Learning Objectives
- Understand the SI system of units and base quantities
- Learn about dimensional analysis and its applications
- Understand significant figures, errors, and accuracy in measurements
- Apply dimensional analysis to check equations and derive relations
- Learn about different types of errors in measurement
Key Concepts
SI Units and Base Quantities
The International System of Units (SI) defines seven base quantities and their units:
- Length: metre (m)
- Mass: kilogram (kg)
- Time: second (s)
- Electric Current: ampere (A)
- Temperature: kelvin (K)
- Amount of Substance: mole (mol)
- Luminous Intensity: candela (cd)
Supplementary units: radian (rad) for plane angle, steradian (sr) for solid angle.
Dimensional Analysis
Every physical quantity can be expressed in terms of fundamental dimensions: [M], [L], [T], [A], [K], [mol], [cd].
Examples of dimensional formulas:
- Force: [M L T⁻²]
- Energy/Work: [M L² T⁻²]
- Power: [M L² T⁻³]
- Pressure: [M L⁻¹ T⁻²]
- Gravitational constant G: [M⁻¹ L³ T⁻²]
- Planck's constant h: [M L² T⁻¹]
Principle of Homogeneity: In a correct equation, the dimensions of each term on both sides must be the same. This is used to check the correctness of equations.
Significant Figures
Rules for counting significant figures:
- All non-zero digits are significant: 1234 has 4 significant figures
- Zeros between non-zero digits are significant: 1002 has 4 significant figures
- Leading zeros are not significant: 0.0025 has 2 significant figures
- Trailing zeros in a decimal number are significant: 3.500 has 4 significant figures
- Trailing zeros without decimal point are ambiguous: 1200 may have 2, 3, or 4
Errors in Measurement
Absolute Error: Δa = |a_measured - a_mean|
Mean Absolute Error: Δa_mean = (1/n) Σ|Δaᵢ|
Relative Error: δa = Δa_mean / a_mean
Percentage Error: (Δa_mean / a_mean) × 100%
Combination of Errors:
- Sum/Difference (Z = A ± B): ΔZ = ΔA + ΔB (absolute errors add)
- Product/Quotient (Z = AB or A/B): ΔZ/Z = ΔA/A + ΔB/B (relative errors add)
- Power (Z = Aⁿ): ΔZ/Z = n(ΔA/A) (relative error multiplied by power)
Applications of Dimensional Analysis
1. Checking correctness of equations: Each term must have the same dimensions.
2. Deriving relationships: If a quantity depends on other quantities, dimensional analysis can determine the form of the relationship.
3. Converting units: n₁u₁ = n₂u₂, where dimensions help convert between systems.
Limitations: Cannot determine dimensionless constants, cannot distinguish between quantities with same dimensions (e.g., work and torque), and cannot handle logarithmic or trigonometric functions.
Summary
The SI system provides a consistent set of units for all physical measurements. Dimensional analysis is a powerful tool for verifying equations, deriving relationships, and converting units. Significant figures indicate the precision of a measurement. Errors are classified as systematic and random, and they combine according to specific rules for sums, products, and powers.
Important Terms
- Physical Quantity: A quantity that can be measured and expressed numerically
- Dimensional Formula: Expression showing how a derived quantity depends on base quantities
- Significant Figures: The number of meaningful digits in a measured value
- Systematic Error: Consistent, repeatable error from faulty instruments or technique
- Random Error: Unpredictable fluctuations in measurements
- Parallax Error: Error due to incorrect positioning of the observer's eye
- Least Count: The smallest measurement a measuring instrument can make
Quick Revision
- 7 base SI units: m, kg, s, A, K, mol, cd
- Dimensional formula of force = [MLT⁻²], energy = [ML²T⁻²]
- Principle of homogeneity: dimensions must match on both sides of an equation
- For Z = AᵖBq/Cʳ: ΔZ/Z = pΔA/A + qΔB/B + rΔC/C
- 1 AU = 1.496 × 10¹¹ m, 1 light year = 9.46 × 10¹⁵ m, 1 parsec = 3.08 × 10¹⁶ m
- 1 angstrom = 10⁻¹⁰ m, 1 fermi = 10⁻¹⁵ m