure.
Parallax error: Error caused by reading a scale from an angle instead of perpendicularly.
Zero error: A non-zero reading when the instrument should read zero.
Accuracy: How close a measurement is to the true value.
Precision: How close repeated measurements are to each other; also the fineness of a scale.
Significant figures: The digits in a number that carry meaning about its precision.

Learn Measurements, units & dimensions step by step, free.
The Cara's Tips app teaches every topic from the very basics, with quick checks, worked examples and exam practice. It works offline.
Open this lesson in the free app Download for Android