OCR Specification focus:
‘Check that physical equations are homogeneous by comparing S.I. base-unit dimensions.’
Dimensional analysis is a fundamental method in physics used to ensure equations are consistent and physically meaningful by comparing their S.I. base-unit dimensions for homogeneity.
The Concept of Dimensional Analysis
Dimensional analysis is the process of expressing physical quantities in terms of their base units to verify the consistency and validity of equations. Each physical quantity can be represented by its dimension, which shows how it depends on the S.I. base quantities: mass (kg), length (m), time (s), electric current (A), temperature (K), and amount of substance (mol).
Representation of the seven S.I. base units as clearly labelled circles. This visual anchors dimensional analysis to its foundational quantities (kg, m, s, A, K, mol, cd). It avoids derived relationships, keeping the focus precisely on base units, as required by the syllabus. Source.
Dimensional analysis is crucial for ensuring that both sides of an equation represent the same type of quantity. If they do not, the equation cannot be physically correct.
Seven S.I. base units with their defining constants (BIPM graphic). The constant references are extra detail beyond the syllabus; students should focus on recognising the base-unit set used to test dimensional consistency. The SVG is crisp at any size for web or print. Source.
Dimensional Analysis: The method of expressing physical quantities in terms of base-unit dimensions to verify equation consistency and derive relationships between quantities.
Understanding Homogeneity of Equations
An equation is said to be homogeneous if the dimensions of all terms on both sides are identical. This is a necessary condition for an equation to be physically valid.
For instance, if one side of an equation represents energy, its dimension must match that of energy on the other side. If they differ, the equation cannot describe a real physical relationship.
Homogeneity: The property of a physical equation where all terms have the same dimensional form, ensuring physical validity.
Importance of Dimensional Consistency
Dimensional homogeneity guarantees that physical equations make sense regardless of the system of units used. It ensures that:
Equations remain valid when expressed in different units (e.g. metres or kilometres).
Physical quantities combined mathematically are of compatible types.
Derived equations and relationships can be trusted to represent real-world behaviour.
Homogeneity is therefore a necessary but not always sufficient condition for correctness—an equation might be dimensionally valid yet still numerically incorrect due to missing constants or coefficients.
Dimensional Representation of Physical Quantities
Each physical quantity can be expressed in terms of base dimensions. The general notation uses M for mass, L for length, and T for time. Quantities involving other base units may also use A, K, or mol when appropriate.
For example:
Velocity has dimensions L T⁻¹.
Acceleration has dimensions L T⁻².
Force has dimensions M L T⁻² (from Newton’s Second Law).
These dimensional forms allow any complex relationship in physics to be broken down and analysed for consistency.
EQUATION
—-----------------------------------------------------------------
Force (F) = mass (m) × acceleration (a)
F = m a
F = M L T⁻²
m = M (kilogram)
a = L T⁻² (metre per second squared)
—-----------------------------------------------------------------
By analysing the equation above, both sides share the same dimensional form, confirming it is homogeneous.
The Process of Checking Equation Homogeneity
To check if an equation is dimensionally homogeneous, follow this systematic method:
Write down the equation clearly.
Express each quantity in terms of its base-unit dimensions.
Simplify the dimensions on both sides of the equation.
Compare the dimensional form of each side.
If they match, the equation is homogeneous.
If they do not, the equation is not physically valid.
This method can be applied to verify standard formulas or to test the validity of newly derived ones.
Common Applications of Dimensional Analysis
Dimensional analysis serves several practical purposes in physics:
Checking derived equations: Ensures that rearranged or derived relationships retain dimensional consistency.
Converting between units: Facilitates conversion by comparing equivalent dimensional forms.
Deriving relationships: Helps deduce possible forms of equations when experimental data or proportional relationships are known.
Estimating unknown quantities: Provides approximate relationships between quantities when constants are unknown.
Derived Unit: A unit expressed as a combination of base units according to the physical relationship between quantities.
Recognising Dimensional Errors
Dimensional analysis can identify errors such as:
Addition of incompatible quantities: For example, adding a length (metres) to a time (seconds) is meaningless.
Incorrect formula rearrangement: Misapplied algebra leading to inconsistent units.
Incorrect use of constants: Omitting or misplacing dimensioned constants such as the gravitational constant G or Planck’s constant h.
By confirming that the dimensions balance, physicists can avoid these fundamental mistakes before or after performing experiments.
Dimensional Equations and Derived Quantities
Many derived quantities can be expressed through dimensional equations that connect them to base quantities.
For instance:
EQUATION
—-----------------------------------------------------------------
Energy (E) = Force (F) × distance (d)
E = M L T⁻² × L = M L² T⁻²
E = Energy (joule, J)
F = Force (newton, N)
d = Distance (metre, m)
—-----------------------------------------------------------------
This dimensional form confirms that energy and work share the same dimensions, showing the consistency of related physical laws.
Dimensional Analysis in Deriving Relationships
When an exact equation is unknown, dimensional analysis can suggest its functional form. For example, by assuming proportionalities between quantities and equating their dimensions, one can determine how they relate, up to a dimensionless constant.
However, dimensional analysis cannot determine such constants’ values, as these arise from experimental or theoretical detail beyond dimensional reasoning.
Dimensionless Constant: A numerical factor without units, determined empirically or theoretically, that cannot be found through dimensional analysis.
Limitations of Dimensional Analysis
While powerful, dimensional analysis has several limitations:
It cannot predict dimensionless quantities, such as refractive index or angles measured in radians.
It cannot provide numerical coefficients (e.g. ½ in kinetic energy formula).
It assumes relationships are multiplicative, not additive, which may not apply to complex or empirical formulas.
It cannot distinguish between quantities with the same dimensions but different meanings (e.g. torque and energy both have M L² T⁻²).
Despite these limitations, dimensional analysis remains a vital tool in ensuring that physics equations conform to fundamental principles of measurement and consistency.
FAQ
Two quantities can share the same dimensions yet describe entirely different physical phenomena. For example, torque and energy both have the dimensional form M L² T⁻², but torque involves a rotational effect (force × perpendicular distance) while energy represents work done.
Dimensional analysis cannot distinguish their nature — it only checks mathematical consistency, not conceptual meaning. Context and definitions within a physical law are essential to interpret such cases correctly.
No — dimensional analysis can only suggest the form of an equation, not establish new laws. It reveals how quantities might relate by comparing their dimensional dependencies.
To derive a new law, you would need:
Empirical evidence or experimental data.
Theoretical reasoning based on physical principles (e.g. conservation laws).
Dimensional analysis acts as a consistency check or a guide, not a discovery tool.
Dimensions express the nature of a quantity, not the scale chosen to measure it. Whether you use metres or kilometres, the underlying dimensional structure — such as L for length — remains constant.
Because base-unit dimensions (M, L, T, etc.) are universal, dimensional analysis ensures an equation’s validity in any coherent unit system, including both S.I. and derived systems like cgs or imperial units.
When analysing experimental results, dimensional analysis helps verify whether proposed relationships are realistic.
You can:
Check if all terms share the same base dimensions.
Identify if any constant or variable is missing or incorrectly expressed.
Detect errors from incorrect unit conversions or power use (e.g. using t instead of t²).
If dimensions do not balance, the relationship cannot represent a valid physical law, even if the numerical data appear to fit.
Dimensionless quantities, such as the Reynolds number or coefficient of friction, have no physical dimensions and remain unchanged across unit systems.
They often compare effects — for instance, inertial vs. viscous forces — rather than represent measurable values.
In dimensional analysis, these act as scaling factors or constants, helping describe similarities between systems without altering dimensional balance. They ensure that physical equations can be applied universally, across different magnitudes and conditions.
Practice Questions
Question 1 (2 marks)
State what is meant by a homogeneous equation in physics.
Explain why checking the homogeneity of an equation is important when performing dimensional analysis.
Mark Scheme
1 mark: Correctly defines homogeneous equation as one in which all terms have the same base-unit dimensions.
1 mark: Explains that homogeneity ensures equations are physically meaningful or dimensionally consistent (e.g. both sides represent the same type of quantity).
Question 2 (5 marks)
A student suggests that the equation
v = u + ½at²
describes the relationship between velocity (v), initial velocity (u), acceleration (a), and time (t).
(a) Use dimensional analysis to check whether this equation is homogeneous.
(b) State whether the equation is physically valid based on your analysis, and explain your reasoning.
Mark Scheme
1 mark: Identifies correct dimensions for each quantity:
v and u = L T⁻¹; a = L T⁻²; t = T.1 mark: Substitutes into the equation and expresses dimensions:
Left-hand side (LHS): L T⁻¹; Right-hand side (RHS): L T⁻¹ + L T⁻² × T² = L T⁻¹ + L.1 mark: Shows that the dimensions of the second term (½at²) simplify to L, not L T⁻¹.
1 mark: States that LHS and RHS do not have the same dimensions.
1 mark: Concludes that the equation is not homogeneous and therefore cannot be correct as written.
