Dimensional Analysis

Fundamental Dimensions and Base Units

Every physical quantity is expressed as a combination of seven fundamental quantities, representing the qualitative nature of the measurement.

$$[Q] = M^a L^b T^c I^d K^e mol^f cd^g$$

The general form of a dimensional formula using the 7 base SI dimensions.

$M$=Mass(kg)

$L$=Length(m)

$T$=Time(s)

$I$=Electric Current(A)

Mechanics

→

Only M, L, and T are typically required.

The Big Three: Mass $[M]$, Length $[L]$, and Time $[T]$ form the core of mechanical dimensional analysis.

Qualitative Nature: Dimensions describe what a quantity is (e.g., a distance), regardless of its unit magnitude (meters vs. miles).

Some physical quantities have no dimensions — their dimensional formula is $M^0 L^0 T^0$. These dimensionless quantities arise as ratios of quantities with identical dimensions.

Common Examples: Strain ($\Delta L / L$), refractive index ($c/v$), relative density, coefficient of friction, and all trigonometric ratios.

Angles: Radian and steradian are dimensionless — they are ratios of lengths ($s/r$) and areas ($A/r^2$) respectively.

Shortcut: If a quantity is defined as a ratio of two quantities with the same dimensions, it is always dimensionless.

Deriving Dimensions for Derived Quantities

Dimensions of derived quantities are obtained by substituting the dimensions of fundamental quantities into their defining algebraic equations.

$$[Force] = [Mass] \times [Acceleration] = M L T^{-2}$$

Force is derived from Newton's Second Law, showing how derived units combine base units.

$F$=Force(newtons (N))

$m$=Mass(kilograms (kg))

$a$=Acceleration(metres per second squared (m/s²))

deriving Pressure units

→

$[P] = [F]/[A] = [MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}] = \text{Pa}$

deriving Energy units

→

$[E] = [F][d] = [MLT^{-2}][L] = [ML^2T^{-2}] = \text{J}$

Square Brackets: Used to denote 'the dimension of' a specific physical variable.

Algebraic Manipulation: Dimensions can be treated as variables in algebraic operations: multiplied, divided, or raised to powers.

Shortcut: Build complex dimensions step by step — velocity is $L/T$, acceleration is $v/T = L/T^2$, force is $m \times a = ML/T^2$.

Common Dimensional Formulas

These are the building blocks for checking and deriving formulas

•

Velocity: $[LT^{-1}]$

•

Acceleration: $[LT^{-2}]$

•

Force: $[MLT^{-2}]$

•

Energy / Work / Torque: $[ML^2T^{-2}]$

•

Power: $[ML^2T^{-3}]$

•

Pressure / Stress: $[ML^{-1}T^{-2}]$

•

Density: $[ML^{-3}]$

•

Momentum: $[MLT^{-1}]$

•

Viscosity (η): $[ML^{-1}T^{-1}]$

•

Gravitational constant (G): $[M^{-1}L^3T^{-2}]$

The Principle of Homogeneity

The Principle of Homogeneity states that only physical quantities with identical dimensions can be added, subtracted, or compared.

$$[LHS] = [RHS]$$

Ensuring equations are physically meaningful by matching dimensions on both sides.

$LHS$=Left Hand Side dimensions(N/A)

$RHS$=Right Hand Side dimensions(N/A)

$A + B = C$

→

Must satisfy $[A] = [B] = [C]$.

Addition Rule: You cannot add 5 meters to 2 seconds; the operation is dimensionally invalid.

Consistency Check: Homogeneity is a necessary, but not sufficient, condition for an equation to be physically correct (it ignores dimensionless constants).

Checking Method: Compute dimensions of every term separately. If any term has different dimensions from the others, the equation is wrong.

Limiting Case Test: Does the formula make sense when $t=0$, $v=0$, or $r \to \infty$?

Applications: Checking and Deriving Relations

Dimensional analysis is used to verify consistency of existing formulas and derive proportionality relations between variables.

$$X \propto a^p b^q c^r$$

The method of undetermined powers for deriving physical laws.

$X$=Dependent variable(varies)

$p, q, r$=Undetermined exponents(dimensionless)

Constants

→

Dimensionless constants (like $2\pi$ or $1/2$) cannot be found using this method.

Checking Correctness: Compare dimensions of all terms in an equation to catch algebraic errors.

Power Method Step-by-Step: (1) Write $X = k \cdot a^p b^q c^r$. (2) Substitute dimensions on both sides. (3) Equate exponents of M, L, T to form simultaneous equations. (4) Solve for $p, q, r$.

Proportionality Shortcut: Once you derive $T \propto \sqrt{l/g}$, doubling $l$ increases $T$ by $\sqrt{2}$; quadrupling $g$ halves $T$ — no need to recalculate from scratch.

Dimensionless Constants Are Invisible: The method yields only the functional form. Constants like $2\pi$, $1/2$, $4/3$ must be found experimentally.

Use Unit Cancellation to quickly eliminate incorrect formulas without full dimensional analysis.

$$\text{Check: Do the units on LHS match RHS?}$$

Quick elimination technique - if units don't match, the formula MUST be wrong.

$\text{LHS units}$=Units of the quantity being calculated(variable)

$\text{RHS units}$=Combined units of all terms on right side(must equal LHS)

checking $v = \sqrt{2gh}$

→

$[v] = \text{m/s}$, RHS: $\sqrt{(\text{m/s}^2)(\text{m})} = \sqrt{\text{m}^2/\text{s}^2} = \text{m/s}$ ✓

checking $F = mv$ (wrong)

→

$[F] = \text{kg m/s}^2$, RHS: $(\text{kg})(\text{m/s}) = \text{kg m/s}$ ✗ Different!

checking $E = mc^2$

→

$[E] = \text{J} = \text{kg m}^2/\text{s}^2$, RHS: $(\text{kg})(\text{m/s})^2 = \text{kg m}^2/\text{s}^2$ ✓

Quick Elimination: In problems with multiple candidate formulas, eliminate 2-3 options just by checking units first.

Dimensionless Quantities: Angle, strain, refractive index, relative density, coefficient of friction have no units.

Limitations of Dimensional Analysis

While powerful, dimensional analysis has inherent limitations that prevent it from being a complete tool for deriving physical laws.

Dimensionless Constants: Cannot determine pure numbers like $2\pi$, $1/2$, or $4/3$ — e.g., the pendulum formula $T = 2\pi\sqrt{l/g}$ gives only $T \propto \sqrt{l/g}$.

Same Dimensions, Different Quantities: Work ($F \cdot d$) and torque ($F \times r$) both have dimensions $[ML^2T^{-2}]$ but are physically different.

Transcendental Functions: Cannot derive formulas involving $\sin$, $\cos$, $\log$, or $e^x$ — their arguments must be dimensionless, but the method cannot predict their presence.

Multiple Variables: If more than 3 unknowns exist (in mechanics), there are not enough equations (M, L, T) to solve for all powers uniquely.

Uncertainty Propagation

When performing calculations with measured values, uncertainties propagate through the result. The rules depend on the mathematical operation involved.

$$\delta R = \sqrt{(\frac{\partial R}{\partial x} \delta x)^2 + (\frac{\partial R}{\partial y} \delta y)^2}$$

General formula for uncertainty propagation (simplified rules below are used in practice).

$\delta R$=Uncertainty in the result(same as R)

$\delta x, \delta y$=Uncertainties in measured quantities(same as x, y)

Addition/Subtraction: Add absolute uncertainties. If $x = a \pm \delta a$ and $y = b \pm \delta b$, then $\delta(x \pm y) = \delta a + \delta b$.

Multiplication/Division: Add percentage uncertainties. For $R = xy$: %uncertainty in $R$ = %uncertainty in $x$ + %uncertainty in $y$.

Power Rule: Multiply percentage uncertainty by the power. For $R = x^n$: %uncertainty in $R$ = $n \times$ %uncertainty in $x$.

Special rules apply for uncertainties in timing experiments and in the average of repeated measurements.

Timing Experiments: Uncertainty in a period is $\delta T = \frac{\text{least count of timer}}{\text{number of vibrations}}$. Counting more vibrations reduces uncertainty.

Example: 30 pendulum swings timed at 54.6 s with a stopwatch (least count 0.1 s) gives $T = 1.82 \pm 0.003$ s, since $\delta T = 0.1/30 = 0.003$ s.

Mean of Repeated Measurements: (1) Calculate the average. (2) Find the absolute deviation of each reading from the average. (3) The mean deviation is the uncertainty in the average.

Example: Six diameter readings 1.20, 1.22, 1.23, 1.19, 1.22, 1.21 mm give mean = 1.21 mm, deviations = 0.01, 0.01, 0.02, 0.02, 0.01, 0, mean deviation = 0.01 mm. Result: $1.21 \pm 0.01$ mm.

Applying uncertainty rules to real measurements requires careful attention to which rule applies at each step of the calculation.

Step 1: Identify whether the operation is addition/subtraction or multiplication/division.

Step 2: For addition/subtraction, compute absolute uncertainties. For multiplication/division, convert to percentage uncertainties first.

Step 3: Combine the uncertainties using the appropriate rule, then convert back to absolute uncertainty for the final answer.

Significant Figures: Uncertainty is quoted to 1 significant figure, and the result is rounded to match the decimal place of the uncertainty.

Combined Operations: When a formula involves both powers and products (e.g., $V = \frac{\pi d^2 l}{4}$), apply the power rule first ($2 \times \%\delta d$), then add percentage uncertainties of all factors.