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SI Units of Derived Quantities
‘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 (partbypart 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

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 platinumiridium 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 cesium133 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 carbon12. (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 10^{12} 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/s^{2}.
· Momentum = mv = (kg) (m/s). So unit of momentum = kg m/s.
· Force (F) = ma. So unit of force = (kg) (m/s^{2}) = 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 & Submultiples)
Multiple

SubMultiple
 
Power of 10

Prefix

Symbol

Power of 10

Prefix

Symbol

10^{1}

Deca

da

10^{–1}

deci

d

10^{2}

hecto

h

10^{–2}

centi

c

10^{3}

kilo

k

10^{–3}

milli

m

10^{6}

mega

M

10^{–6}

micro


10^{9}

giga

G

10^{–9}

nano

n

10^{12}

tetra

T

10^{–12}

pico

p

10^{15}

peta

P

10^{–15}

femto

f

10^{18}

exa

E

10^{–18}

atto

a

Some Practical Units of Length & Other Quantities
Light year = 9.46 x 10^{15} m

Parsec = 3.084 x 10^{16} m

Fermi = 10^{–15} m

Angstrom (A^{O}) = 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 10^{11} m

Quantity

Unit(s)

Mass

Solar Mass = 2 x 10^{30 }kg,
Dalton = 1.66 x 10^{–27} kg,
Chander Shekhar (Lunar mass) = 1.4 times the solar mass

Pressure

Pascal = 1Nm^{–2},
Bar = 10^{5} 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/s^{2} 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 & submultiples. For example: 1/10^{9} 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: Mm^{3} means (10^{6}m)^{3} and not 10^{6}m^{3}.
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}’.
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