SI units, prefixes and homogeneity
Key idea: Units are part of the physics, not decoration added after a calculation. They help you convert data, build derived quantities and reject equations that cannot be correct.
Continue where you stopped
The core idea
H1 Physics 8867 · Lesson 1 of 3
Check your understandingBy the end of this lesson, you should be able to
- Recall the six SI base quantities used in H1 Physics and their units.
- Convert prefixes safely and express derived units in base units.
- Use base units to test whether an equation is homogeneous without claiming that homogeneity proves the model.
Learn the idea
Big question: How can units expose a mistake before any physics calculation is completed?
Start from the six base quantities
The H1 base quantities are mass (kg), length (m), time (s), electric current (A), temperature (K) and amount of substance (mol). Other quantities in this course are built from these, so a named unit such as the newton or joule can always be unpacked when needed.
Know the prefix ladder: pico p (10⁻¹²), nano n (10⁻⁹), micro μ (10⁻⁶), milli m (10⁻³), centi c (10⁻²), deci d (10⁻¹), kilo k (10³), mega M (10⁶), giga G (10⁹) and tera T (10¹²). Upper- and lower-case symbols matter.
Check your understanding: Which symbol means 10⁹ and why must its case be kept?
G means giga, 10⁹. Lower-case g is not the giga prefix and is commonly used for gravitational field strength or gram in context.
Read a unit as a physical recipe
A derived unit tells you how a quantity is built. Since force equals mass times acceleration, N becomes kg m s⁻². Since energy transferred by a force is force times displacement, J becomes kg m² s⁻². Rebuilding a unit this way is safer than memorising an isolated string of symbols.
Powers apply to the prefix and the unit together. A square measuring 3.0 mm by 3.0 mm has area 9.0 mm², which is 9.0 × 10⁻⁶ m²—not 9.0 × 10⁻³ m².
Check your understanding: Why is 1 cm³ equal to 10⁻⁶ m³?
Because 1 cm = 10⁻² m, so cubing the complete conversion gives (10⁻²)³ = 10⁻⁶.
Use homogeneity as a filter, not a proof
Terms joined by addition or subtraction must have identical dimensions. The two sides of an equation must also match. This can reject an impossible equation immediately, but two different physical models can share the same dimensions, so matching units cannot prove which model describes nature.
Check your understanding: The expression v = u + at² has matching numerical-looking terms. What does the unit check show?
u has unit m s⁻¹, but at² has unit (m s⁻²)(s²) = m. Quantities with different dimensions cannot be added, so the equation is impossible.
Key ideas
- Prefix symbols are case-sensitive: m means milli while M means mega.
- Convert the whole measurement, including powers of its unit.
- Compare dimensions term by term, especially when quantities are added.
Relationships to know
1 N = 1 kg m s⁻²1 J = 1 kg m² s⁻²1 W = 1 kg m² s⁻³
Follow the reasoning
Worked example
Find the dimensions of an unknown constant
Question: Air-resistance power is modelled by P = kρAv³, where ρ is density, A is area and v is speed. Determine the dimensions of k and state what the result can establish.
Step 1: Replace each quantity by base dimensions
Why: The unknown can be isolated only after every known quantity uses the same dimensional language.
Working: [P] = M L² T⁻³, [ρ] = M L⁻³, [A] = L² and [v³] = L³ T⁻³.
Step 2: Combine the known factors
Why: Multiplying quantities means adding their powers of M, L and T.
Working: [ρAv³] = (M L⁻³)(L²)(L³ T⁻³) = M L² T⁻³.
Step 3: Compare with the left side
Why: P and ρAv³ already have identical dimensions, so k supplies no remaining dimension.
Working: [k] = [P]/[ρAv³] = 1.
Answer: k is dimensionless. The equation is homogeneous, so it passes this necessary check, but the check does not prove that the drag model or the value of k is correct.
Check: Substituting the result leaves both sides with kg m² s⁻³, the base-unit form of a watt.
Now try it with support
Practise with support
Convert 3.6 cm² to m², then express the pascal, Pa = N m⁻², in SI base units.
Hints
- Convert centimetres before squaring, or square the factor 10⁻².
- Replace N by kg m s⁻² and simplify.
View the guided answer
3.6 cm² = 3.6 × 10⁻⁴ m². Since 1 Pa = 1 N m⁻², 1 Pa = 1 kg m⁻¹ s⁻².
Your turn
Practise independently
A proposed drag-power model is P = CρAr²v³. Determine the dimensions of C and decide whether the expression can represent power.
Check your answer
[ρAr²v³] = (M L⁻³)(L²)(L²)(L³T⁻³) = M L⁴T⁻³. Power is M L²T⁻³, so [C] = L⁻². With that dimension the expression can be homogeneous, but the check cannot establish that it is the correct physical model.
Common mistakes and exam guidance
Watch out for
- Treating cm² as 10⁻² m² instead of 10⁻⁴ m².
- Writing ‘the equation is correct’ after only checking its units.
In an exam
- Show the base-unit substitution before cancelling powers.
- For a homogeneity question, finish with what the check can and cannot establish.
Put the ideas together
Exam-style practice [4 marks]
The period T of a mass–spring system is proposed to be T = C√(m/k), where m is mass, k is spring constant and C is a constant. Use base units to determine whether C has dimensions. State one limitation of your conclusion.
Plan before you answer
- Find the base units of k from F = kx.
- Simplify √(m/k).
- Separate dimensional consistency from physical proof.
View the marking points and model answer
Marking points
- Uses [k] = M T⁻².
- Obtains [m/k] = T².
- Concludes that C is dimensionless because √(m/k) has dimension T.
- States that homogeneity does not prove the formula or determine C.
Model answer
From F = kx, [k] = (M L T⁻²)/L = M T⁻². Hence [m/k] = M/(M T⁻²) = T², so √(m/k) has dimension T. C is dimensionless. This shows that the proposal is homogeneous, but it cannot prove the model or give the numerical value of C.
Finish from memory
Three-question recap
Express one watt in SI base units.
Check
1 W = 1 kg m² s⁻³.
Convert 2.5 cm² to m².
Check
2.5 × 10⁻⁴ m².
What can a failed homogeneity check prove?
Check
It proves that the proposed equation is wrong as written.
Continue with the next resource in this course.
Course and syllabus information
- Course
- GCE A-Level H1 Physics
- Edition
- GCE A-Level H1 Physics 2027