Units and Measurement (NEET Master Page)

Physics and Measurement (Unit-1) NEET examination

Encyclopedia of Physical Measurement

UNITS AND MEASUREMENT

1.1 INTRODUCTION

Measurement forms the bedrock of physics—it is how we transform qualitative observations into quantitative knowledge. When we measure any physical quantity, we are essentially comparing it against a predefined reference standard. This reference standard, accepted internationally, is called a unit.

Every measurement result must be expressed as a numerical value accompanied by its unit—the number tells us “how many,” and the unit tells us “of what.” For instance, saying a table is “5” is meaningless; saying it is “5 metres” conveys the actual measurement.

One of the elegant features of physics is that although we encounter countless physical quantities in our study, we don’t need an equal number of independent units. This is because physical quantities are interconnected through mathematical relationships. For example:

• Speed is defined as distance divided by time
• Force is mass times acceleration
• Energy is force times distance

These interrelations mean we can reduce all units to a small set of fundamental or base units—those for quantities that cannot be expressed in terms of other quantities. All other units are called derived units because they are combinations of these base units. A system of units is simply the complete collection of both base and derived units.

Consider how this works in practice: if we know the base units for length (metre), mass (kilogram), and time (second), we can derive the unit for speed as metre per second (m/s), for force as kilogram-metre per second squared (kg·m/s²), and for energy as kilogram-metre squared per second squared (kg·m²/s²). This hierarchical structure is what makes systems of measurement coherent and manageable.

1.2 THE INTERNATIONAL SYSTEM OF UNITS

Historically, different regions and scientific communities developed their own measurement systems, leading to considerable confusion. Until recently, three systems dominated scientific work:

Imagine trying to share experimental results when one scientist reports density in g/cm³, another in kg/m³, and a third in lb/ft³—conversions become tedious and error-prone. The need for international standardisation became increasingly urgent as science and technology grew more global.

The SI System: Today, the universally accepted system is the Système International d’Unités (International System of Units), abbreviated as SI. This system, developed by the Bureau International des Poids et Mesures (BIPM) in France, was established in 1971 and underwent a significant revision in November 2018. The 2018 revision was historic because it redefined the base units in terms of fundamental constants of nature rather than physical artefacts—for instance, the kilogram is no longer defined by a physical platinum-iridium cylinder in Paris but by the Planck constant. This ensures that the definitions remain stable and reproducible anywhere in the universe.

The SI system has a decimal structure, meaning larger and smaller units are related by powers of ten. For example, 1 kilometre = 1000 metres, 1 millimetre = 0.001 metres. This decimal nature makes conversions within the system straightforward—move the decimal point rather than remember conversion factors like 12 inches in a foot or 5280 feet in a mile.

Base Quantities and Their Units

The SI system is built on seven base units, each defined with extraordinary precision:

1. Length — metre (m): Defined using the speed of light in vacuum. The metre is the distance light travels in 1/299,792,458 of a second. This definition connects length to an invariant property of the universe—the speed of light—making the metre reproducible in any laboratory with sufficient precision, without reference to a physical metre bar.

2. Mass — kilogram (kg): Defined by the Planck constant h = 6.62607015 × 10⁻³⁴ J·s. The kilogram is perhaps the most significant example of the 2018 revision—it was the last base unit defined by a physical object. The new definition uses quantum mechanics to define mass, allowing for much more precise and stable measurements.

3. Time — second (s): Defined by the frequency of radiation from the caesium-133 atom. Specifically, the second is the duration of 9,192,631,770 oscillations of the radiation corresponding to the transition between two hyperfine levels of the ground state of caesium-133. This atomic clock definition makes time measurement extraordinarily precise—accurate to better than one second in 100 million years.

4. Electric Current — ampere (A): Defined by the elementary charge e = 1.602176634 × 10⁻¹⁹ C. The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length and negligible circular cross-section, placed one metre apart in vacuum, would produce a force of 2 × 10⁻⁷ N per metre of length.

5. Thermodynamic Temperature — kelvin (K): Defined by the Boltzmann constant k = 1.380649 × 10⁻²³ J/K. The kelvin is now defined in terms of the triple point of water, but more fundamentally, it links temperature to molecular motion through the Boltzmann constant.

6. Amount of Substance — mole (mol): Defined by the Avogadro constant NA = 6.02214076 × 10²³ mol⁻¹. One mole contains exactly this many elementary entities (which could be atoms, molecules, ions, electrons, or other specified particles). When we say “one mole of sodium chloride,” we mean 6.022 × 10²³ formula units of NaCl.

7. Luminous Intensity — candela (cd): Defined by the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, Kcd = 683 lm/W. The candela is the luminous intensity of a source emitting monochromatic radiation of frequency 540 × 10¹² Hz with a radiant intensity of 1/683 W/sr in a given direction.

Supplementary Units

Two additional quantities—plane angle and solid angle—while not base quantities, are commonly used and have defined SI units:

Plane angle — radian (rad): The angle subtended at the centre of a circle by an arc whose length equals the radius. Mathematically, dθ = ds/r, where ds is arc length and r is radius. Since this is the ratio of two lengths, the radian is dimensionless.

Solid angle — steradian (sr): The solid angle subtended at the centre of a sphere by a surface area equal to the square of the radius. Mathematically, dΩ = dA/r², where dA is area on the sphere’s surface and r is radius. Like the radian, it’s dimensionless.

Both these quantities are ratios, which is why they have no dimensions in the fundamental system—they are pure numbers.

Beyond the Seven Base Units

The derived units are countless, but they are all combinations of the seven base units. For instance:

Some derived units have been given special names in honour of scientists who made significant contributions to physics—the newton (force), joule (energy), watt (power), pascal (pressure), hertz (frequency), and many others. These special names make the vocabulary of physics more efficient; it’s more convenient to say “10 newtons” than “10 kilogram-metres per second squared.”

Units Retained for General Use

While the SI system is comprehensive, some traditional units are still widely used in specific contexts:

The angstrom, for example, is commonly used in atomic physics and chemistry, while the electronvolt is standard in nuclear and particle physics. These units are acceptable for general use alongside SI units.

1.3 SIGNIFICANT FIGURES

Every measurement inherently carries uncertainty. No instrument is infinitely precise, and no measurement is perfectly reproducible. The way we report a measurement should honestly reflect this uncertainty—neither claiming more precision than we actually have nor hiding the precision we do have.

Significant figures are the digits in a measurement that carry meaningful information about its precision. They include all digits that are known with certainty plus one additional digit that is estimated or uncertain.

Consider measuring the length of a table with a ruler marked in millimetres. If we report the length as 287.5 cm, we are saying:

• The digits 2, 8, and 7 are certain (we could read these from the ruler)
• The digit 5 is estimated (we judged the table end fell between two millimetre marks)

Thus, this measurement has four significant figures. Reporting “287.50 cm” would be misleading—it would imply we could estimate to 0.01 cm (0.1 mm), which our ruler cannot do.

Why significant figures matter: The number of significant figures tells others how precisely you measured. If you report the same length as “287 cm” (three significant figures), you’re saying you measured only to the nearest centimetre. If you report it as “287.5 cm” (four significant figures), you’re claiming millimetre precision. Both could be correct—depending on your measuring instrument.

Rules for Identifying Significant Figures

Rule 1: All non-zero digits are significant.

• 123.45 has five significant figures
• 7.891 has four significant figures

Rule 2: Zeroes between non-zero digits are significant.

• 2.308 cm has four significant figures (2, 3, 0, 8)
• 1002 kg has four significant figures
• 5.003 × 10² has four significant figures

Rule 3: Zeroes to the left of the first non-zero digit (leading zeros) are NOT significant. They only indicate the position of the decimal point.

• 0.0023 m has two significant figures (2 and 3)
• 0.000805 has three significant figures (8, 0, 5)

Rule 4: Zeroes to the right of a decimal point (trailing zeros) ARE significant because they indicate precision of measurement.

• 3.500 m has four significant figures (3, 5, 0, 0)
• 0.06900 has four significant figures (6, 9, 0, 0)
• 2.000 has four significant figures

Rule 5: Trailing zeros in a number without a decimal point are NOT significant (unless indicated otherwise).

• 123 m = three significant figures
• 4700 m = two significant figures (4 and 7)

The ambiguity problem: The last rule can be problematic. Suppose a length is reported as 4.700 m. The zeros here are significant because they indicate the measurement was made to 0.001 m precision. But if we convert this to different units:

• 4.700 m = 470.0 cm (four significant figures)
• 4.700 m = 4700 mm (according to Rule 5, this would suggest only two significant figures, which is wrong!)

The solution is to use scientific notation, which eliminates all ambiguity:

Scientific Notation: Every measurement should be expressed as a × 10^b, where 1 ≤ a < 10 and b is an integer exponent. All digits in the coefficient a are significant.

• 4.700 m = 4.700 × 10^0 m (four significant figures)
• 4.700 m = 4.700 × 10² cm (four significant figures)
• 4.700 m = 4.700 × 10³ mm (four significant figures)
• 4.700 m = 4.700 × 10⁻³ km (four significant figures)

Order of Magnitude: When we only need a rough idea, we can approximate a to 1 (if a ≤ 5) or 10 (if a > 5). The exponent b then gives the order of magnitude.

• Diameter of Earth = 1.28 × 10⁷ m → order of magnitude 10⁷
• Diameter of hydrogen atom = 1.06 × 10⁻¹⁰ m → order of magnitude 10⁻¹⁰
• The Earth is 17 orders of magnitude larger than a hydrogen atom

Arithmetic with Significant Figures

When performing calculations with measured values, the result cannot be more precise than the least precise measurement used in the calculation.

Rule for Multiplication and Division: The result should have the same number of significant figures as the quantity with the fewest significant figures.

Example: Calculate density when mass = 4.237 g (4 sig figs) and volume = 2.51 cm³ (3 sig figs).

Density = 4.237 g / 2.51 cm³ = 1.68804780876... g/cm³

We report this as 1.69 g/cm³ (3 sig figs—limited by volume measurement).

Example: Speed of light = 3.00 × 10⁸ m/s (3 sig figs), one year = 365.25 days = 3.1557 × 10⁷ s (5 sig figs).

Light year = (3.00 × 10⁸) × (3.1557 × 10⁷) = 9.47 × 10¹⁵ m (3 sig figs—limited by speed of light measurement).

Rule for Addition and Subtraction: The result should have the same number of decimal places as the quantity with the fewest decimal places.

Example: Add 436.32 g, 227.2 g, and 0.301 g.

436.32 g
227.2 g  (only 1 decimal place - least precise)
0.301 g
---------
663.821 g → report as 663.8 g (1 decimal place)

Example: Subtract 0.304 m from 0.307 m.

0.307 m - 0.304 m = 0.003 m = 3 × 10⁻³ m

Important: Do NOT apply the multiplication/division rule to addition/subtraction problems. The rules are different and reflect different ways uncertainties combine in arithmetic operations.

Rounding Off

When the result of a calculation has too many digits, we must round it off to the appropriate number of significant figures.

Standard rounding rules:

1. If the digit to be dropped is less than 5, the preceding digit stays unchanged.

• 1.743 → 1.74 (dropping 3, less than 5)

1. If the digit to be dropped is greater than 5, the preceding digit increases by 1.

• 2.746 → 2.75 (dropping 6, greater than 5)

1. If the digit to be dropped is exactly 5:

• If the preceding digit is even, drop the 5 (leave the preceding digit unchanged)
  • 2.745 → 2.74 (preceding digit 4 is even)
• If the preceding digit is odd, increase it by 1
  • 2.735 → 2.74 (preceding digit 3 is odd)

This “round half to even” convention minimizes cumulative rounding errors in long calculations.

Practical guideline for complex calculations: In multi-step calculations, keep one extra digit in intermediate steps and round only at the final result. This prevents rounding errors from accumulating.

For example, if we compute 1/9.58:

• Rounding too early: 1/9.58 = 0.104 (3 sig figs)
• Then 1/0.104 = 9.62 (3 sig figs) — we lost the original 9.58!
• Better: 1/9.58 = 0.1044 (4 sig figs kept)
• Then 1/0.1044 = 9.58 (original value recovered)

Determining Uncertainty in Results

Beyond just counting significant figures, we should understand how uncertainties propagate through calculations:

Example with area calculation:

• Length = 16.2 ± 0.1 cm → relative uncertainty = (0.1/16.2) × 100% = 0.6%
• Breadth = 10.1 ± 0.1 cm → relative uncertainty = (0.1/10.1) × 100% = 1.0%
• Area = 16.2 × 10.1 = 163.62 cm²
• Percentage uncertainty in area = 0.6% + 1.0% = 1.6%
• Absolute uncertainty = 163.62 × 0.016 = 2.6 cm²
• Final result: 164 ± 3 cm²

Note on subtraction: When subtracting numbers, the number of significant figures can decrease dramatically:

• 12.9 g – 7.06 g = 5.84 g → but properly reported as 5.8 g
• The uncertainty from subtraction combines in a way that reduces precision

Relative uncertainty depends on the value itself:

• 1.02 g ± 0.01 g has relative uncertainty = (0.01/1.02) × 100% ≈ 1.0%
• 9.89 g ± 0.01 g has relative uncertainty = (0.01/9.89) × 100% ≈ 0.1%

This means that for the same absolute uncertainty, larger measurements are relatively more precise.

1.4 DIMENSIONS OF PHYSICAL QUANTITIES

The dimensions of a physical quantity describe its fundamental nature—what basic quantities it is composed of. We denote the seven base dimensions with square brackets:

• [L] for length
• [M] for mass
• [T] for time
• [A] for electric current
• [K] for thermodynamic temperature
• [cd] for luminous intensity
• [mol] for amount of substance

In mechanics (where we typically begin our study), only [L], [M], and [T] are needed. The dimensions of any mechanical quantity are written as powers of these three.

Example: Volume

• Volume = length × breadth × height = [L] × [L] × [L] = [L³]
• Dimension: [M⁰ L³ T⁰] (zero dimension in mass and time)

Example: Velocity

• Velocity = displacement / time = [L]/[T] = [L T⁻¹]
• Dimension: [M⁰ L T⁻¹]

Example: Acceleration

• Acceleration = change in velocity / time = [L T⁻¹]/[T] = [L T⁻²]
• Dimension: [M⁰ L T⁻²]

Example: Force

• Force = mass × acceleration = [M] × [L T⁻²] = [M L T⁻²]
• Dimension: [M L T⁻²]

Example: Energy/Work

• Energy = force × distance = [M L T⁻²] × [L] = [M L² T⁻²]
• Dimension: [M L² T⁻²]

Important observation: Different physical quantities can have the same dimensions. For instance:

• Work and torque both have dimensions [M L² T⁻²]
• Angular velocity (rad/s) and frequency (1/s) both have dimensions [T⁻¹]

This is why dimensional analysis alone cannot distinguish between such quantities—we need definitions and relationships to tell them apart.

1.5 DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS

The dimensional formula of a physical quantity expresses its dimensions in terms of the base quantities. It is typically written in the form [Mᵃ Lᵇ Tᶜ …].

A dimensional equation is obtained by equating a physical quantity with its dimensional formula:

• [Velocity] = [M⁰ L T⁻¹]
• [Force] = [M L T⁻²]
• [Density] = [M L⁻³ T⁰]

These equations are not algebraic equations in the usual sense—they represent dimensional relationships, not numerical equalities.

1.6 DIMENSIONAL ANALYSIS AND ITS APPLICATIONS

Dimensional analysis has three main applications: checking dimensional consistency, deducing relationships between quantities, and converting units.

1.6.1 Checking Dimensional Consistency

The principle of homogeneity of dimensions states that every term in a physically valid equation must have the same dimensions. Only quantities with identical dimensions can be added, subtracted, or equated.

Why this is useful: A dimensionally inconsistent equation is guaranteed to be wrong. This provides a quick way to catch errors.

Example: Check the equation for displacement under uniform acceleration:

x = x₀ + v₀t + (1/2)at²

Dimensions of each term:

• [x] = [L]
• [x₀] = [L]
• [v₀t] = [L T⁻¹][T] = [L]
• [(1/2)at²] = [L T⁻²][T²] = [L]

All terms have dimensions of [L]. The equation is dimensionally consistent.

Example: Check the equation for kinetic energy:

K = (1/2)mv² = mgh

Left side: [M] × [L² T⁻²] = [M L² T⁻²]
Right side: [M] × [L T⁻²] × [L] = [M L² T⁻²]

Both sides have the same dimensions. The equation is dimensionally consistent (though we’d need to verify the factor 1/2 through other means).

Example: Check if force can be expressed as mass × velocity:

F = mv

Left: [M L T⁻²]
Right: [M] × [L T⁻¹] = [M L T⁻¹]

These are different—the equation is dimensionally incorrect and cannot be physically valid.

Important limitation: Dimensional consistency is a necessary condition but not a sufficient one. A dimensionally consistent equation could still be wrong (e.g., missing a dimensionless factor, using the wrong combination of variables). However, a dimensionally inconsistent equation is definitely wrong.

Special functions: The arguments of trigonometric (sin, cos, tan), logarithmic (log, ln), and exponential (e^x) functions must be dimensionless:

• sin(θ): θ must be dimensionless (radians are dimensionless)
• log(3.5): 3.5 must be dimensionless
• e^(kt): kt must be dimensionless (k would have dimension T⁻¹)

This is because mathematical series expansions for these functions require all terms to have the same dimensions.

1.6.2 Deducing Relations Among Physical Quantities

Dimensional analysis can help derive relationships between physical quantities when we know which quantities the result depends on. The technique is called method of dimensions or dimensional analysis.

Example: Simple Pendulum
Suppose we want to find the time period T of a simple pendulum. We know it may depend on:

• Length of pendulum (l)
• Mass of bob (m)
• Acceleration due to gravity (g)

We hypothesize that T ∝ lˣ gʸ mᶻ (product relationship)

Writing dimensions:

[T] = [L]ˣ [L T⁻²]ʸ [M]ᶻ
[M⁰ L⁰ T¹] = [Mᶻ Lˣ⁺ʸ T⁻²ʸ]

Equating exponents:

• For M: z = 0
• For L: x + y = 0
• For T: -2y = 1 → y = -1/2

Solving: y = -1/2, x = 1/2, z = 0

Therefore: T = k × l^(1/2) × g^(-1/2) × m^0 = k√(l/g)

The constant k cannot be determined by dimensional analysis—it must come from experiment or more detailed theory. In fact, k = 2π.

The power of this method: Without solving any differential equations, we’ve derived the correct functional form for the pendulum’s period. We know:

• The period is independent of mass (surprising but true for small oscillations)
• T ∝ √l (longer pendulum → longer period)
• T ∝ 1/√g (weaker gravity → longer period)

Limitations of dimensional analysis:

1. Cannot determine dimensionless constants (π, 2, 1/2, etc.)
2. Cannot distinguish between quantities with the same dimensions
3. Fails if the relationship involves trigonometric, exponential, or logarithmic functions
4. Requires knowing all relevant variables—if you miss one, you get the wrong relationship

Example where dimensional analysis fails:

• The period of a pendulum at large amplitudes depends on the angle θ, but sin θ is dimensionless—dimensional analysis cannot detect this dependence

Example of what dimensional analysis can and cannot distinguish:
Both kinetic energy (1/2mv²) and torque (r × F) have dimensions [M L² T⁻²], but they are very different physical quantities. Dimensional analysis alone cannot tell them apart.

Practical tips for using dimensional analysis:

1. Use scientific notation to avoid ambiguity in significant figures
2. Keep one extra digit in intermediate steps
3. Remember that exact numbers (2, π, constants) have infinite significant figures
4. In dimensional analysis, ignore dimensionless factors—they don’t affect dimensions

1.6.3 Unit Conversion Using Dimensional Analysis

Dimensional analysis is also useful for converting between units. The key idea is that multiplying by 1 (in the form of a ratio of equivalent quantities) doesn’t change the value.

Example: Convert 60 km/h to m/s.

60 km/h = 60 × (1000 m / 1 km) × (1 h / 3600 s)
        = 60 × 1000/3600 m/s
        = 16.7 m/s

Example: Convert density 2.7 g/cm³ to kg/m³.

2.7 g/cm³ = 2.7 × (1 kg / 1000 g) × (100 cm / 1 m)³
          = 2.7 × (1/1000) × (100³)
          = 2.7 × 10³ kg/m³
          = 2700 kg/m³

This method works because we’re systematically cancelling units like algebraic symbols—essentially applying the same dimensional analysis principles we’ve discussed.

Summary of Key Points:

1. Measurement requires a number and a unit—both are essential
2. SI system has 7 base units and numerous derived units
3. Significant figures reflect measurement precision—rules govern their identification and use in calculations
4. Dimensional analysis checks equation consistency and can derive relationships
5. The principle of homogeneity requires all terms in an equation to have the same dimensions
6. Limitations of dimensional analysis include the inability to find dimensionless constants and distinguish between quantities with the same dimensions