Unit I: Units and Measurements

Sub-topics: Need for measurement; Units of measurement; Systems of units; SI units, fundamental and derived units; Significant figures; Dimensions of physical quantities; Dimensional analysis and its applications.

Part 1: The Need for Measurement & Units

1.1 Why Do We Measure? (The Need for Measurement)

Physics is a quantitative science. Every observation must be expressed in terms of number + units. This brings clarity, objectivity, and universality to our understanding of natural phenomena. To establish laws of physics through experiments, which require precise measurements of quantities like length, mass, time, etc.

Example:

• We don’t just say “the object is heavy”, we say “the object has a mass of 10 kg”.
• Saying “the rod is long” is vague, but saying “the rod is 2 m long” is meaningful.
• Imagine a recipe saying “add some sugar” vs. “add 250 grams of sugar.” Which one is more reliable? Measurement turns vague ideas into precise, reproducible data.

1.2 Units of Measurement

A unit is a standard reference used to measure a physical quantity.

Example: The word “meter” is a unit, and “10” is the numerical magnitude. So, “10 meters” means the length is 10 times the standard length of 1 meter.

1.3 Systems of Units

Note: The SI system is an extended and refined version of the MKS system. It was adopted in 1960 by the General Conference on Weights and Measures to create a unified, coherent global standard.

Part 2: The International System (SI) of Units

It has 7 fundamental (base) units, 2 supplementary units and many derived units.

2.1 Fundamental (Base) Units

These are the independently defined units for fundamental quantities. All other units can be derived from these. There are 7 base SI units.

• Metre (m): The length of the path travelled by light in vacuum in a time interval of 1/299,792,458 of a second.
• Kilogram (kg): Defined by taking the fixed numerical value of the Planck constant *h* to be 6.62607015 × 10⁻³⁴ when expressed in the unit J s (kg m² s⁻¹).
• Second (s): The duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.

2.2 Derived Units

These are units for physical quantities that can be expressed in terms of the base units through the mathematical operations of multiplication and division.

Examples:

• Area: m² (m × m)
• Volume: m³ (m × m × m)
• Speed or Velocity: m/s (m ÷ s) or m s⁻¹
• Acceleration: m/s² (m ÷ s ÷ s) or m s⁻²
• Force: Newton (N) = kg m s⁻²
• Work/Energy: Joule (J) = N m = kg m² s⁻²

2.3 SI Prefixes

Used to express very large or very small numbers compactly.

Example: 1 nm (nanometer) = 10⁻⁹ m, 1 Mg (Megagram) = 10⁶ g = 1000 kg.

2.4 Supplementary Units

Part 3: Significant Figures

3.1 What are Significant Figures?

The reliable digits in a measured number, known with certainty, plus the first uncertain digit. They indicate the precision of a measurement. A measurement written with more significant figures is more precise. The last digit in any measurement is always an estimate.

3.2 Rules for Determining Significant Figures

1. All non-zero digits (1-9) are significant. (e.g., 123.45 has 5 significant figures)
2. All zeros between non-zero digits are significant. (e.g., 100.5 has 4 significant figures)
3. Leading zeros (zeros to the left of the first non-zero digit) are NOT significant. They just position the decimal. (e.g., 0.0025 has 2 significant figures)
4. Trailing zeros (zeros to the right of non-zero digits) ARE significant ONLY if they are after the decimal point. (e.g., 12.500 has 5 significant figures; 12500 is ambiguous, it has 3, 4, or 5? Use scientific notation to avoid ambiguity: 1.2500 × 10⁴ has 5)
5. In numbers without a decimal, trailing zeros are generally not significant. (e.g., 2500 is assumed to have 2 significant figures unless specified otherwise).

3.3 Rules for Arithmetic Operations with Significant Figures

• Addition & Subtraction: The final result should have as many decimal places as the number with the least decimal places.
  • Example: 12.5 + 3.24 = 15.74 → Round to 15.7 (one decimal place)
• Multiplication & Division: The final result should have as many significant figures as the number with the least significant figures.
  • Example: 12.5 × 3.24 = 40.5 → Round to 40.5 (3 significant figures). But 12.5 × 3.2 = 40 → Write as 4.0 × 10¹ (2 significant figures).
• Rounding Off: If the digit to be dropped is more than 5, increase the preceding digit by 1. If it’s less than 5, leave it. If it’s exactly 5, make the preceding digit even.

Part 4: Dimensions & Dimensional Analysis

4.1 Dimensions of Physical Quantities

The power (exponent) to which the fundamental quantities (Mass, Length, Time, etc.) are raised to represent a physical quantity. Dimensions of mass are [M], length [L], time [T], electric current [A], etc.

Examples:

• Area: Length × Length = [L] × [L] = [L²]
• Volume: Length × Length × Length = [L] × [L] × [L] = [L³]
• Speed: Distance/Time = [L]/[T] = [L T⁻¹]
• Acceleration: Velocity/Time = [L T⁻¹]/[T] = [L T⁻²]
• Force: Mass × Acceleration = [M] × [L T⁻²] = [M L T⁻²]

4.2 Dimensional Analysis and its Applications

This is a powerful tool to check the correctness of equations, derive relations, and convert units.

1. Dimensional Homogeneity (Principal of Homogeneity)

Every correct physical equation must be dimensionally homogeneous. This means the dimensions on the Left-Hand Side (LHS) must be equal to the dimensions on the Right-Hand Side (RHS).

Example:

Check the equation s = ut + (1/2)at²

• LHS: [s] = [L]
• RHS: [ut] = [L T⁻¹][T] = [L]; [(1/2)at²] = [L T⁻²][T²] = [L]
• Since [L] = [L] + [L], the equation is dimensionally correct.

Note: Dimensional correctness does NOT guarantee the equation is physically correct. The numerical constant (1/2) is dimensionless and cannot be checked this way.

2. Deriving Relations

If you know how a physical quantity depends on others, you can find the relation dimensionally.

Example: Derive the time period (T) of a simple pendulum.

Time period (T) of a pendulum depends on (i) Length of pendulum (l) → [L] and acceleration due to gravity (g) → [L T⁻²]

• Assume, T ∝ lᵃ gᵇ
• Write dimensions: [T] = [L]ᵃ [L T⁻²]ᵇ = [Lᵃ⁺ᵇ T⁻²ᵇ]
• Equate powers on both sides: (i) for [T]: 1 = -2b → b = -1/2 and (ii) for [L]: 0 = a + b → a = -b = 1/2

So, T ∝ √(l/g) ⇒ T = k √(l/g). Experiment shows k = 2π

3. Conversion of Units

Example: Convert 1 Joule (SI unit of energy) into ergs (CGS unit).

Dimension of Energy is [M L² T⁻²] will be the same for all unit systems. Therefore,

{n_1}\left[ {{M_1}\;L_1^2\;T_1^{ - 2}} \right]\; = \;{n_2}\left[ {{M_2}\;L_2^2\;T_2^{ - 2}} \right]\;\;\;\; \Rightarrow \;{n_2} = \;{n_1}{\left[ {{M_1}/{M_2}} \right]^1}\;{\left[ {{L_1}/{L_2}} \right]^2}{\left[ {{T_1}/{T_2}} \right]^{ - 2}}

For conversion from SI units to CGS units:

{M_1}\left( {kg} \right)/{M_2}\left( g \right) = 1000/1 = {10^3};\;{L_1}\left( m \right)/{L_2}\left( {cm} \right) = 100/1 = {10^2};\;{T_1}\left( s \right)/{T_2}\left( s \right) = 1/1 = 1

{n_2}\left( {erg} \right) = 1\left( J \right) \times \left( {{{10}^3}} \right) \times {\left( {{{10}^2}} \right)^2} \times {\left( 1 \right)^{ - 2}} = {10^7}

So, 1 J = 10⁷ erg