Introduction

Measurement is the foundation of physics. When we say an object has a mass of 5 kg or a length of 3 metres, we are expressing physical quantities using numbers and units. This chapter establishes the standard framework for all measurements in physics.

💡 Key Idea / मुख्य विचार

\( \text{Measurement} = \text{Numerical Value} \times \text{Unit} \)
Everything we measure in physics requires both a number and a unit to be meaningful. "5" alone is incomplete — "5 metres" is a complete measurement.

Physical Quantities

A physical quantity is anything that can be measured and expressed as a number with a unit. All physical quantities are classified into two types:

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Classification of Physical Quantities
TypeDefinitionExamples
Fundamental (Base)Independent quantities — not derived from othersMass, Length, Time, Temperature, Current, Luminous Intensity, Amount of substance
DerivedDerived from two or more fundamental quantitiesForce, Velocity, Pressure, Energy, Density

The SI System of Units

The International System of Units (SI) — Système International d'Unités — is the globally accepted standard for measurement. It defines 7 base units:

Physical QuantitySI UnitSymbol
LengthMetrem
MassKilogramkg
TimeSeconds
Electric CurrentAmpereA
Thermodynamic TemperatureKelvinK
Amount of SubstanceMolemol
Luminous IntensityCandelacd
⚠️ Exam Trap

Temperature is measured in Kelvin (K) in the SI system — NOT Celsius or Fahrenheit. Remember: 0°C = 273.15 K. This is a common MCQ trap!

Dimensions and Dimensional Formulae

The dimensions of a physical quantity are the powers to which the fundamental quantities must be raised to represent it. We use square brackets to denote dimensions:

$$[M] = \text{Mass} \quad [L] = \text{Length} \quad [T] = \text{Time} \quad [A] = \text{Current}$$ $$[K] = \text{Temperature} \quad [\text{mol}] = \text{Amount} \quad [\text{cd}] = \text{Luminosity}$$

Important Dimensional Formulae

Physical QuantityFormulaDimensional Formula
Velocity\( v = d/t \)\( [M^0 L T^{-1}] \)
Acceleration\( a = v/t \)\( [M^0 L T^{-2}] \)
Force\( F = ma \)\( [M L T^{-2}] \)
Work/Energy\( W = F \cdot d \)\( [M L^2 T^{-2}] \)
Power\( P = W/t \)\( [M L^2 T^{-3}] \)
Pressure\( P = F/A \)\( [M L^{-1} T^{-2}] \)
Momentum\( p = mv \)\( [M L T^{-1}] \)

Dimensional Analysis

Dimensional analysis is a powerful tool used to:

  • Check the correctness of a physical equation
  • Derive relationships between physical quantities
  • Convert units from one system to another

Principle of Homogeneity

A physical equation is dimensionally correct only if the dimensions on both sides of the equation are equal. For example, for the equation v = u + at:

$$[v] = [LT^{-1}]$$ $$[u] = [LT^{-1}]$$ $$[at] = [LT^{-2}][T] = [LT^{-1}]$$ $$\therefore \text{LHS} = \text{RHS} \Rightarrow \text{Equation is dimensionally correct} \checkmark$$
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⚠️ Limitation

Dimensional analysis cannot determine dimensionless constants (like π, 2, ½) in an equation. It also fails when quantities are added or subtracted.

Significant Figures

Significant figures indicate the precision of a measurement. All non-zero digits are significant. Zeros may or may not be significant depending on their position.

Rules for Significant Figures

  1. All non-zero digits are significant: 2345 has 4 significant figures.
  2. Zeros between non-zero digits are significant: 2005 has 4 sig. figs.
  3. Leading zeros are NOT significant: 0.0025 has only 2 sig. figs.
  4. Trailing zeros after a decimal are significant: 2.500 has 4 sig. figs.
  5. Trailing zeros in a whole number are ambiguous: 2500 could have 2, 3, or 4 sig. figs.

Errors in Measurement

No measurement is perfectly accurate. The difference between the measured value and the true value is called error.

Types of Errors
  • Systematic Error: Consistent, reproducible error caused by faulty instruments or techniques. Can be corrected.
  • Random Error: Unpredictable errors due to environmental disturbances. Reduced by taking multiple readings.
  • Gross Error: Human mistakes — like reading a scale incorrectly.

Error Formulae

$$\Delta a = |\text{True Value} - \text{Measured Value}|$$ $$\bar{\Delta a} = \frac{\Delta a_1 + \Delta a_2 + \cdots + \Delta a_n}{n}$$ $$\text{Relative Error} = \frac{\bar{\Delta a}}{\bar{a}}$$ $$\text{Percentage Error} = \frac{\bar{\Delta a}}{\bar{a}} \times 100\%$$
Code — Numerical Calculation
🐍 Python Example

Calculate mean absolute error programmatically:

# Error Calculation — Units & Measurements
measurements = [4.23, 4.25, 4.21, 4.24]  # in metres

# Step 1: Calculate mean
mean = sum(measurements) / len(measurements)
print(f"Mean value: {mean:.4f} m")

# Step 2: Calculate absolute deviations
deviations = [abs(x - mean) for x in measurements]
print(f"Deviations: {[round(d,4) for d in deviations]}")

# Step 3: Mean absolute error
mae = sum(deviations) / len(deviations)
print(f"Mean Absolute Error: {mae:.4f} m")
print(f"Result: ({mean:.2f} ± {mae:.2f}) m")
print(f"Percentage Error: {(mae/mean)*100:.2f}%")

Solved Examples

Example 1 — Dimensional Analysis

Q: Check if the equation \( v^2 = u^2 + 2as \) is dimensionally correct.

Solution:

$$[v^2] = [LT^{-1}]^2 = [L^2T^{-2}]$$ $$[u^2] = [L^2T^{-2}]$$ $$[2as] = [LT^{-2}][L] = [L^2T^{-2}]$$ $$\therefore \text{All terms have the same dimensions} \Rightarrow \text{Equation is correct} \checkmark$$
Example 2 — Significant Figures

Q: How many significant figures are in 0.00820?

Solution: The leading zeros (0.00) are NOT significant. The digits 8, 2, and 0 (trailing zero after decimal) are significant.

Answer: 3 significant figures

Example 3 — Error Calculation

Q: If measurements of a length are 4.23, 4.25, 4.21, 4.24 m, find mean absolute error.

$$\bar{a} = \frac{4.23 + 4.25 + 4.21 + 4.24}{4} = 4.2325 \approx 4.23 \text{ m}$$ $$\Delta a_i = |0.00|, |0.02|, |0.02|, |0.01|$$ $$\bar{\Delta a} = \frac{0.00 + 0.02 + 0.02 + 0.01}{4} = 0.0125 \approx 0.01 \text{ m}$$ $$\boxed{\text{Result: } (4.23 \pm 0.01) \text{ m}}$$