Variables, Units &
Dimensional Analysis
Exam-ready reference with crystal-clear formulas, definitions, worked examples, and full dimensional formulae — covering every key concept.
Variables & Their Types
Every scientific study involves at least two variables — one intentionally changed and one measured. Understanding variable types is fundamental to designing valid experiments and drawing reliable conclusions.
Types of Variables
Independent → (causes change in) → Dependent, while all Control Variables remain constant. This is the foundation of the Scientific Method.
Importance of Control Variables
Why Control Variables Are Critical
Eliminates Confounding Factors
Without control variables, multiple factors could change simultaneously, making it impossible to identify which factor caused the observed result. Control variables isolate the effect of the independent variable.
Ensures Experimental Validity
A valid experiment measures what it intends to measure. Holding control variables constant ensures the design is sound and results are not contaminated by extraneous factors.
Enables Reproducibility
Scientific experiments must be reproducible. When control variables are documented and maintained, other researchers can replicate the experiment and verify results — the hallmark of good science.
Allows Fair Comparison
When conducting multiple trials, control variables ensure all conditions (except the independent variable) are identical — making comparisons between trials fair and meaningful.
Improves Accuracy of Results
By minimizing uncontrolled variation, control variables reduce experimental error and increase precision, leading to more reliable conclusions.
Real-World Example: Ohm’s Law Experiment
Voltage (V) — varied by adjusting the power supply across different trials.
Current (I) — measured with an ammeter as voltage changes.
Resistance, Temperature, Wire material, Length — all held constant.
Beginners confuse control variable with control group. A control group doesn’t receive the experimental treatment; a control variable is any quantity kept constant throughout the experiment.
Fundamental vs Derived Units
Measurement is the backbone of physics. To communicate measurements consistently worldwide, scientists use standardized units classified into two broad categories: Fundamental (Base) Units and Derived Units.
- Defined independently — not derived from other units
- Internationally agreed upon by the BIPM
- Exactly 7 base quantities in the SI system
- Form the foundation of all physical measurement
- Examples: metre (m), kilogram (kg), second (s)
- Derived mathematically from base units
- Can have special names (Newton, Joule, Pascal)
- Infinite possible derived units exist
- Change if the underlying base units change
- Examples: m/s, N, J, Pa, W, V, Ω, Hz
The 7 SI Base (Fundamental) Units
| # | Quantity | SI Unit Name | Symbol | Dimension | Modern Definition |
|---|---|---|---|---|---|
| 1 | Length | Metre | m | [L] | Distance light travels in 1/299,792,458 seconds |
| 2 | Mass | Kilogram | kg | [M] | Based on Planck constant h = 6.626×10⁻³⁴ J·s |
| 3 | Time | Second | s | [T] | 9,192,631,770 oscillations of caesium-133 atom |
| 4 | Electric Current | Ampere | A | [I] | Based on elementary charge e = 1.602×10⁻¹⁹ C |
| 5 | Temperature | Kelvin | K | [Θ] | Absolute zero = 0 K; based on Boltzmann constant |
| 6 | Amount of Substance | Mole | mol | [N] | 6.022×10²³ elementary entities (Avogadro’s number) |
| 7 | Luminous Intensity | Candela | cd | [J] | Based on luminous efficacy at 540 THz |
Common Derived Units with Formulas
| Quantity | Unit Name | Symbol | Base Unit Expression | Formula |
|---|---|---|---|---|
| Velocity / Speed | — | m/s | m · s⁻¹ | v = d/t |
| Acceleration | — | m/s² | m · s⁻² | a = Δv/Δt |
| Force | Newton | N | kg · m · s⁻² | F = ma |
| Work / Energy | Joule | J | kg · m² · s⁻² | W = F · d |
| Power | Watt | W | kg · m² · s⁻³ | P = W/t |
| Pressure | Pascal | Pa | kg · m⁻¹ · s⁻² | P = F/A |
| Electric Charge | Coulomb | C | A · s | Q = I · t |
| Voltage (EMF) | Volt | V | kg · m² · s⁻³ · A⁻¹ | V = W/Q |
| Resistance | Ohm | Ω | kg · m² · s⁻³ · A⁻² | R = V/I |
| Frequency | Hertz | Hz | s⁻¹ | f = 1/T |
| Momentum | — | kg·m/s | kg · m · s⁻¹ | p = mv |
| Density | — | kg/m³ | kg · m⁻³ | ρ = m/V |
Fundamental units are the root — defined by physical constants and irreducible. Derived units are branches — built from fundamental units using mathematical operations defined by physical laws.
Dimensional Analysis
The “dimension” of a quantity refers to the type of fundamental quantity it represents — not its numerical value. Dimensional analysis lets us check equation consistency, derive relationships, and convert between unit systems.
The 7 Fundamental Dimensional Symbols
Dimensional Formulae of Common Quantities
| Physical Quantity | Derived From | Dimensional Formula | SI Unit |
|---|---|---|---|
| Velocity | Length / Time | [M⁰ L¹ T⁻¹] | m/s |
| Acceleration | Velocity / Time | [M⁰ L¹ T⁻²] | m/s² |
| Force | Mass × Acceleration | [M¹ L¹ T⁻²] | N |
| Work / Energy | Force × Distance | [M¹ L² T⁻²] | J |
| Power | Work / Time | [M¹ L² T⁻³] | W |
| Momentum | Mass × Velocity | [M¹ L¹ T⁻¹] | kg·m/s |
| Pressure | Force / Area | [M¹ L⁻¹ T⁻²] | Pa |
| Density | Mass / Volume | [M¹ L⁻³ T⁰] | kg/m³ |
| Frequency | 1 / Time | [M⁰ L⁰ T⁻¹] | Hz |
| Gravitational Constant G | F·r²/(m₁·m₂) | [M⁻¹ L³ T⁻²] | N·m²/kg² |
| Surface Tension | Force / Length | [M¹ L⁰ T⁻²] | N/m |
| Angular Velocity | Angle / Time | [M⁰ L⁰ T⁻¹] | rad/s |
| Torque | Force × Arm | [M¹ L² T⁻²] | N·m |
| Specific Heat Capacity | Energy/(mass×temp) | [M⁰ L² T⁻² Θ⁻¹] | J/(kg·K) |
Analysis of Equations Using Dimensions
Dimensional analysis is one of the most powerful tools in physics. It lets us verify equations, derive new relationships, and convert units systematically. The central rule is the Principle of Homogeneity of Dimensions.
Applications of Dimensional Analysis
① Checking Equation Correctness
Verify whether a physical equation is dimensionally consistent. If dimensions don’t match, the equation is definitely wrong. If they match, it may be correct.
A dimensionally consistent equation passes the homogeneity test — a basic but powerful validity check.
② Deriving Relations
When the mathematical form of a physical law is unknown, dimensional analysis can suggest the correct relationship (up to a dimensionless constant).
Period of pendulum: T depends on L and g → T = k√(L/g) derived purely from dimensions.
③ Unit Conversion
Convert quantities between unit systems (CGS, MKS, SI, FPS) using dimensional formulae. Essential in scientific and engineering calculations.
④ Dimensionless Quantities
Quantities with [M⁰ L⁰ T⁰] are dimensionless — pure numbers with no units. They appear as constants in equations.
Refractive index, strain, angle (in radians), Reynolds number, coefficient of friction.
Worked Examples — Step by Step
Verify: v = u + at
LHS — Final Velocity (v)
[v] = [M⁰ L¹ T⁻¹]
RHS — Initial velocity (u)
[u] = [M⁰ L¹ T⁻¹]
RHS — Acceleration × Time (at)
[a][t] = [M⁰ L¹ T⁻²] × [T¹] = [M⁰ L¹ T⁻¹]
Conclusion
LHS = [M⁰ L¹ T⁻¹] = RHS ✅ Dimensionally Homogeneous. Equation is valid.
Verify: F = mv²/r (Centripetal Force)
LHS — Force (F)
[F] = [M¹ L¹ T⁻²]
RHS — mv²/r
[m][v²]/[r] = [M¹] × [L¹T⁻¹]² / [L¹] = [M¹ L² T⁻²] / [L] = [M¹ L¹ T⁻²]
Conclusion
LHS = [M¹ L¹ T⁻²] = RHS ✅ Dimensionally Homogeneous. Equation is valid.
Derive: Period of Simple Pendulum T = f(L, g)
Assume
T = k · Lᵃ · gᵇ, where k is a dimensionless constant
Write dimensional equations
[T¹] = [L¹]ᵃ × [L¹T⁻²]ᵇ = Lᵃ⁺ᵇ · T⁻²ᵇ
Compare exponents
For T: 1 = −2b → b = −½ | For L: 0 = a + b → a = ½
Result
T = k · L^(½) · g^(−½) = k√(L/g) — experiment confirms k = 2π
Limitations of Dimensional Analysis
- Cannot determine dimensionless constants — values like 2π, ½, or any pure number are invisible to dimensional analysis.
- Cannot handle trigonometric/exponential functions — quantities inside sin θ or eˣ must be dimensionless; analysis fails inside such functions.
- Cannot distinguish same-dimension quantities — Torque and Work both have [M¹L²T⁻²] but are physically different concepts.
- Fails for multi-term additions — if a formula has addition of terms from different origins, dimensional analysis cannot isolate correct forms.
- Cannot confirm numerical accuracy — even if dimensions match, missing constants can make the equation numerically wrong.
Dimensional analysis gives a necessary but not sufficient condition. Passing the check means the equation could be correct, not that it is correct. Always combine with physical reasoning and empirical verification.
Quick Reference — Key Formulas & Dimensional Check
Chapter Summary
📐 Variables
- Independent: What you change (cause)
- Dependent: What you measure (effect)
- Control: What you keep constant
- Discrete: Countable, specific values only
- Continuous: Any value within a range
🔒 Control Variables
- Eliminate confounding factors
- Ensure experimental validity
- Enable reproducibility of results
- Allow fair comparison between trials
📏 Units
- 7 SI base units: m, kg, s, A, K, mol, cd
- Fundamental: Independent, irreducible
- Derived: Combined from base units
- Examples: N, J, W, Pa, V, Ω, Hz
🧮 Dimensional Analysis
- Based on Principle of Homogeneity
- Symbols: M, L, T, I, Θ, N, J
- Applications: verify, derive, convert units
- Cannot determine dimensionless constants