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 ...

In order to enhance your confidence there are Solved 'India Study Solution Test Series’ each set containing 10 important questions with hints & solutions for practice from the chapter Units, Dimensions, Measurements and Error Analysis

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

Physical World and Quantities - Fundamental or Basic, Derived, Supplementary quantities

Units - International System of Units, SI Units, Properties and Definitions, Norms for writing Unit Symbols

Notes on Measurements

Significant Figures and its Rules, Rounding Off a Digit

Types and Definition of Errors, Errors in Measurement, Error Analysis

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.

360^O = 2p rad

∴ 1^O = (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’.    

<< Physical World and Quantities (Fundamental or Basic, Derived, Supplementary)           

Notes on Measurements >>

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

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Units, Dimensions, Measurements and Error Analysis - Notes on Physical World and Quantities - fundamental or basic, derived, supplementary quantities

Hi Friends,

You may be aware that this chapter ‘Units, Dimensions, Measurements and Error Analysis’ though sounds preliminary, is not only important but has direct implications with all other chapters of Physics. Moreover, you will have at least some questions from this chapter whether in NEET, IIT JEE Main or JEE Advanced, AIPMT or in any other Medical and Engineering Entrance Exams or NTSE, KVPY etc.

Section wise notes (part-by-part from this & other chapters) containing important study materials (Tables, Formulae, etc.) have been/will be published in sequence.

Additionally, you have -

Solved 'India Study Solution Test Series’ each set containing exclusively selected 10 expected questions with hints & solutions from the chapter Units,Dimensions, Measurements and Error Analysis (click on the links given at the end of this post).

Also, scroll below to find a common syllabus for this chapter given at the end.    

Physics and Physical World

The word 'Science' has come from a Lain word, 'Scientica' which means, 'to know'. Science is the subject of knowledge achieved by systematic approach through observation and experience, aimed at understanding natural phenomena in as much detail as possible.

A scientific approach (method) involves various interconnected steps - 

Other Topics from this Chapter

Physical World and Quantities - Fundamental or Basic, Derived, Supplementary quantities

Units - International System of Units, SI Units, Properties and Definitions, Norms for writing Unit Symbols

Notes on Measurements

Significant Figures and its Rules, Rounding Off a Digit

Types and Definition of Errors, Errors in Measurement, Error Analysis

Dimensions, Dimensional Formulae, Dimensional Equation

Dimensional Analysis and its Applications

1.     Systematic Observation

2.     Controlled Experiment: A controlled experiment involves setting up two experimental conditions that are exactly same except for a single factor that the scientist manipulates.

3.     Rationalisation or Qualitative and Quantitative Reasoning

4.     Mathematical Modelling

5.     Interpretation: Statement based on above observations and some scientific theory. It can be scientifically altered (falsifiable).

6.     Verification or Falsification

Physics is one of the many disciplines of science originated from a Greek word ‘Fusis’, which means ‘nature’.

The two main thrusts in Physics are -

1.     Unification: that is explaining diverse physical phenomena in terms of concepts and laws. For example, the same law of gravitation discovered by Newton also explains planetary motion, motion of moons around a planet as well as a body falling on the earth.

2.     Reduction: It is deriving the properties of a bigger or more complex system from the properties and interaction of its constituent parts. For example, in thermodynamics the temperature is related to the average kinetic energy of molecules of the system.

The two main domains of interest in Physics are -

1.     Macroscopic Domain: This includes Classical Physics which is study of objects of finite size that can be in a laboratory, in a terrestrial scale or even on astronomical scale. For example, mechanics, optics, electrodynamics and thermodynamics.

2.     Microscopic Domain: It includes studies involving atomic, molecular and nuclear phenomena, also interaction of elementary particles like electron, protons.  

Technology	Scientific Principle
Steam / Diesel / Petrol Engines:	Laws of Thermodynamics
Refrigerator:	Laws of Thermodynamics
Aeroplane:	Bernoulli’s Principle in Fluid Dynamics
Radio & Television:	Principle of Communication Systems using Electromagnetic Waves
Rocket Propulsion:	Newton’s Laws of Motion
Computer:	Digital Logic
Orbital Motion of Satellites:	Kepler’s Laws of Planetary Motion
Electron Microscope:	Wave Nature of Matter
Cyclotron:	Electromagnetic Force on Charged Particles
Nuclear Reactor:	Nuclear Fission
Electric Generator, Transformer:	Faraday’s Law of Electromagnetic Induction
Sonar	Reflection of Ultrasonic Waves
Pressure Cooker	Rising of Boiling Point of water by increasing pressure
Photocell	Photoelectric Effect
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Physical Quantities

1.     Those quantities which can be measured by an instrument and by which we can describe the laws of physics are called physical quantities. For example - mass, length, time, velocity, force, density etc.

2.     Measurement is necessary to determine magnitude of a physical quantity so that we can compare two similar physical quantities and also prove physical laws or equations.

3.     A Physical Quantity is completely specified if, it has -

·  Numerical value only (ratio): e.g. refractive index, dielectric constant etc.

·  Magnitude only (scalar): e.g. mass, electric charge etc.

·  Magnitude & Direction (vector): e.g. displacement, torque etc.

4.     There are a few physical quantities which are not specified even by unit, magnitude or direction. Such physical quantities are known as Tensors for example, Moment of Inertia, Stress, Strain, Thermal Conductivity, Magnetic Susceptibility and Electrical Permittivity, etc.

5.     Physical Quantity = Numerical Value (n) x Unit (u). Since the Physical Quantity will be a definite or constant value so, if the unit(u) changes, the magnitude(n) will also change but product 'nu' will remain same. That means magnitude of a physical quantity and units are inversely proportional to each other. Larger the unit, smaller will be the magnitude.

6.     There are three types of Physical Quantities: (a) Fundamental or Basic Quantities. (b) Derived Quantities. (c) Supplementary Quantities.

Fundamental (Basic) Quantities: 

There are a large number of physical quantities out of which only a few are elementary quantities. We define them as fundamental or basic quantities which are independent of all other quantities and cover the entire span of physics. All other quantities can be derived or expressed in terms of the fundamental quantities by multiplication or division. Fundamental quantities are total 7 in numbers: Length (L), Time (T), Mass (M), Temperature (K), Electric current (A), Luminous intensity (Cd), Amount of substance (mol).

Derived Quantities:

Physical quantities which can be derived from or expressed in terms of basic quantities are called derived quantities. For example - Momentum, Speed etc.

Supplementary Quantities:

Apart from 7 fundamental quantities, there are 2 supplementary quantities:

1.     Plane angle (It is angle between two lines or the angle subtended by an arc of a circle at its center. Its SI unit is ‘Radian’).   

2.     Solid angle (Angle subtended by a given surface area of a spherical surface at its centre is called a solid angle. Its SI unit is ‘Steradian’).

Syllabus 

Need for measurement of physical quantities, Units for measurement and Dimensions, System of Units - SI, Fundamental and Derived Units. Dimensional Formula and Dimensional Equations. Dimensional Analysis and its Applications. Significant figures and rounding off the numbers. Measurement of length, mass, and time. Accuracy, Precision of Instruments and error analysis. 

                  System of Units, Definitions, Properties, Norms >>

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

• Physics MCQ Test Series with Solutions for NEET, IITJEE, BTech, MBBS and BDS Entrance Exams - Units, Dimensions, Measurements and Error Analysis
• Units, Dimensions, Measurements and Error Analysis - Physics MCQ test series and practice questions for BTech, MBBS, BDS admission tests  
• Free online solved Test Series on Units, Dimensions, Measurements and Error Analysis - Physics questions answers for IITJEE, NEET