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SI base and derived units

The six SI base units used in A-level physics are the metre, kilogram, second, ampere, kelvin and mole, and every other unit derives from them. The newton, the joule and the pascal are each a combination of base units. Reducing both sides of an equation to base units tests it for homogeneity and can expose a wrong formula.

SI units, prefixes and conversions

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Year 12AQA 3.1.1 CIE 1.1.1, 1.2.1, 1.2.2, 1.2.3, 15.1.1 OCR A 2.1.1(a), 2.1.2(a), 2.1.2(b), 2.1.2(c), 2.1.2(d), 3.2.1(b)EDEXCEL 1

SI units and prefixes, part 1 of 2. Part 2 is Prefixes, labels and unit conversions.

IN THIS TOPIC

Name the six SI base quantities used at A-level, and unpack any derived unit back into them.
Check an equation for homogeneity by reducing every term to base units, and say what a failed check proves and what a passed one does not.

COMMON MISCONCEPTION

Units are labels you attach once the number is finished.

Units are part of the physics, not labels added afterwards. Carry them through the calculation to check the algebra; an answer with the wrong unit is wrong.

Six units underneath everything

The SI system rests on seven base units. A-level physics uses six of them: the kilogram for mass, the metre for length, the second for time, the ampere for electric current, the kelvin for temperature and the mole for amount of substance. The specification leaves out the seventh, the candela for luminous intensity, and you are not expected to recall formal definitions of any of them.

One of the six catches people out. Amount of substance is a base quantity in its own right, on the same footing as mass and length, and its base unit is the mole. A question that asks for the base units of the molar gas constant expects you to treat mol like any other base unit.

Six labelled cards, one per SI base quantity used in this course: mass in kilograms, length in metres, time in seconds, current in amperes, temperature in kelvin and amount of substance in moles. The SI's seventh base quantity, luminous intensity in candela, is not needed at A-level. — FIG. 1The six SI base quantities used in this course and their units; the SI's seventh, luminous intensity in candela, is not used at A-level. Every other unit in the subject is assembled from these.

Derived units

A derived unit is a product of powers of the base units. From $F = ma$ , 1 N = 1 kg m s⁻². Consequently, 1 Pa = 1 N m⁻² and 1 J = 1 N m. Named units are abbreviations for these base-unit combinations.

Three cards for kilogram, metre and second to the minus two, with arrows merging into a single newton card: one newton is one kilogram metre per second squared. — FIG. 2One newton is one kilogram metre per second squared. The name is an abbreviation for the base-unit combination, not a separate thing.

A few quantities come out with no unit at all. Divide one length by another and the metres cancel, which is why strain is a bare number, and the same happens to efficiency, refractive index, relative permittivity and every other ratio of two like quantities. The units are not missing; they cancelled, and the correct thing to write on the answer line is the number by itself. An angle in radians is the same trick, an arc length over a radius, which is why the radian is not one of the six.

Carry units through every line of a calculation rather than attaching them at the end. The unit of the answer then follows from the working, and a final unit that does not match the quantity being calculated shows that an error occurred earlier.

Checking an equation for homogeneity

An equation says that two quantities are the same thing, so both sides must carry the same units. Reduce every term to SI base units and compare them. An equation whose terms all agree is homogeneous; one whose terms disagree is wrong, and you have proved that without knowing anything about where the equation came from.

The method is three steps. Replace each symbol by its base units. Collect the powers and cancel. Then compare, not only the two sides but every term that is added to another, since adding a length to a time means nothing whatever the numbers are. Pure numbers such as 2, π and ½ carry no units and vanish at the first step.

WORKED EXAMPLE

A formula that fails the check

Under pressure a student writes the period of a simple pendulum as $T = 2\pi\sqrt{g/l}$ , with l the length of the pendulum. Decide from base units alone whether it can be right.

The left-hand side is a time, so it has to come out in s. On the right, 2π contributes nothing, so everything hangs on what is inside the root.

g is in m s⁻² and l is in m, so g/l has base units (m s⁻²)/m = s⁻². Take the square root and the right-hand side is in s⁻¹.

s⁻¹ against s, so the formula is not homogeneous and cannot be right. Turn the fraction over: l/g gives m/(m s⁻²) = s², whose root is s. The correct version is $T = 2\pi\sqrt{l/g}$ , and the check found it with no derivation and no booklet.

A homogeneity check is a filter, not a proof. It catches an inverted fraction, a wrong power and a quantity in the wrong place, and it is blind to anything without units. $E_{k} = mv^{2}$ is perfectly homogeneous and still wrong, because the missing ½ leaves no trace. Failing the check proves an equation wrong; passing it only means you may carry on.

GUIDED PRACTICE

A pressure, checked

Pressure has base units kg m⁻¹ s⁻². Show that ρgh, with ρ a density, h a depth and g the gravitational field strength, has the same, and say what that does and does not establish.

Show the working

Density is kg m⁻³, g is m s⁻² and h is m. Multiplying gives kg m⁻³ × m s⁻² × m = kg m⁻¹ s⁻².

The two match, so ρgh is homogeneous with a pressure. That does not prove the formula, because a stray numerical factor would survive the check untouched. It does rule out every rearrangement that fails, and ρg/h and ρh/g both fail.

Prefixes and standard form has moved to the next part of this lesson.

Labelling columns and axes has moved to the next part of this lesson.

Converting between units of the same quantity has moved to the next part of this lesson.

ASSESSMENT FOCUS

Run a half-remembered or freshly rearranged formula past its base units before you trust it. A mismatch proves the formula wrong; a match only licenses you to continue, because a missing ½ or 2π leaves the units untouched.
Compare every term that is added to another, not only the two sides. Adding a length to a time is meaningless whatever the numbers are, and it is the error a two-sided check walks past.
Write the unit on the answer line whenever the quantity has one: a numerical answer without its unit is incomplete. The exceptions are the ratios, strain and efficiency and refractive index among them, which are pure numbers and take no unit.
Amount of substance is a base quantity like any other. A question asking for the base units of the molar gas constant expects mol to appear in the answer.

CHECK YOURSELF

The drag force on a sphere moving slowly through a fluid is written $F = 6\pi\eta r v$ , with r a radius and v a speed. (a) Determine the SI base units of the viscosity $\eta$ . (b) State what the check would have proved had the units failed to match, and what it does not prove now that they do.

Show a hint

Put every symbol into base units, then make the two sides balance.

Show the answer

(a) Force is kg m s⁻², radius is m and speed is m s⁻¹, and $6\pi$ is a pure number that carries nothing. So kg m s⁻² = [ $\eta$ ] × m × m s⁻¹, which leaves [ $\eta$ ] = kg m⁻¹ s⁻¹.

(b) A mismatch would have proved the equation wrong. The match proves only that nothing is inconsistent: a missing $6\pi$ , or a 2 in the wrong place, would leave every unit exactly where it is.

That asymmetry is the whole value of the check. It is a cheap way to reject a rearrangement, and no evidence at all that one is right.

Every unit in the subject is built from the six base units.

A failed homogeneity check proves an equation wrong; a passed one proves nothing.

WORKBOOK

Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.

9 questions on this topicAnswer them one at a time and mark yourself against the mark scheme.Practise this topic

Or read them with their mark schemes on the SI base and derived units questions page.

7 flashcards on this topicDefinitions, off-sheet equations and a spot-the-error card, scheduled by spaced repetition in your browser.Revise with flashcards

WHERE TO GO NEXT

Standard form and prefixes is the maths this lesson leans on, worked through step by step.
Significant figures, units and checking is the maths this lesson leans on, worked through step by step.

USED LATER IN

CHECK YOUR PROGRESS

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Name the six SI base quantities used at A-level, and unpack any derived unit back into them.
Check an equation for homogeneity by reducing every term to base units, and say what a failed check proves and what a passed one does not.

Open the full revision checklist to track your progress across the whole unit.

SI base and derived units

Six units underneath everything

Derived units

Checking an equation for homogeneity

Question workbook

Question workbook

Answer book