use SI base units to check the homogeneity of physical equations

SI Units and the Homogeneity of Physical Equations (Cambridge International AS & A Level Physics 9702)

1. Why Dimensional Homogeneity Matters

  • Every physically valid equation must be dimensionally homogeneous: each term on both sides carries exactly the same combination of SI base units.

  • Checking homogeneity is a quick diagnostic for algebraic or conceptual mistakes, for verifying derived formulae, and for confirming that experimental relations are used correctly.

  • It underpins error‑propagation – the way uncertainties combine follows the same algebraic rules as the exponents of the units.

2. Syllabus Outcomes Linked to This Topic

OutcomeRelevant Syllabus BlockWhat You’ll Use Here
1.1 – Identify and use SI unitsPhysical quantities & unitsTables 2 & 3, dimensional symbols
1.2 – Apply dimensional analysisAll blocks (kinematics, dynamics, …)Procedure (section 8) and worked examples
1.3 – Propagate uncertaintiesUncertainty & error analysisSection 7
1.4 – Use vectors correctlyVectors (topic 1.4)Section 6
2–10 – Apply equations in each physics topicKinematics, dynamics, …, particle physicsExpanded “Core equations” table (section 5)

3. The Seven SI Base Units

QuantitySymbolSI Base UnitUnit Symbol
Lengthℓmetrem
Massmkilogramkg
Timetseconds
Electric currentIampereA
Thermodynamic temperatureTkelvinK
Amount of substancenmolemol
Luminous intensityIvcandelacd

4. Frequently Used Derived Units (full syllabus list)

QuantitySymbolDerived Unit (SI)Unit SymbolBase‑unit expression
Velocityvmetre per secondm s⁻¹m s⁻¹
Accelerationametre per second squaredm s⁻²m s⁻²
ForceFnewtonNkg m s⁻²
PressureppascalPakg m⁻¹ s⁻²
Densityρkilogram per cubic metrekg m⁻³kg m⁻³
Energy / WorkE, WjouleJkg m² s⁻²
PowerPwattWkg m² s⁻³
Electric chargeQcoulombCA s
VoltageVvoltVkg m² s⁻³ A⁻¹
ResistanceRohmΩkg m² s⁻³ A⁻²
CapacitanceCfaradFkg⁻¹ m⁻² s⁴ A²
Magnetic fluxΦweberWbkg m² s⁻² A⁻¹
Magnetic flux densityBteslaTkg s⁻² A⁻¹
InductanceLhenryHkg m² s⁻² A⁻²
FrequencyfhertzHzs⁻¹
Angular frequencyωradian per secondrad s⁻¹s⁻¹
Momentumpkilogram metre per secondkg m s⁻¹kg m s⁻¹
ImpulseJnewton secondN skg m s⁻¹
StressσpascalPakg m⁻¹ s⁻²
Strainεdimensionless––
Intensity (light)Iluxlxcd sr m⁻²

5. Dimensional Symbols (for analysis)

When performing dimensional analysis we replace each physical quantity by its dimensional symbol:

  • Length → [L]

  • Mass → [M]

  • Time → [T]

  • Electric current → [I]

  • Temperature → [Θ]

  • Amount of substance → [N]

  • Luminous intensity → [J]

6. Procedure for Checking Homogeneity

  1. Write the equation in symbolic form (include any constants).

  2. Replace each quantity with its dimensional symbol or base‑unit expression.

  3. Apply the algebraic rules:

    • [AB] = [A][B]

    • [A/B] = [A][B]⁻¹

    • [Aⁿ] = [A]ⁿ

  4. Simplify both sides; compare the resulting dimensional products.

  5. If they differ, locate the missing factor, sign error, or inappropriate constant.

  6. For derivatives, remember d/dt adds [T]⁻¹; for integrals, ∫dt adds [T].

7. Core Syllabus Equations – Dimensional Check

TopicEquationDimensional FormHomogeneous?
Kinematicsv = u + at[L][T]⁻¹ = [L][T]⁻¹ + [L][T]⁻²·[T]Yes
s = ut + ½at²[L] = [L][T] + [L][T]⁻²·[T]²Yes
a = (v – u)/t[L][T]⁻² = ([L][T]⁻¹)/[T]Yes
DynamicsF = ma[M][L][T]⁻² = [M]·[L][T]⁻²Yes
p = mv (momentum)[M][L][T]⁻¹ = [M]·[L][T]⁻¹Yes
J = FΔt (impulse)[M][L][T]⁻¹ = [M][L][T]⁻²·[T]Yes
Δp = FΔt (impulse‑momentum theorem)[M][L][T]⁻¹ = [M][L][T]⁻²·[T]Yes
Forces, Density & PressureWeight: W = mg[M][L][T]⁻² = [M]·[L][T]⁻²Yes
Pressure: p = F/A[M][L]⁻¹[T]⁻² = [M][L][T]⁻²·[L]⁻²Yes
Density: ρ = m/V[M][L]⁻³ = [M]·[L]⁻³Yes
Work, Energy & PowerW = Fd cosθ[M][L]²[T]⁻² = [M][L][T]⁻²·[L]Yes
Gravitational PE: E_g = mgh[M][L]²[T]⁻² = [M]·[L][T]⁻²·[L]Yes
Kinetic E: E_k = ½mv²[M][L]²[T]⁻² = [M]·([L][T]⁻¹)²Yes
Power: P = W/t = Fv[M][L]²[T]⁻³ = [M][L][T]⁻²·[L][T]⁻¹Yes
Efficiency: η = (useful E / input E) ×100 %dimensionless (no units)Yes
Deformation of SolidsHooke’s law: F = kx[M][L][T]⁻² = [M][T]⁻²·[L]Yes
Stress: σ = F/A[M][L]⁻¹[T]⁻² = [M][L][T]⁻²·[L]⁻²Yes
Strain: ε = ΔL/LdimensionlessYes
WavesWave speed: v = fλ[L][T]⁻¹ = [T]⁻¹·[L]Yes
Angular frequency: ω = 2πf[T]⁻¹ = [T]⁻¹Yes
Wave equation (1‑D): ∂²y/∂t² = v²∂²y/∂x²[L][T]⁻² = [L]²[T]⁻²·[L]⁻²Yes
Intensity–Amplitude: I = ½ρvω²A²[M][T]⁻³ = [M][L]⁻³·[L][T]⁻¹·[T]⁻²·[L]²Yes
SuperpositionAllowed wavelengths: λ = 2L/n[L] = [L]/(dimensionless)Yes
Fundamental frequency of a string: f₁ = v/2L[T]⁻¹ = ([L][T]⁻¹)/[L]Yes
Electricity & DC CircuitsOhm’s law: V = IR[M][L]²[T]⁻³[I]⁻¹ = [I]·[M][L]²[T]⁻³[I]⁻²Yes
Power: P = IV[M][L]²[T]⁻³ = [I]·[M][L]²[T]⁻³[I]⁻¹Yes
Resistance of a uniform wire: R = ρℓ/A[M][L]²[T]⁻³[I]⁻² = [M][L]³[T]⁻³[I]⁻²·[L]·[L]⁻²Yes
Capacitance: C = Q/V[I]²[T]⁴[M]⁻¹[L]⁻² = [I][T]/[M][L]²[T]⁻³[I]⁻¹Yes
Time constant: τ = RC[T] = [M][L]²[T]⁻³[I]⁻²·[M]⁻¹[L]⁻²[T]⁴[I]²Yes
MagnetismMagnetic flux: Φ = BA[M][L]²[T]⁻²[I]⁻¹ = [M][T]⁻²[I]⁻¹·[L]²Yes
Force on a current‑carrying conductor: F = BIL sinθ[M][L][T]⁻² = [M][T]⁻²[I]⁻¹·[I]·[L]Yes
Induced emf (Faraday): ε = –dΦ/dt[M][L]²[T]⁻³[I]⁻¹ = [M][L]²[T]⁻²[I]⁻¹·[T]⁻¹Yes
Particle PhysicsE = mc²[M][L]²[T]⁻² = [M]·([L][T]⁻¹)²Yes
Relativistic momentum: p = E/c[M][L][T]⁻¹ = [M][L]²[T]⁻²·[L]⁻¹[T]Yes

8. Vector Refresher (Syllabus 1.4)

  • Notation: \(\mathbf{A}=A{x}\hat{i}+A{y}\hat{j}+A_{z}\hat{k}\).

  • Magnitude: \(|\mathbf{A}|=\sqrt{A{x}^{2}+A{y}^{2}+A_{z}^{2}}\).

  • Addition: \(\mathbf{A}+\mathbf{B}= (A{x}+B{x})\hat{i}+ (A{y}+B{y})\hat{j}+ (A{z}+B{z})\hat{k}\).

  • Scalar (dot) product: \(\mathbf{A}\!\cdot\!\mathbf{B}=AB\cos\theta\) – units are the product of the units of \(\mathbf{A}\) and \(\mathbf{B}\).

  • Vector (cross) product: \(\mathbf{A}\!\times\!\mathbf{B}=AB\sin\theta\,\hat{n}\) – same units as the dot product, direction given by the right‑hand rule.

  • When checking homogeneity, treat the vector as a whole; only the magnitude’s units matter (e.g. \(\mathbf{F}\) has units of newtons).

9. Errors, Uncertainties and Units

In experimental work (Paper 3 & 5) the propagation of uncertainties follows the same algebraic rules as units:

  • If \(Z = aX^{p}Y^{q}\) (where \(a\) is a pure number), then

    \[

    \frac{\Delta Z}{Z}=|p|\frac{\Delta X}{X}+|q|\frac{\Delta Y}{Y}.

    \]

  • Multiplication or division adds the exponents of the units; the same exponents appear in the uncertainty formula.

  • Always keep track of significant figures after the dimensional check – a homogeneous equation does not guarantee numerical correctness.

10. Worked Examples

Example 1 – Newton’s Second Law

Equation: \(F = ma\)

\[

[F]=[M][L][T]^{-2},\qquad [m]=[M],\qquad [a]=[L][T]^{-2}

\]

Since \([M][L][T]^{-2}= [M]\times[L][T]^{-2}\), the equation is homogeneous.

Example 2 – Kinetic Energy

Equation: \(E_{k}= \tfrac12 mv^{2}\)

\[

[E_{k}]=[M][L]^{2}[T]^{-2},\qquad [m]=[M],\qquad [v]^{2}=([L][T]^{-1})^{2}=[L]^{2}[T]^{-2}

\]

Both sides give \([M][L]^{2}[T]^{-2}\); the equation is dimensionally consistent.

Example 3 – Wave Speed – Common Mistake

Incorrect proposal: \(v = \dfrac{f}{\lambda}\)

\[

[v]=[L][T]^{-1},\qquad [f]=[T]^{-1},\qquad [\lambda]=[L]

\]

Right‑hand side gives \([T]^{-1}[L]^{-1}\), which is \([L]^{-1}[T]^{-1}\) – the opposite of the required dimensions. The correct form is \(v = f\lambda\).

Example 4 – Time Constant of an RC Circuit

Equation: \(\tau = RC\)

\[

[R]=[M][L]^{2}[T]^{-3}[I]^{-2},\qquad [C]=[M]^{-1}[L]^{-2}[T]^{4}[I]^{2}

\]

\[

[RC]=[M][L]^{2}[T]^{-3}[I]^{-2}\times[M]^{-1}[L]^{-2}[T]^{4}[I]^{2}= [T]

\]

Both sides have units of seconds – the equation is homogeneous.

Example 5 – Hooke’s Law for a Spring

Equation: \(F = kx\)

\[

[F]=[M][L][T]^{-2},\qquad [k]=[M][T]^{-2},\qquad [x]=[L]

\]

\[

[kx]=[M][T]^{-2}\times[L]=[M][L][T]^{-2}

\]

Units match, confirming the correctness of the constant’s dimensions.