Notes on SI Units and Measurements

1.1 Introduction

Physics is a quantitative science in which we measure various physical quantities during experiments.
In daily life we measure many quantities: size of objects, volume of liquids, amount of matter, weight of produce, body temperature, length of cloth, etc.
A measurement involves a comparison with a standard measuring unit that is internationally accepted.
- Examples of units: 1 kg, 500 g, etc.
The measured quantity is expressed as a number followed by its unit, e.g., the length of a wire is written as $5\text{ m}$ where m is the unit and 5 is the value in that unit.
Different quantities use different units, e.g., length in metres (m), time in seconds (s), mass in kilograms (kg).
The standard measure of any quantity is called the unit of that quantity.

1.2 System of Units

Historical systems include CGS (Centimetre–Gram–Second), MKS (Metre–Kilogram–Second), FPS (Foot–Pound–Second).
SI (System International) was adopted following recommendations of the 14th International General Conference on Weights and Measures in 1971.
SI uses the decimal system, making conversions within the system simple and convenient.
Recall prompts (for self-review):
- 1) What is a unit?
- 2) Which units have you used in the lab for measuring (i) length, (ii) mass, (iii) time, (iv) temperature?
- 3) Which system of units have you used?

1.2.1 Fundamental Quantities and Units

Seven fundamental quantities (quantities that do not depend on any other quantities for their measurement):
- Length
- Mass
- Time
- Temperature
- Electric current
- Luminous intensity
- Amount of substance
Fundamental units (units for the fundamental quantities):
- Length: metre, symbol $ext{m}$
- Mass: kilogram, symbol $ext{kg}$
- Time: second, symbol $ext{s}$
- Temperature: kelvin, symbol $ext{K}$
- Electric current: ampere, symbol $ext{A}$
- Luminous intensity: candela, symbol $ext{cd}$
- Amount of substance: mole, symbol $ext{mol}$
Table (fundamental quantities with SI units and symbols):
- Length → metre → $m$
- Mass → kilogram → $kg$
- Time → second → $s$
- Temperature → kelvin → $K$
- Electric current → ampere → $A$
- Luminous Intensity → candela → $cd$
- Amount of substance → mole → $mol$

1.2.2 Derived Quantities and Units

Derived quantities depend on the seven fundamental quantities and can be expressed in terms of them.
Derived units are expressed in terms of the fundamental units. Examples:
- Velocity (speed): displacement per unit time
- $\text{Unit: } v = \dfrac{\Delta s}{\Delta t} \quad \Rightarrow\quad [v] = [L][T]^{-1} = \text{m s}^{-1}$
- Momentum: product of mass and velocity
- $p = mv\quad\Rightarrow\quad [p] = [M][L][T]^{-1} = \text{kg m s}^{-1}$
- Force: mass times acceleration; SI unit: newton (N) = $\text{kg m s}^{-2}$
- Energy/Work: $J = \text{N} \cdot \text{m} = \text{kg m}^2\text{/s}^2$
- Impulse: $J = F \Delta t = \text{kg m s}^{-1}$
Supplementary (non-fundamental) units:
- Plane angle: radian (rad)
- Solid angle: steradian (sr)
Definitions:
- Plane angle: $d\theta = \dfrac{ds}{r}\,,$ angle subtended by an arc at the centre of a circle; unit is radian.
- Solid angle: $d\Omega = \dfrac{dA}{r^2}$ ; measured in steradians; the solid angle subtended by a sphere of radius $r$ is $\Omega = 4\pi\text{ sr}$ .
Example: Solid angle subtended by the Moon
- Moon diameter ≈ 3474 km; distance Earth–Moon ≈ 3.84×10^8 m
- Angular diameter α ≈ (diameter)/(distance) ≈ 3.474×10^6 / 3.84×10^8 ≈ 0.00904 rad
- Solid angle ≈ $\Omega_{\text{moon}} \approx \pi(\alpha/2)^2 = \frac{\pi \alpha^2}{4} \approx 6.4\times 10^{-5}\ \text{sr}$
Radian-to-degree relation: $\pi\text{ rad} = 180^{\circ}$
Do you know? (conventions on radians and degrees and related relations)

1.2.3 Conventions for the use of SI Units

Rules for writing units and symbols:
- (1) The symbol represents the unit of any physical quantity; use symbol for the unit rather than the full name in formulas.
- (2) Full names start with a lowercase letter, e.g., newton, joule; but symbols for units named after persons are capitalized, e.g., N (Newton), J (Joule).
- (3) Symbols for units do not take plural forms: e.g., 20 N, not 20 Newtons.
- (4) Symbols do not end with a full stop.
- (5) Units in numerator and denominator should be written as a single ratio, e.g., acceleration: $\text{m s}^{-2}$ , not $\text{m}/\text{s}/\text{s}$ .
- (6) Avoid combining units with symbols when expressing a quantity by a combination of two units; e.g., prefer $\text{J kg}^{-1} \text{K}^{-1}$ to writing words; avoid $\text{J/kg K}$ as incorrect usage.
- (7) Prefix symbols (e.g., k, m, µ) are used before unit symbols to form new unit symbols; prefixes can be raised to powers of 10; prefixes multiply the unit by a power of 10.
Examples: kilowatt (kW), millisecond (ms), microsecond (µs), etc., and note about the use of prefixes and powers.

1.3 Measurement of Length

The metre (m) is the SI unit for length.
Historical definition changes:
- 1960: metre based on the wavelength of orange-red light from krypton.
- 1983: metre defined as the length of the path travelled by light in vacuum during a time interval of $1/299{,}792{,}458\text{ s}$ .
Speed of light in vacuum: $c = 299{,}792{,}458\ \text{m s}^{-1}$ .
Typical distances (order of magnitude examples):
- Distances across galaxies, stars, planetary scales, Earth dimensions, etc., spanning from 10^22 m down to 10^-15 m (atomic scales and smaller).
1.3.1 Measurements of Large Distance: Parallax method
- Parallax: apparent shift in position of a nearby object due to a change in the observer’s position.
- Simple demonstration: hold a hand in front of you, close one eye, then the other; the hand seems to shift against the background.
- Parallax angle θ is measured using two observer positions; distance OP can be determined from the baseline E1E2 as $OP = \dfrac{E1E2}{\theta}$ (θ in radians).
- For distant planets/stars, θ is very small; AB is treated as an arc of length b on a circle of radius D: $AB = b,\ AS = BS = D,\ \theta \approx \dfrac{AB}{D}$ , hence $D = \dfrac{b}{\theta}$ .
1.3.2 Measurement of Distance to Stars
- The Sun is the closest star; the next closest star is at about 4.29 light-years away.
- Parallax for stars measured from two farthest points on Earth is too small; instead parallax is measured using points 2 AU apart (Earth's orbit).
1.3.3 Measurement of the Size of a Planet or a Star
- If d is the diameter of a planet and D is its distance from Earth, the angular diameter α measured by a telescope is the angle subtended by the diameter at Earth.
- For large D, $d = D\alpha$ with α in radians.
1 astronomical unit (AU) = $1.496 \times 10^{11} \text{ m}$ ; 1 light year = $9.46 \times 10^{15} \text{ m}$ ; 1 parsec (pc) ≈ $3.08 \times 10^{16} \text{ m}$ ≈ 3.26 light years.
- A light year is the distance light travels in one year; 1 AU is the mean Earth–Sun distance; 1 pc is the distance at which 1 AU subtends 1 arcsecond (1″).
1.3.4 Measurement of Very Small Distances
- For atoms/molecules, conventional tools like Vernier calipers are inadequate.
- Electron microscopes (∼0.6 Å) or tunnelling electron microscopes are used to measure atomic sizes (≈1 Å).
Example 1.2: A star is 5.5 light years away from the Earth. What parallax in arcsec will it subtend when viewed from two opposite points along Earth’s orbit? (Two opposite points are 2 AU apart.)
- Solution (from text): θ ≈ AB/distance = 2 AU / distance; with distance in AU, θ in radians; conversion to arcseconds gives ≈ 1.186 arcsec.
1.3.3 Size-distance relationship (recap):
- For very large distances, angular relationships help infer sizes and separations; units used include AU, ly, pc.
1.3.4 Useful size and distance references (order of magnitude):
- 1 AU ≈ 1.496×10^{11} m; 1 pc ≈ 3.086×10^{16} m; 1 ly ≈ 9.46×10^{15} m.
- Distances across the Solar System and to stars vary across many orders of magnitude.
1.4 Measurement of Mass
- 1889: the kilogram was defined as the mass of a platinum–iridium alloy cylinder kept in a controlled environment at the International Bureau of Weights and Measures.
- 2019: the kilogram definition was revised (the transcript notes a shift toward electromagnet-based balance; historically this led to the modern redefinition based on fundamental constants such as Planck’s constant). The transcript states the kilogram could be described in terms of the current required to balance an electromagnet to balance a standard kilogram mass; this reflects the historical description prior to the 2019 redefinition.
- Atomic-scale masses are easier to compare using atomic mass units (amu).
- 1 amu = $1.6605402 \times 10^{-27}\ \text{kg}$ with an uncertainty in the last digits (the text notes an uncertainty of two last digits).
1 amu context:
- Atomic mass unit provides a convenient scale for atoms and molecules relative to a standard atom (C-12).
1.5 Measurement of Time
- SI unit of time is the second (s).
- Earlier, the mean solar day (24 h) was used as reference, but the solar day length varies due to Earth’s rotation.
- The current standard: a cesium atomic clock.
- A second is defined as the time taken for 9{,}192{,}631{,}770 vibrations of the radiation emitted during a transition between two hyperfine states of the Cs-133 atom:
- $\text{second} = 9{,}192{,}631{,}770\ \text{vibrations}$ of that transition.
1.5.1 Symbol conventions for quantities
- The text notes standard symbols: $L$ for length, $M$ for mass, $T$ for time, $K$ for temperature, $I$ for current, $C$ for luminous intensity, $mol$ for amount of substance.
- Dimensions of a quantity are the powers to which the fundamental units must be raised to obtain the unit of that quantity. This is called the dimensional formula and is written with square brackets, without commas, e.g., for velocity: $[v] = [L][T]^{-1}$ .
1.6 Dimensions and Dimensional Analysis
1.6.1 Uses of Dimensional Analysis
- (i) Check dimensional correctness of equations (principle of homogeneity).
- Example: v = u + a t
  - Dimensions: $[v] = [L][T]^{-1}$ ; $[u] = [L][T]^{-1}$ ; $[a t] = [L][T]^{-2} [T] = [L][T]^{-1}$
  - Since LHS and RHS have the same dimensions, the equation is dimensionally correct.
- (ii) Establish relationships between related quantities. Example: pendulum period T depends on length l and gravity g.
- Suppose $T \propto l^a g^b$ . Dimensional analysis yields: $[M]^0[L]^0[T]^1 = [L]^a [L T^{-2}]^b = [L]^{a+b} [T]^{-2b}$
- Equating exponents gives: $a+b = 0$ and $-2b = 1\Rightarrow b=-\tfrac{1}{2}, a=\tfrac{1}{2}$
- Hence $T = k \sqrt{\dfrac{l}{g}}$ with a dimensionless constant k. Experimentally, $k = 2\pi$ , giving
  $T = 2\pi \sqrt{\dfrac{l}{g}}.$
- (iii) Conversion factors between unit systems. Example: 1 J = ? erg. Dimensional analysis shows:
- Work has dimensions $[M][L]^2[T]^{-2}$ ; in SI, $L=M^?$ , and in CGS units, etc., leading to the classic relation
  $1\ \text{J} = 10^7\ \text{erg}.$
1.6.2 Limitations of Dimensional Analysis
- (1) Some dimensionless constants cannot be derived from dimensions alone; experiments determine their numeric value.
- (2) Dimensional analysis cannot derive relations involving trigonometric, exponential, or logarithmic functions (dimensionless quantities) alone.
- (3) If the equation contains a dimensionful constant that is not dimensionless, the method may fail (e.g., gravitation F = G m1 m2 / r^2; G is not dimensionless).
- (4) If the correct equation contains extra terms with the same dimensions, dimensional analysis alone may not reveal their presence (example: S = ut + (1/2)at^2 is dim. correct but not obvious from a simpler form).
1.7 Accuracy, Precision and Uncertainty in Measurement
- Accuracy: how close a measurement is to the true value.
- Precision: how reproducible or how close repeated measurements are to each other.
- A result can be accurate, precise, both, or neither.
- Common sources of uncertainty include instrument quality, observer skill, method, and environmental conditions.
- Example prompt: if ten students measure a length with a meter scale to the nearest mm, results may differ due to parallax and reading errors.
1.8 Errors in Measurements
1.8.1 Estimation of Error
- Suppose repeated readings for a quantity are $a1, a2, \ldots, a_n$ .
- Arithmetic mean (most probable value):
- $a{\text{mean}} = \frac{1}{n}\sum{i=1}^n a_i$
- Absolute error for a reading: $\Delta ai = |ai - a_{\text{mean}}|$
- Mean absolute error: $\Delta a{\text{mean}} = \frac{1}{n}\sum{i=1}^n \Delta a_i$
- Result representation: $a = a{\text{mean}} \pm \Delta a{\text{mean}}$
- Relative error: $\text{relative error} = \frac{\Delta a{\text{mean}}}{a{\text{mean}}}$
- Percentage error: $\% ext{ error} = \left(\frac{\Delta a{\text{mean}}}{a{\text{mean}}}\right) \times 100\%$
1.8.2 Combination of Errors
- Errors in sum/difference: If $Z = A \pm \Delta A$ and $W = B \pm \Delta B$ , then for $Z + W$ , the maximum absolute error is
- $\Delta Z = \Delta A + \Delta B$ ;
- For $Z = A \pm B$ , the maximum absolute error is also $\Delta Z = \Delta A + \Delta B$ .
- Errors in product/division: If $Z = AB$ with $A \pm \Delta A\,, B \pm \Delta B$ , then
- $\frac{\Delta Z}{Z} \approx \frac{\Delta A}{A} + \frac{\Delta B}{B}$ (neglecting the product $\Delta A \Delta B$ ).
- For powers: If $Z = A^p$ , then
- $\frac{\Delta Z}{Z} \approx |p| \frac{\Delta A}{A}$ .
- General rule: for $Z = A^p B^q C^r$ ,
- $\frac{\Delta Z}{Z} \approx |p| \frac{\Delta A}{A} + |q| \frac{\Delta B}{B} + |r| \frac{\Delta C}{C}$ .
Example 1.5 (illustrative): compute mean, mean absolute error, and percentage error for a set of radius measurements; use the given values to obtain the most probable value (mean), etc.
1.8.2 (additional): Combination of errors including scenarios with powers and multiple quantities.
1.8.3 (power exponent section): If a measured quantity Z = A^3, the relative error is three times the relative error in A; generalization to multiple factors is given via (1.9) above.
1.9 Significant Figures
- Least count of an instrument limits the precision.
- Significant digits are those that carry meaning contributing to precision.
- Rules:
- (1) All nonzero digits are significant.
- (2) Zeros between nonzero digits are significant.
- (3) Leading zeros to the left of the first nonzero digit are not significant (e.g., 0.001405 has four significant digits).
- (4) Trailing zeros to the right of the last nonzero digit are significant if a decimal point is present (e.g., 1.500 or 0.01500 have four significant figures).
- When reporting a measurement with ambiguous digits, scientific notation helps: write as A × 10^n with 0.5 ≤ A < 5.
- Examples and demonstrations include the Earth radius, elementary charges, etc., in order-of-magnitude terms.
Definitions of SI Units (context for the 2019 redefinitions)
- Before 2019, the kilogram was defined by a physical artifact (a platinum–iridium prototype).
- In 2018–2019, the General Conference on Weights and Measures defined SI units by fixing fundamental constants.
- Seven base units, each defined using a fundamental constant (with exact values):
- The Planck constant, $h = 6.62607015 \times 10^{-34}\ \text{J s}$ (kg m^2 s^{-1}).
- The elementary charge, $e = 1.602176634 \times 10^{-19}\ \text{C}$ (A s).
- The Boltzmann constant, $k = 1.380649 \times 10^{-23}\ \text{J K}^{-1}$ (kg m^2 s^{-2} K^{-1}).
- The Avogadro constant, $N_A = 6.02214076 \times 10^{23}\ \text{mol}^{-1}$ .
- The speed of light in vacuum, $c = 299{,}792{,}458\ \text{m s}^{-1}$ .
- The Cs-133 hyperfine transition frequency, $\Delta\nu_{Cs} = 9{,}192{,}631{,}770\ \, \text{Hz}$ .
- The luminous efficacy of monochromatic radiation of frequency $540 \times 10^{12}\ \text{Hz}$ , $K_{cd} = 683\ \text{lm W}^{-1} = 683\ \text{cd sr}^{-1}$ .
- The metre is defined via the speed of light and the second as above; the second and mole definitions are anchored to their constants; other units depend on these base units via the defining relationships.
Figures illustrate the dependencies among units and constants (new SI definitions vs. old definitions).

Exercises (key prompts and problems from the chapter)

Exercises 1: Multiple choice
- i) The dimensional formula $[L^1 M^1 T^{-2}]$ corresponds to which quantity? (Options: Velocity, Acceleration, Force, Work)
- ii) If the error in measuring the sides of a rectangle is 1%, the error in its area is: (A) 1%, (B) 1/2%, (C) 2%, (D) None of the above.
- iii) Light year is a unit of: (A) Time, (B) Mass, (C) Distance, (D) Luminosity.
- iv) Dimensions of kinetic energy are the same as those of: (A) Force, (B) Acceleration, (C) Work, (D) Pressure.
- v) Which is not a fundamental unit? (A) cm, (B) kg, (C) centigrade, (D) volt.
Exercises 1: Short questions
- i) If Star A is farther than Star B, which star has a larger parallax angle?
- ii) What are the dimensions of the quantity ll/g (l is length, g is gravitational acceleration)?
- iii) Define absolute error, mean absolute error, relative error, and percentage error.
- iv) Describe what significant figures are and what order of magnitude means.
- v) If measured values are A ± ΔA and B ± ΔB, what is the maximum possible error in A ± B? Show that for Z = A ± B, ΔZ = ΔA + ΔB (in the worst case).
- vi) Derive the formula for kinetic energy from dimensional analysis.
Exercises 2: Numerical problems (selected examples)
- i) Masses 15.7 ± 0.2 kg and 27.3 ± 0.3 kg; find total mass and its error. (Answer: 43 kg, ± 0.5 kg)
- ii) Distance 100 ± 1 s; measured distance 5.2 ± 0.1 m; compute speed and its relative error. (Answer: 0.052 m s^{-1}, ± 0.0292 m s^{-1})
- iii) Electron in a uniform magnetic field B with velocity perpendicular to B; find dimensions of B from the magnetic force formula; (Answer: $[L^0 M^1 T^{-2} I^{-1}]$ )
- iv) A hollow sphere of radius 2 m made from a rope with square cross-section edge 4 mm; estimate total rope length to nearest order of magnitude (≈ 10^6 m or 10^3 km).
- v) Nuclear radius R ∝ A^{1/3} with R = 1.3×10^{-16} A^{1/3} m; for A = 125, estimate R in metres (order of magnitude: around 10^{-15} m).
- vi) Vernier calipers with least count 0.01 cm measuring a plate; measurements 3.11, 3.13, 3.14, 3.14 cm; compute mean, mean absolute error, and percentage error (Answer: 3.13 cm, 0.01 cm, 0.32%).
- vii) Percentage error in kinetic energy for mass 60.0 ± 0.3 g moving at 25.0 ± 0.1 cm s^{-1} (Answer: 1.3%).
- viii) Ohm’s law measurements: unknown resistances 6.12 Ω, 6.09 Ω, 6.22 Ω, 6.15 Ω; compute mean absolute error, relative error, and percentage error (Answers: 0.04 Ω, 0.0065 Ω, 0.65%).
- ix) Dimensional analysis: a freely falling body velocity after distance h is v; prove v = k g h^{1/2} dimensionally (factor form varies; the standard result is v ∝ sqrt(gh)).
- x) If v0 is initial velocity, formulate dimensional relations for a, b, c in v = a t + b t^2 + c t^3 (dimensionally consistent forms and their results).
- xi) A rectangular sheet has dimensions 4.234 m × 1.005 m and thickness 2.01 cm; compute area and volume with correct significant figures (Answers: 4.255 m^2, 8.552 m^3).
- xii) If length l = 4.00 ± 0.001 cm, radius r = 0.0250 ± 0.001 cm, mass m = 6.25 ± 0.01 g; compute percentage error in density.
- xiii) Jupiter’s angular diameter measured at distance of 824.7 million km to be 35.72″; compute Jupiter’s diameter (Answer: ≈ 1.428×10^5 km).
- xiv) If a quantity X = a^4 b^3 c^1 d^2 with percentage errors in a, b, c, d of 2%, 3%, 3%, and 4% respectively, compute percentage error in X (Answer: 20%).
- xv) Significance figures for: 0.003 m^2, 0.1250 g cm^{-2}, 6.4×10^6 m, 1.6×10^{-19} C, 9.1×10^{-31} kg (Answers: 1, 4, 2, 2, 2).
- xvi) Diameter of a sphere: 2.14 cm; compute volume to correct significant figures (Answer: 5.13 cm^{3}).
These exercises reinforce concepts of dimensional analysis, errors, significant figures, and SI unit definitions, and they illustrate practical calculations for lab measurements and estimations.

Notes on commonly used constants and SI definitions (summary)

Fundamental constants and exact values (as per the new SI definitions):
- Planck constant: $h = 6.62607015\times 10^{-34}\ \text{J s}$
- Elementary charge: $e = 1.602176634\times 10^{-19}\ \text{C}$
- Boltzmann constant: $k = 1.380649\times 10^{-23}\ \text{J K}^{-1}$
- Avogadro constant: $N_A = 6.02214076\times 10^{23}\ \text{mol}^{-1}$
- Speed of light: $c = 299{,}792{,}458\ \text{m s}^{-1}$
- Caesium-133 hyperfine frequency: $\Delta\nu_{Cs} = 9{,}192{,}631{,}770\ \text{Hz}$
- Luminous efficacy: $K_{cd} = 683\ \text{lm W}^{-1} = 683\ \text{cd sr}^{-1}$
The metre, second, and mole are tied to these constants; other units are defined via their relationships to these base units.
Practical reminders for measurement and reporting:
- Always report units using symbols; prefer SI units; be mindful of prefixes and powers when expressing quantities.
- When reporting measurements, indicate significant figures consistent with the instrument’s least count and the uncertainty estimates.
- Use dimensional analysis as a quick consistency check for equations, but remember its limitations when constants are not dimensionless or higher-order terms are involved.
Summary of key formulas (LaTeX format):
- Velocity: $[v] = [L][T]^{-1}$ ; $v = \dfrac{\Delta s}{\Delta t}$ ; $\text{Unit: } \text{m s}^{-1}$
- Momentum: $[p] = [M][L][T]^{-1}$ ; $p = mv$
- Force: $[F] = [M][L][T]^{-2}$ ; $F = ma$ ; $\text{Unit: N} = \text{kg m s}^{-2}$
- Energy/Work: $[J] = [M][L]^2[T]^{-2}$ ; $J = F \cdot s$
- Power: (derived from energy per unit time) not explicitly listed, but related to $P = \dfrac{dW}{dt}$
- Density: $\rho = \dfrac{M}{V}$ ; $[\rho] = [M][L]^{-3}$ ; Unit: $\text{kg m}^{-3}$
- Acceleration: $a = \dfrac{\Delta v}{\Delta t}$ ; $[a] = [L][T]^{-2}$
- Angular relationships: $\alpha = \dfrac{\Delta \theta}{\Delta t}$ ; radian as unit for plane angle; $\Omega$ for solid angle with units sr.
Note: The transcript contains several misprints or typographical inconsistencies in some table values (e.g., distances in the distance table and some exponents). The essential concepts, definitions, and relationships have been preserved, with standard accepted values used where appropriate for clarity.