Unit & Measurement

Table of Contents
1. Measurement
2. Unit
3. The International System of Units (SI)
4. The Seven Base Units of Measurement
5. Derived units
6. Units of long distance
7. Units of short distance
8. Practical notes and applications
9. Summary
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Measurement

Measurement is the process of determining the magnitude of a physical quantity by comparing it with an agreed standard. A measurement yields a numerical quantity together with a unit that identifies the kind of physical quantity measured.

• Measurement requires an instrument and a procedure that together produce a numerical result and an associated unit.
• Modern science, engineering and commerce commonly use the metric system; its internationally agreed modern form is the International System of Units (SI).
• Use of SI simplifies communication, design, calculation and manufacture by providing standard units, prefixes and rules for combining them.

Unit

A unit is a particular physical quantity, defined and adopted by convention, with which other quantities of the same kind are compared to express their values. A unit must be reproducible in practice so that different observers obtain consistent values for the same measurement.

The International System of Units (SI)

The International System of Units (SI) (from the French Système International d'Unités) is the internationally accepted system of measurement for science, technology, industry and everyday use. SI defines a small set of base units from which derived units are formed by multiplication and division.

Historical context

Before the international adoption of SI, many different systems of units were in use; this diversity produced confusion in science, engineering and trade. Notable historical systems include:

• CGS system: Centimetre-gram-second system. It was widely used in physics and had useful variants (for example, Gaussian units) for electromagnetism.
• FPS / Imperial system: Foot-pound-second and related units. It persisted in some English-speaking countries for civil and engineering work.
• MKS system: Metre-kilogram-second system. It scaled better for practical engineering and became the principal precursor to SI.

Formation and adoption of SI

The General Conference on Weights and Measures (CGPM) and the International Bureau of Weights and Measures (BIPM) develop and maintain SI. SI evolved during the 20th century and has been updated periodically to reflect advances in measurement science. SI provides a coherent set of base units, prefixes and rules for forming derived units so that quantities expressed in SI are consistent and internationally comparable.

The Seven Base Units of Measurement

The SI is founded on seven base quantities and their corresponding base units. Where possible, these base units are defined by fixing the numerical values of fundamental physical constants, making the definitions stable, reproducible and independent of physical artefacts.

Length - Metre (m)

• The metre is defined in terms of the speed of light in vacuum. The metre is the distance travelled by light in vacuum in a specified fraction of a second.

Definition in constant form:

\[c = 299\,792\,458\ \mathrm{m\,s^{-1}}\]

The metre is the distance light travels in vacuum during \\(1/299\\,792\\,458\\) of a second.

Time - Second (s)

• The second is defined by a property of the caesium-133 atom: it equals the duration of a specified number of periods of the radiation corresponding to a particular atomic transition.

The defining number is:

\[9\,192\,631\,770\]

Thus, one second is 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. Time is realised using highly precise atomic clocks that measure this transition frequency.

Mass - Kilogram (kg)

• Historically the kilogram was defined by a platinum-iridium cylinder kept at the BIPM; national copies were used as standards. That artefact-based definition has been replaced by a constant-based definition.
• Since 20 May 2019 the kilogram is defined by fixing the numerical value of the Planck constant.

The fixed value is:

\[h = 6.62607015\times 10^{-34}\ \mathrm{J\,s}\]

With this definition, mass is linked to the Planck constant and can be realised experimentally (for example using a Kibble balance) without reference to a physical prototype.

Electric current - Ampere (A)

• Historically the ampere was defined by the force between two parallel conductors carrying current. Since 20 May 2019 the definition is based on a fixed value of the elementary charge.
• The ampere is the flow of one coulomb of charge per second; the coulomb is derived from the defined value of the elementary charge.

The fixed value is:

\[e = 1.602176634\times 10^{-19}\ \mathrm{C}\]

Thermodynamic temperature - Kelvin (K)

• Historically the kelvin was defined via the triple point of water. Since 20 May 2019 the kelvin is defined by fixing the Boltzmann constant.
• This links temperature directly to energy at the microscopic scale and improves reproducibility across different laboratory methods.

The fixed value is:

\[k = 1.380649\times 10^{-23}\ \mathrm{J\,K^{-1}}\]

Amount of substance - Mole (mol)

• Historically a mole was defined by reference to 0.012 kilogram of carbon-12. Since 20 May 2019 the mole is defined by a fixed numerical value of the Avogadro constant.
• One mole of a specified elementary entity (atoms, molecules, ions, electrons, etc.) contains exactly the fixed number of such entities, regardless of the substance.

The fixed value is:

\[N_{A} = 6.02214076\times 10^{23}\ \mathrm{mol^{-1}}\]

Luminous intensity - Candela (cd)

• The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of a specified frequency and has a specified radiant intensity in that direction.
• The definition connects photometric quantities (perceived brightness) to radiometric quantities (physical power) by a numerical factor chosen at a reference frequency.

The defining frequency and factor are:

\[\text{frequency} = 540\times 10^{12}\ \mathrm{Hz}\]

\[\text{radiant intensity} = \frac{1}{683}\ \mathrm{W\,sr^{-1}}\]

Derived units

SI derived units are units for quantities that can be expressed algebraically in terms of the seven base units. When useful, some derived units are given special names and symbols.

• Derived units are expressed as products of base units raised to integer powers; for example, velocity has units \(\mathrm{m\,s^{-1}}\) and force has units \(\mathrm{kg\,m\,s^{-2}}\).
• When a derived unit is named after a person, the unit symbol uses an uppercase initial letter while the full name is written in lower case (for example, hertz - symbol Hz; newton - symbol N).
• Common derived units include the newton (N) for force, the joule (J) for energy, the pascal (Pa) for pressure, the watt (W) for power and the coulomb (C) for electric charge.

Examples expressed in base units:

\[\text{force (newton)}:\ \mathrm{N} = \mathrm{kg\,m\,s^{-2}}\]\[\text{energy (joule)}:\ \mathrm{J} = \mathrm{N\,m} = \mathrm{kg\,m^{2}\,s^{-2}}\]

Units of long distance

• Light year: The distance that light travels in vacuum in one Julian year (365.25 days). A light year is commonly used in astronomy to express interstellar and intergalactic distances.

Approximate value:

\[1\ \text{light year} \approx 9.46\times 10^{15}\ \mathrm{m}\]

Parsec: The distance at which one astronomical unit subtends an angle of one arcsecond. The parsec is widely used in astrometry and galactic astronomy.

Relation to light years:

\[1\ \text{parsec} \approx 3.26\ \text{light years}\]

Thus parsecs and light years are convenient for communicating very large distances in astronomy.

Units of short distance

For microscopic and atomic scales, the following metric multiples and submultiples are commonly used; these make it straightforward to express lengths spanning many orders of magnitude.

• Micrometre (micron, µm):  \[1\ \mathrm{µm} = 10^{-6}\ \mathrm{m}\]
• Nanometre (nm): \[1\ \mathrm{nm} = 10^{-9}\ \mathrm{m}\]
• Angström (Å) - commonly used in atomic physics, chemistry and crystallography (not an SI base unit but widely used for convenience)

•  \[1\ \mathrm{Å} = 10^{-10}\ \mathrm{m}\]

• Picometre (pm)

•  \[1\ \mathrm{pm} = 10^{-12}\ \mathrm{m}\]

Practical notes and applications

Understanding SI and units is essential across government examinations, engineering, science and everyday life. Key practical points:

• Always quote both a numerical value and its unit when reporting a measurement (for example, \\(5\\ \\mathrm{m}\\) not just \\(5\\)).
• Use SI prefixes (kilo-, mega-, milli-, micro-, nano-, etc.) to express quantities comfortably across many orders of magnitude.
• When converting between systems, carry units through each step of a calculation to avoid errors; dimensional analysis is a powerful check on formulae and results.
• Realisation of SI units in laboratories uses experimental techniques (atomic clocks, Kibble balances, quantum electrical standards, etc.) that link macroscopic measurements to invariant constants of nature.

Summary

The International System of Units (SI) provides a coherent, internationally agreed framework of base and derived units. Since 2019, the seven base units are defined by fixing exact values of fundamental physical constants. This modern approach improves reproducibility and connects measurement to invariant properties of nature, supporting science, technology and trade.