Units and Measurements
'Measurement is the first step that leads to control and eventually to improvement.' — James Harrington
1. Chapter Overview
Physics is a QUANTITATIVE science. Every measurement consists of a NUMERICAL VALUE and a UNIT. This chapter establishes the SYSTEM OF UNITS (SI), teaches you how to DERIVE RELATIONSHIPS using dimensional analysis, handle ERRORS in measurements, and report results with correct SIGNIFICANT FIGURES.
2. Physical Quantities and Units
- Physical Quantity: A property that can be MEASURED (length, mass, time, temperature, etc.)
- Unit: A STANDARD of measurement for a physical quantity
- Fundamental Quantities: Base quantities that are INDEPENDENT of others
- Derived Quantities: Expressed in terms of fundamental quantities
The SI System (Système International)
| Base Quantity | SI Unit | Symbol |
|---|---|---|
| Length | metre | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | ampere | A |
| Thermodynamic Temperature | kelvin | K |
| Amount of Substance | mole | mol |
| Luminous Intensity | candela | cd |
Supplementary Units
- Plane Angle: radian (rad)
- Solid Angle: steradian (sr)
3. Dimensional Analysis
Dimensions of Physical Quantities
- Dimensions are expressed in terms of M (mass), L (length), T (time), A (current), K (temperature)
- Example: Velocity = [LT⁻¹]; Force = [MLT⁻²]; Energy = [ML²T⁻²]
Uses of Dimensional Analysis
- Check correctness of a formula: Dimensions on both sides must match (Principle of Homogeneity)
- Derive relationship between quantities: If a quantity depends on several variables, we can find the form of the equation
- Convert units from one system to another: e.g., converting newton to dyne
Limitations
- Cannot determine dimensionless constants (like k in F = k m₁m₂/r²)
- Cannot derive formulas involving trigonometric, logarithmic, or exponential functions
- Cannot distinguish between quantities with the same dimensions (torque and energy both [ML²T⁻²])
Worked Problem
Q: Check whether the equation v² = u² + 2as is dimensionally correct. Solution:
- LHS: v² has dimension [LT⁻¹]² = [L²T⁻²]
- RHS: u² has dimension [LT⁻¹]² = [L²T⁻²], 2as has dimension [LT⁻²][L] = [L²T⁻²]
- All terms have [L²T⁻²], so the equation is DIMENSIONALLY CORRECT.
4. Errors in Measurement
Types of Errors
| Error Type | Description |
|---|---|
| Systematic Error | Repeats consistently; due to instrument fault, personal bias, or environmental conditions |
| Random Error | Fluctuates irregularly; minimised by taking multiple readings and averaging |
| Gross Error | Careless mistakes — parallax, incorrect recording |
Absolute, Relative, and Percentage Error
- Absolute Error Δa = |a₀ — aᵢ| (difference between true value and measured value)
- Mean Absolute Error Δā = (|Δa₁| + |Δa₂| + ... + |Δaₙ|) / n
- Relative Error = Δā / ā
- Percentage Error = (Δā / ā) × 100%
Propagation of Errors
| Operation | Error Formula |
|---|---|
| Sum Z = A + B | ΔZ = ΔA + ΔB |
| Difference Z = A — B | ΔZ = ΔA + ΔB |
| Product Z = A × B | ΔZ/Z = ΔA/A + ΔB/B |
| Quotient Z = A/B | ΔZ/Z = ΔA/A + ΔB/B |
| Power Z = Aⁿ | ΔZ/Z = n(ΔA/A) |
5. Significant Figures
- Rules: All non-zero digits are significant; zeros between digits are significant; trailing zeros after decimal are significant
- Example: 0.00452 has 3 significant figures (4, 5, 2)
- Rounding off: If digit after last significant digit > 5, round up; if < 5, leave; if = 5, round to nearest even
Arithmetic with Significant Figures
- Addition/Subtraction: Result rounded to the LEAST number of decimal places
- Multiplication/Division: Result rounded to the LEAST number of significant figures
6. Common Mistakes
- Confusing dimension with units: Dimensions are abstract ([L]), units are concrete (m, cm)
- Applying homogeneity incorrectly: ALL terms in an equation must have the same dimensions
- Significant figures in measured vs exact values: Exact values (π, constants in formulas) have INFINITE significant figures
- Using wrong error propagation: Errors ALWAYS ADD, never cancel
7. CBSE Exam Focus
- Dimensional formula of physical quantities (frequently asked MCQ)
- Checking dimensional correctness of given equations (3-mark)
- Error propagation problems (3-mark numerical)
- Significant figures in calculations (1-mark)
- Convert units using dimensional analysis (5-mark)
8. Key Formulas
- Dimension of velocity: [LT⁻¹]
- Dimension of force: [MLT⁻²]
- Dimension of energy/work: [ML²T⁻²]
- Percentage error: (Δā / ā) × 100%
- Maximum error in product: ΔZ/Z = ΔA/A + ΔB/B
9. Self-Test (5+ Q&A)
Q1: Find the dimensions of the universal gravitational constant G (from F = Gm₁m₂/r²). A: [M⁻¹L³T⁻²]
Q2: The period of a pendulum T depends on length L and acceleration g. Use dimensional analysis to derive the relation. A: T = k√(L/g). Dimensionally: [T] = [L]ᵃ[LT⁻²]ᵇ → a = 1/2, b = -1/2 → T ∝ √(L/g)
Q3: In an experiment, five measurements of time are: 2.02s, 2.04s, 1.98s, 2.00s, 2.01s. Find mean absolute error. A: Mean = 2.01s. Absolute errors: 0.01, 0.03, 0.03, 0.01, 0.00. Mean absolute error = 0.016s.
Q4: How many significant figures in 0.005060? A: 4 (the digits 5, 0, 6, 0 — trailing zero after decimal is significant).
Q5: A wire's length is 5.2 cm and radius is 0.12 cm. Find volume with correct significant figures. A: V = πr²h = π(0.12)²(5.2) = 3.14 × 0.0144 × 5.2 = 0.235... → Rounded to 2 significant figures = 0.24 cm³.
10. Conclusion
Units and measurements are the LANGUAGE of physics. Dimensional analysis is your MOST POWERFUL tool for checking formulas. Mastering error analysis separates a careful scientist from a careless one. Grip these fundamentals firmly — every subsequent physics chapter depends on them.
