# Units, Dimensions, Measurements and Error Analysis - Notes on International System of Units, SI Units, Properties and Definitions, Norms for writing unit symbols

Hi Friends,
‘Units, Dimensions, Measurements and Error Analysis’ is an important chapter as it has direct connection with all other chapters of Physics. And there will be at least a few questions from this chapter either in NEET, IIT JEE Main or JEE Advanced, AIPMT or in any other Medical and Engineering Entrance Exams or even in NTSE, KVPY.
As promised section wise notes (part-by-part from this & other chapters) containing important study materials (Tables, Formulae, etc.) have been/will be published in sequence, here it goes ...

### Units

Measurement of any physical quantity is expressed in terms of an internationally accepted, certain basic standard called ‘Unit’. For the measurement of a physical quantity a definite magnitude of quantity is taken as the standard and the name given to this standard is called ‘Unit’.
Properties of Unit
 Other Topics from this Chapter Dimensions, Dimensional Formulae, Dimensional Equation Dimensional Analysis and its Applications
1.       The unit should be well defined.
2.       The unit should be of some suitable and proper size.
3.       The unit should be easily reproducible.
4.       The unit should not change with time.
5.       The unit should not change with physical or surrounding conditions.

The International System of Units -
A complete set of both base and derived units is called the System of Units. If a physical quantity involves only length, mass and time then its units can be written in:
1.       CGS system: centimetre, gram and second are used as the base units for length, mass and time respectively.
2.       FPS system: foot, pound and second are used as the base units for length, mass and time respectively.
3.       MKS system: meter, kilogram and second are used as the base units for length, mass and time respectively.
4.       SI system: In 1971 an international organization ‘CGPM’ (General Conference on Weights and Measures) put forward a new system of units which was then internationally accepted as SI units or International System of Units. The SI system is a decimal system, also known as metric system, a modernised and extended form of CGS and MKS systems. In SI system, there are 7 base units and 2 supplementary units. (see Table below)

SI Units of Basic Quantities
 Base Quantity SI Units Name Symbol Definition Length meter m The meter is the length of the path travelled by light in vacuum during a time interval of 1/299792458 of a second. (1983) Mass kilogram kg The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at International Bureau of Weights and Measures, at Sevres near Paris, France. (1889) Time second s The 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 cesium-133 atom. (1967) Electric Current ampere A One ampere is that constant current which when maintained in two straight, thin and parallel conductors of infinite length, and placed 1 meter apart in vacuum, would produce between these conductors of a force of 2 x 10–7 newton per meter of length. (1948) Thermodynamic Temperature kelvin K One kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967) (The triple point of water is the temperature at which ice, water and water vapour can coexist). Amount of Substance mole mol The mole is the amount of substance of a system, which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon-12. (1972) Luminous Intensity candela cd One candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979)

SI Units of Supplementary Quantities
 Supplementary Quantities SI Units Name Symbol Definition Plane Angle radian rad One radian is the plane angle subtended by arc of length equal to the radius, at the center of the circle. ** Solid Angle steradian sr On steradian is the solid angle subtended at the center of a sphere by an arc of its surface equal to the square of radius of the sphere.
** Total angle subtended by a circle at its center is 2p radian. Another unit of plane angle is degree.
1O = (p / 180) rad = 60 minutes = 3600 seconds
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SI Units of Derived Quantities
·          Velocity (v) = displacement (meter) ÷ time (sec). So, unit of velocity will be m/s.
·          Acceleration (a) = change in velocity (m/s) ÷ time (s). So, unit of acceleration will be m/s2.
·          Momentum = mv = (kg) (m/s). So unit of momentum = kg m/s.
·          Force (F) = ma. So unit of force = (kg) (m/s2) = kgms–2 = newton (N).
·          Work = force (F) x displacement (S). So unit of work = Nm = joule (J).
·          Power = work ÷ time. So, unit of power = J/s = watt (W).
In the next posts see Formulae Chart (Dimensional formulae with their SI units for various other physical quantities).

Common SI Prefixes and Symbols (Multiples & Sub-multiples)
 Multiple Sub-Multiple Power of 10 Prefix Symbol Power of 10 Prefix Symbol 101 Deca da 10–1 deci d 102 hecto h 10–2 centi c 103 kilo k 10–3 milli m 106 mega M 10–6 micro  109 giga G 10–9 nano n 1012 tetra T 10–12 pico p 1015 peta P 10–15 femto f 1018 exa E 10–18 atto a

Some Practical Units of Length & Other Quantities
 Light year = 9.46 x 1015 m Parsec = 3.084 x 1016 m Fermi = 10–15 m Angstrom (AO) = 10–10 m Micron or Micrometer = 10–6 m Nano meter = 10–9 m Pico meter = 10–12 m Astronomical unit (A.U.) = 1.496 x 1011 m

 Quantity Unit(s) Mass Solar Mass = 2 x 1030 kg, Dalton = 1.66 x 10–27 kg, Chander Shekhar (Lunar mass) = 1.4 times the solar mass Pressure Pascal = 1Nm–2, Bar = 105 Nm–2, Time Shake = 10–8 sec Radio Activity Becquerel Radiation Dose for Cancer Rontgen

How to write SI Units in Symbols / Guidelines or Norms to follow for writing symbols of SI Units -

1.     Standard unit symbols are written in lower case roman type. For example: ‘meter’ has symbol ‘m’; ‘kilogram’ has symbol ‘kg’.
2.      Unit names are never capitalised. A unit symbol is capitalised only if it is named after a scientist. For example units like ‘newton’, ‘joule’ ‘ampere’ ‘volt’ etc. have symbols N, J, A, V respectively.
3.     If a unit which is named after a scientist contain two letters in symbol, then the initial letter of the symbol is written in capital. For example: ‘Hz’ for hertz, ‘Wb’ for weber, ‘Pa’ for pascal etc.
4.     Symbols are never followed by a full stop.
5.     Unit symbols are never written in plural. For example: 50 meters to be written as 50 m and not as 50 ms.
6.     Only single solidus (/) to be used for example, the SI unit of acceleration should be written as m/s2 or ms–2 and never as m/s/s.
7.     For extremely large or small value of quantities prefixes are used with their units to indicate actual multiples & sub-multiples. For example: 1/109 m or 10–9 m can be written as nano meter or ‘nm’.
8.     Prefix symbol is written very close to the unit symbol without any space between them. However, when the unit of a physical quantity is obtained by multiplying the units of two or more quantities, then these unit symbols are written with a space between them. For example: ‘ms–1’ indicates millisecond and ‘m s–1’ indicates meter per second.
9.     When a prefix is written before a unit symbol, the combined prefix and symbol should be as a one symbol which can be raised to a positive or negative power without any bracket. For example: Mm3 means (106m)3 and not 106m3.
10.  The use of double prefixes should be avoided as far as possible.
11.  Unit names and unit symbols are not to be used together such as, unit of linear momentum (= mass x velocity) should either be written as ‘kg m s–1’ or as ‘kilogram meter per second’, never as ‘kg meter s–1’.

Units, Dimensions, Measurements and Error Analysis - Solved Test Series, Practice Questions