Dimensional Equations

A dimensional equation is a mathematical statement that expresses a physical quantity in terms of the fundamental dimensions of the SI system: mass [M][M], length [L][L], time [T][T], electric current [A][A], temperature [Θ][\Theta], amount of substance [N][N], and luminous intensity [J][J]. Rather than working with specific units like metres or kilograms, dimensional equations strip every quantity down to its most basic building blocks, allowing physicists to check whether a formula is physically meaningful before a single measurement is taken. The power of dimensional equations lies in the principle of dimensional homogeneity: every term on both sides of a valid physical equation must have identical dimensions. If you derive a formula and one side has dimensions of [MLT−2][MLT^{-2}] while the other has [ML2T−2][ML^{2}T^{-2}], you know immediately that an error has been made, without needing experimental data. This acts as a powerful error-checking tool and is used constantly by professional physicists and engineers during derivation work. Dimensional equations also enable a technique called dimensional analysis, which can reveal the functional form of unknown relationships. By assuming a quantity depends on several others and demanding dimensional consistency, students can derive well-known formulae — such as the period of a pendulum or the speed of waves — purely from the logic of dimensions. This article focuses on how dimensional equations are written, interpreted, and applied.

Key takeaways
  • Identify the Fundamental Dimensions
  • Write the Dimensional Formula
  • Apply the Principle of Dimensional Homogeneity
  • Use Dimensional Equations to Derive Relationships
  • Recognise the Limitations

Identify the Fundamental Dimensions

Every physical quantity can be expressed using the seven SI base dimensions. The most common in mechanics are mass [M][M], length [L][L], and time [T][T]. For example, velocity has dimension [LT−1][LT^{-1}] because it is displacement (length) divided by time. Acceleration is [LT−2][LT^{-2}]. Always begin by listing the dimensions of every quantity involved.

Write the Dimensional Formula

A dimensional formula expresses a derived quantity as a product of powers of base dimensions: [Q]=MaLbTc[Q] = M^a L^b T^c. For force, Newton's second law gives F=maF = ma, so [F]=[M][LT−2]=MLT−2[F] = [M][LT^{-2}] = MLT^{-2}. For energy (work = force × distance): [E]=[MLT−2][L]=ML2T−2[E] = [MLT^{-2}][L] = ML^2T^{-2}. For pressure (force per area): [P]=MLT−2L2=ML−1T−2[P] = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2}.

Apply the Principle of Dimensional Homogeneity

For an equation A=B+CA = B + C to be physically valid, [A]=[B]=[C][A] = [B] = [C]. You cannot add a force to an energy — they have different dimensions. Consider the kinematic equation v2=u2+2asv^2 = u^2 + 2as. Check: [v2]=L2T−2[v^2] = L^2T^{-2}, [u2]=L2T−2[u^2] = L^2T^{-2}, [2as]=[LT−2][L]=L2T−2[2as] = [LT^{-2}][L] = L^2T^{-2}. All three terms match, confirming dimensional consistency.

Use Dimensional Equations to Derive Relationships

Suppose the period TpT_p of a simple pendulum depends on its length ll, mass mm, and gravitational acceleration gg. Assume Tp=k lambgcT_p = k \, l^a m^b g^c where kk is a dimensionless constant. Writing dimensions: [T]=[L]a[M]b[LT−2]c=MbLa+cT−2c[T] = [L]^a [M]^b [LT^{-2}]^c = M^b L^{a+c} T^{-2c}. Comparing exponents: for TT: 1=−2c⇒c=−121 = -2c \Rightarrow c = -\tfrac{1}{2}; for MM: 0=b⇒b=00 = b \Rightarrow b = 0; for LL: 0=a+c⇒a=120 = a + c \Rightarrow a = \tfrac{1}{2}. Therefore Tp∝l/gT_p \propto \sqrt{l/g}, which is exactly the correct result.

Recognise the Limitations

Dimensional analysis cannot determine dimensionless constants like 2π2\pi in Tp=2πl/gT_p = 2\pi\sqrt{l/g}. It also cannot distinguish between quantities with the same dimensions, such as torque and energy (both ML2T−2ML^2T^{-2}), and it fails when a relationship involves dimensionless ratios like angles (in radians) or logarithmic functions. These limitations mean dimensional equations are a consistency check, not a replacement for full derivation or experiment.

The Key Equation

$$[F] = M^1 L^1 T^{-2}
- **[F]** — Dimension of force (kg m s⁻² (newton)): The dimensional formula for force, derived from Newton's second law $F = ma$. - **M** — Mass dimension (kg): The base dimension representing the amount of matter; exponent is 1 for force. - **L** — Length dimension (m): The base dimension representing spatial extent; exponent is 1 for force. - **T^{-2}** — Time dimension (s): The base dimension representing duration; force involves time to the power $-2$ because acceleration is $\Delta v / \Delta t$. ## Worked Example **Problem.** Use dimensional analysis to find how the speed $v$ of a wave on a stretched string depends on the tension $F_T$ (SI unit: N) and the linear mass density $\mu$ (mass per unit length, SI unit: kg m⁻¹). Assume $v = k F_T^a \mu^b$ where $k$ is a dimensionless constant.
v=FTμv = \sqrt{\dfrac{F_T}{\mu}}
∗∗Answer.∗∗Thewavespeedonastringisproportionaltothesquarerootoftensiondividedbylinearmassdensity.**Answer.** The wave speed on a string is proportional to the square root of tension divided by linear mass density.
Figure
Diagram
Dimensional Equations: Homogeneity CheckEquation under test: $v^2 = u^2 + 2as$(Kinematic equation: final velocity squared)Term 1v²[L²T⁻²]Term 2u²[L²T⁻²]Term 32as[LT⁻²][L]=[L²T⁻²]All dimensions match!Equation is dimensionally consistentDimensional homogeneity is a necessary (not sufficient) condition for a correct equation.
Origins · 1 minJoseph Fourier and how this was discovered
JF
Joseph Fourier
1768–1830 · French

Joseph Fourier is widely credited with being the first physicist to formalise the principle of dimensional homogeneity. In his 1822 masterwork 'Théorie Analytique de la Chaleur' (The Analytical Theory of Heat), Fourier explicitly stated that every term in a physical equation must have the same dimensions, and he systematically assigned dimensional exponents to physical quantities. This foundational insight transformed dimensional reasoning from an informal habit into a rigorous mathematical tool, laying the groundwork for modern dimensional analysis. Fourier spent years on Napoleon's Egyptian campaign and became so fascinated by the desert heat that it inspired his life's work on heat conduction — ultimately leading him to the dimensional methods that physicists still use today.

  • SI Base Units — Dimensional equations are written using the seven SI base dimensions; understanding the base units is a prerequisite for constructing any dimensional formula.
  • Buckingham Pi Theorem — The Buckingham Pi theorem is the formal generalisation of dimensional equations, providing a systematic method for reducing a set of dimensional variables into a smaller set of dimensionless groups.
  • Significant Figures and Measurement Error — Dimensional consistency ensures an equation is meaningful; significant figures and error analysis ensure the numerical values substituted into that equation are reliable.
  • Newton's Second Law — Newton's second law F=maF = ma provides the most commonly used example for deriving dimensional formulas, and it underpins the dimensional equations for force, energy, pressure, and momentum.