SI Base Units and Quantities

Physics is, fundamentally, a quantitative science. Every physical statement worth making ultimately reduces to a measurement — a number paired with a unit. The OCR A-Level Physics A specification (H556) places Module 2: Foundations of Physics at the very start of the course for exactly this reason: without a watertight understanding of units, quantities and their manipulation, no meaningful physics can follow.

This lesson introduces the Système International d'Unités (SI), the seven base units, and the essential principle that every physical quantity is expressed as:

physical quantity = numerical value × unit

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Why We Need a Standard System

Before 1960, scientists and engineers worked in a bewildering variety of units — inches, feet, pounds, ergs, calories, horsepower, atmospheres, and dozens more. Communication was difficult, and expensive mistakes were common. The most famous example is the 1999 NASA Mars Climate Orbiter, which was lost because one team used imperial pound-seconds for thrust while another assumed newton-seconds. The mismatch sent the probe too close to the Martian atmosphere, where it burned up. The entire $327 million mission failed because of a unit error.

The SI system, adopted in 1960 and continually refined, provides a single coherent framework. It is:

• Coherent — derived units follow directly from base units without numerical factors.
• Decimal — all prefixes are powers of ten.
• Universal — used for scientific work in every country.
• Reproducible — every base unit is defined in terms of fundamental physical constants (since the 2019 SI redefinition).

At A-Level, you must know the seven base quantities and their units, and be able to express any derived quantity in terms of them.

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The Seven SI Base Quantities

There are exactly seven base quantities in the SI system. Every other physical quantity — no matter how complicated — can be built up from these seven.

Base Quantity	Symbol for Quantity	SI Base Unit	Unit Symbol
Mass	m	kilogram	kg
Length (distance)	l	metre	m
Time	t	second	s
Electric current	I	ampere	A
Thermodynamic temperature	T	kelvin	K
Amount of substance	n	mole	mol
Luminous intensity	I_v	candela	cd

At A-Level Physics, you use the first six regularly. Candela is rarely examined but you should know it exists.

Exam Tip: The OCR specification explicitly requires you to "be able to recall" the base units of mass, length, time, current, temperature and amount of substance. Expect a short-answer question such as "State the SI base unit of mass" worth one mark. The answer is kilogram (kg) — never grams, never "kilo".

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Why the Kilogram (Not the Gram)?

One curiosity of the SI system is that the base unit of mass is the kilogram, not the gram. This is historical: the original definition (1795) was based on a physical artefact — a platinum-iridium cylinder kept in Sèvres, France. Since 2019, the kilogram has been defined via Planck's constant, h = 6.62607015 × 10⁻³⁴ J s (exactly).

This has a subtle implication for unit prefixes: you can say "1 gram = 10⁻³ kg", but you do not use prefixes on the gram as if it were the base. For example, "1 microgram" means 10⁻⁶ g = 10⁻⁹ kg, and "megagram" (10⁶ g = 10³ kg) is more commonly written as tonne (t) in engineering.

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Quantity Equals Number Times Unit

At the heart of every physics calculation is the rule:

A physical quantity = a pure number × a unit

If a rod has a length of 2.5 metres, we write:

l = 2.5 m

The 2.5 is dimensionless — a pure number. The m is the unit. Together they specify the physical quantity "length of the rod".

This has several important consequences for how you work.

Consequence 1 — You Can Do Algebra With Units

Units behave exactly like algebraic symbols. If you multiply a length by a length, you get an area:

l × l = 2.5 m × 2.5 m = 6.25 m²

If you divide a distance by a time, you get a speed:

(100 m) / (20 s) = 5 m s⁻¹

The unit that comes out of such a calculation tells you whether your equation makes sense — a point we will return to in Lesson 2 when discussing homogeneity.

Consequence 2 — Units Must Be Converted Before Substitution

A common mistake is to substitute quantities in mixed units into an equation. Consider Newton's second law, F = ma. If you are given m = 500 g and a = 2 m s⁻², then:

F = 0.500 kg × 2 m s⁻² = 1.00 N

You cannot write "500 × 2 = 1000 N". You would be 1000 times too large. The base unit of mass is the kilogram, and equations involving newtons must take mass in kilograms.

Common Exam Mistake: Forgetting to convert grams to kilograms, centimetres to metres, or minutes to seconds before substituting. Always convert first, then substitute.

Consequence 3 — Numerical Answers Must Carry Units

A bare number is not an answer. "F = 20" is meaningless — 20 what? Pounds? Kilonewtons? Dynes? In every answer you write, include the unit, and make it an SI unit unless the question asks otherwise.

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Worked Example 1 — Expressing a Quantity

A student measures the mass of a trolley as 1.2 kilograms and its velocity as 3.0 metres per second. State each quantity as (number × unit) and identify the base quantities involved.

Solution:

• m = 1.2 kg — mass (base quantity, SI base unit kg)
• v = 3.0 m s⁻¹ — velocity (derived quantity, combining length and time)

The velocity involves two base quantities: length (m) and time (s). Velocity itself is not a base quantity.

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Worked Example 2 — Unit Conversion Before Substitution

A 250 g apple falls from a tree. Taking g = 9.81 m s⁻², calculate its weight.

Solution:

W = mg

You must convert 250 g to kilograms:

m = 250 g = 0.250 kg

Then substitute:

W = 0.250 × 9.81 = 2.4525 N ≈ 2.45 N (3 s.f.)

If you had substituted m = 250 directly, you would obtain 2452.5 N — the weight of a small car, not an apple. This is a classic unit slip.

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Worked Example 3 — Using the Base Unit of Current

A current of 2.0 A flows through a wire for 5.0 minutes. Calculate the total charge transferred.

Solution:

Q = It

Convert time to seconds:

t = 5.0 × 60 = 300 s

Substitute:

Q = 2.0 × 300 = 600 C

The coulomb (C) is a derived unit equal to 1 A s, so the base-unit form of the answer is 6.0 × 10² A s.

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SI Units and Physical Constants

Since 2019, the seven SI base units have been defined by assigning exact numerical values to seven physical constants:

Constant	Symbol	Exact Value	Defines
Speed of light in vacuum	c	299 792 458 m s⁻¹	metre
Hyperfine transition frequency of caesium-133	Δν_Cs	9 192 631 770 Hz	second
Planck constant	h	6.626 070 15 × 10⁻³⁴ J s	kilogram
Elementary charge	e	1.602 176 634 × 10⁻¹⁹ C	ampere
Boltzmann constant	k	1.380 649 × 10⁻²³ J K⁻¹	kelvin
Avogadro constant	N_A	6.022 140 76 × 10²³ mol⁻¹	mole
Luminous efficacy	K_cd	683 lm W⁻¹	candela

You do not need to memorise these exact values — they are provided in your data sheet. But you should appreciate that the base units are no longer defined by physical artefacts; they are defined so that fundamental constants have exact values. This gives the SI system extraordinary precision and universality.

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Writing Units Correctly

OCR examiners deduct marks for sloppy notation. Learn the conventions.

Rules for Symbols

• Unit symbols are not pluralised: "5 kg", never "5 kgs".
• A space separates the number and the unit: "5 m", not "5m".
• Unit symbols named after a person begin with a capital letter when abbreviated, but are lower case when spelled out: "N" and "newton", "Pa" and "pascal", "K" and "kelvin".
• Do not add a full stop to a unit symbol unless it is at the end of a sentence.
• Use a space between units being multiplied: "m s⁻¹", not "ms⁻¹" (which would mean "millisecond inverse").
• Use negative exponents, not slashes, in formal answers: write "m s⁻¹", not "m/s", in physics equations. Both are accepted at A-Level but the negative-exponent form is the formal convention used in mark schemes.

Rules for Quantity Symbols

• Quantity symbols are in italics in textbooks (the italic font is part of the convention).
• Unit symbols are in upright (roman) font.
• So: v = 5 m s⁻¹ — the v is italic, the "m s⁻¹" is upright.

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Common Exam Mistakes

1. Confusing "gram" with "kilogram" as the base unit. The base unit is the kilogram. Grams are a sub-multiple.
2. Writing "Kg" instead of "kg". The kilogram's symbol is lower-case k, lower-case g — because Kelvin uses the capital K.
3. Writing "Sec" or "sec" instead of "s". The SI symbol for seconds is simply "s".
4. Pluralising unit symbols. Never write "5 Ns" — that would be ampere-seconds. Write "5 N".
5. Forgetting to convert. Always reduce mixed-unit questions to SI base units before substituting.
6. Omitting the unit from a numerical answer. An unadorned "5" is not a physical quantity.
7. Using "°K". Kelvin is written without the degree symbol: "300 K", not "300 °K".

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Summary

• The SI system has seven base quantities and seven base units.
• You must recall: mass (kg), length (m), time (s), current (A), temperature (K), amount (mol).
• Every physical quantity has the form (number × unit).
• Units behave algebraically — they multiply, divide and cancel like symbols.
• Always convert to SI base units before substituting into equations.
• Numerical answers without units are incomplete and lose marks.

In the next lesson, we build derived units from these base units and learn how to check whether an equation is physically sensible using the principle of homogeneity.