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Chapter Review

Measurements

Units and Errors · Dimensional Analysis

SI Base and Derived Units

The SI system defines seven base quantities (length, mass, time, current, temperature, luminous intensity, amount of substance) from which all derived units are constructed by algebraic combination.

Key Points

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    Seven base units: m, kg, s, A, K, cd, mol
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    Derived units combine base units: e.g., 1 N = 1 kg·m·s⁻²
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    When a prefixed unit is raised to a power, the power applies to the entire prefix+unit: 1 km² = 10⁶ m²
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    Compound prefixes (e.g., µµF) are forbidden — use the single correct prefix
  • •
    Supplementary units (radian, steradian) are dimensionless ratios of geometric quantities
Formula

Precision vs. Accuracy

Accuracy measures closeness to the true value (limited by systematic error), while precision measures consistency of repeated readings (limited by random error and least count).

Key Points

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    Systematic error shifts all readings in one direction — caused by faulty calibration, zero error, or parallax
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    Random error causes scatter around the mean — reduced by averaging multiple readings
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    Zero error correction: subtract positive error, add the magnitude of negative error
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    Parallax error is avoided by reading scales with the eye perpendicular to the marking
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    A precise instrument can still be inaccurate if it has uncorrected systematic error

Uncertainty Propagation

Uncertainties in measured quantities propagate through calculations according to rules that depend on whether the operation is addition/subtraction or multiplication/division.

Key Points

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    Addition/Subtraction: add absolute uncertainties — δ(A ± B) = δA + δB
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    Multiplication/Division: add percentage uncertainties — %δ(AB) = %δA + %δB
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    Powers: multiply percentage uncertainty by the power — %δ(xⁿ) = |n| × %δx
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    For timing experiments, δT = least count / number of oscillations counted
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    Uncertainty is quoted to 1 significant figure; the result is rounded to match
Formula

Significant Figures

Significant figures include all reliably known digits plus the first doubtful digit, indicating the precision of a measurement.

Key Points

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    Leading zeros are never significant (0.002 → 1 sig fig)
  • •
    Trapped zeros are always significant (2.002 → 4 sig figs)
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    Trailing zeros are significant only if a decimal point is present (2.30 → 3, but 230 → 2)
  • •
    Scientific notation removes ambiguity: 8.00 × 10³ clearly has 3 sig figs
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    Multiplication/division: result keeps sig figs of the least accurate factor
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    Addition/subtraction: result keeps decimal places of the least precise term

Rounding: The Even-Odd Rule

When the digit to be dropped is exactly 5, rounding toward the nearest even digit prevents systematic upward bias in large datasets.

Key Points

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    If last retained digit is even, round down (2.345 → 2.34)
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    If last retained digit is odd, round up (2.355 → 2.36)
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    This rule only applies when the dropped digit is exactly 5 with no further nonzero digits
  • •
    Standard rounding (always up on 5) introduces statistical bias over many calculations

Dimensions and Dimensional Formulas

Dimensions represent the qualitative nature of a quantity using base symbols M, L, T, independent of the unit system, and are manipulated algebraically to analyse physical equations.

Key Points

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    Core mechanical dimensions: Mass M, Length L, Time T
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    Derived dimensions built step-by-step: velocity LT⁻¹, force MLT⁻², energy ML²T⁻²
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    Square brackets denote 'the dimension of': F = MLT⁻²
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    Same dimensions do not guarantee same quantity — work and torque both have ML²T⁻²
  • •
    Dimensionless quantities (strain, refractive index, angles) have M⁰L⁰T⁰ but may still carry units
Formula

Principle of Homogeneity

A physically valid equation requires every term that is added, subtracted, or equated to have identical dimensions.

Key Points

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    Check correctness by computing dimensions of each term separately
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    Homogeneity is necessary but not sufficient — it cannot detect missing dimensionless constants
  • •
    Arguments of sin, cos, log, and exponentials must be dimensionless
  • •
    Limiting-case tests (t → 0, v → 0, r → ∞) complement dimensional checks
Formula

Deriving Relations by Dimensional Analysis

The method of undetermined powers lets you derive proportionality relations by equating exponents of M, L, and T on both sides of an assumed power-law dependence.

Key Points

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    Write X = k · aᵖ bᵍ cʳ, substitute dimensions, equate exponents of M, L, T
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    Yields functional form only — dimensionless constants (2π, ½) must come from experiment
  • •
    Cannot handle transcendental functions (sin, log, eˣ)
  • •
    Limited to 3 unknowns in mechanics (one equation per base dimension)
  • •
    Quick elimination: if units of LHS ≠ RHS, the formula is definitely wrong
Formula

Formulas

Percentage Uncertainty

Uncertainty as a fraction of the measured value.

Uncertainty in Powers

Percentage uncertainty scales with the exponent.

Uncertainty in Products/Quotients

Add percentage uncertainties for multiplication or division.

Uncertainty in Sums/Differences

Add absolute uncertainties for addition or subtraction.

Dimensions of Force

Fundamental building block for mechanical dimensional analysis.

Dimensions of Energy/Work

Applies to kinetic, potential, and all forms of mechanical energy.

Dimensions of Pressure

Force per unit area; same dimensions as stress.

Dimensions of Viscosity

Derived from Stokes' law: F = 6πηrv.