Units and Dimensions

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Physical Quantities

The physical phenomena which can be measured are called Physical Quantities.

$\to$ Length, Time, Speed, and Magnetic Field are Physical Quantities as they can be measured.

$\to$ Smell, Taste are not Physical Quantities as they can not be measured in a general sense.

$\to$ There are two types of physical quantities: Fundamental Quantities and Derived Quantities.

Fundamental Quantities:

$\to$ The quantities that are independent of other quantities are called fundamental quantities.

$\to$ There are seven fundamental quantities. They are Length, Mass, Time, Temperature, Electric Current, Luminous Intensity, and Amount of Substance.

Derived Quantities:

$\to$ The quantities that are dependent on other quantities are called derived quantities.

$\to$ Area = Length $\times$ Length and Speed = $\dfrac{\text{Distance}}{\text{time}}$ . Here, area depends on length and speed depends on length and time hence they are derived quantities.

$\to$ Other derived quantities are Volume, Force, Power, Intensity of sound, Magnetic Flux, etc.

Units

Unit is the standard reference in which physical quantity is expressed.

$\to$ Length is expressed in meters and speed is expressed in meters/second. The meter and meter/second are the units.

System of Units

$\to$ The standard references can vary based on which unit Mass, Length, Time, and other fundamental quantities are measured. On the basis of that, there are four systems of units. They are CGS, FPS, MKS, and SI systems.

Quantity	CGS System	FPS System	MKS System
1. Length	Centimeter	Foot	Meter
2. Mass	Gram	Pound	Kilogram
3. Time	Second	Second	Second

$\to$ SI system is an extension of the MKS system in which units of all fundamental quantities are included.

Quantity	SI Unit	Symbol
1. Length	meter	m
2. Mass	Kilogram	kg
3. Time	Second	s
4. Electric Current	Ampere	A
5. Temperature	Kelvin	K
6. Luminous Intensity	Candela	Cd
7. Amount of Substance	Mole	mol

Types of Units

$\to$ There are three types of units. Namely, Fundamental units, Derived Units, and Supplementary Units.

$\to$ Fundamental Units: The units of fundamental quantities which are independent of other units are called fundamental units. There are seven fundamental units. eg. m,kg, s, K, Cd, Mol, A.

$\to$ Supplementary Units: The units of plane angle and solid angle which are two purely geometric angles are called supplementary units.

Quantity	Unit	Symbol
1. Plane Angle	Radian	Rad
2. Solid Angle	Steradian	Sr

$\to$ Derived Units: The units of derived quantities which are dependent on fundamental units and supplementary units are called derived units. For eg. m/s, kgms $^{-2}$ , m/s $^{2}$ etc.

To cover all the scientific measurements in terms of SI unit, the prefixes are used for the powers of ten.

Powers of Ten	Prefixes	Symbol	Powers of Ten	Prefixes	symbol
$10^{24}$	Yotta	Y	$10^{-24}$	Yocto	y
$10^{21}$	Zetta	Z	$10^{-21}$	zepto	z
$10^{18}$	Exa	E	$10^{-18}$	Atto	a
$10^{15}$	Peta	P	$10^{-15}$	femto	f
$10^{12}$	Tera	T	$10^{-12}$	Pico	p
$10^{9}$	Giga	G	$10^{-9}$	nano	n
$10^{6}$	Mega	M	$10^{-6}$	micro	$\mu$
$10^{3}$	Kilo	K	$10^{-3}$	Milli	m
$10^{2}$	Hecto	h	$10^{-2}$	centi	c
$10^{1}$	deca	da	$10^{-1}$	deci	d

The size of the nucleus is in order of $10^{-14}$ m and the size of an atom is in order of $10^{-10}$

Common Non-SI Units

SI units are commonly used worldwide and cover all scientific measurements. Due to historical, political, and situational significance, there are common Non-SI units in use.

Length

Common Non-SI unit	Symbol	Value in SI unit	Significance
1 Astronomical Unit	1 AU	1.496 $\times$ 10 $^{11}$ m	The distance from Earth to the Sun
1 Light Year	1 Light Year	9.46 $\times$ 10 $^{15}$ m	The distance covered by light in one year
1 Parallactic Second	1 Parsec	3.084 $\times$ 10 $^{16}$ m/(3.26 Light Year)	the distance at which the length of one astronomical unit subtends an angle of one second of an arc.
1 Angstrom	1 $\AA$	1 $\times$ 10 $^{-10}$ m	named after 19th-century Swedish physicist Anders Jonas Ångström
1 Fermi	1 Fermi	1 $\times$ 10 $^{-15}$ m	The smallest unit of distance used in nuclear physics/ named after Enrico Fermi
1 X-ray Unit	xu	1 $\times$ 10 $^{-13}$ m	used to quote the wavelength of X-rays and gamma rays.
1 Inch	1 "	0.0254 m	Imperial System
1 Foot	1 ft/ 1'	0.3048 m	Imperial System
1 Yard	1 yd	0.9144 m	Imperial System
1 mile	1 mile	1609.344 m	Imperial System
1 Nautical mile	1 Nm	1852 m	Used in air, marine, and space navigation

Mass

Common Non-SI unit	Symbol	Value in SI unit	Significance
1 Pound	1 lb	0.4536 kg	Imperial System
1 Slug	1 slug	14.59 kg	British system based on standard gravity
1 Quintal	1 Q	100 kg	historical unit of mass
1 Metric tonne	1 t	1000 kg	historical unit of mass
1 amu	1 amu	1.66 $\times$ 10 $^{-27}$ kg	precisely 1/12 the mass of an atom of carbon-12

Time

Common Non-SI unit	Symbol	Value in SI unit	Significance
1 Minute	1 min	60 s	Convenience
1 Hour	1 hr	= 60 min = 3600 s	Convenience
1 Day	1 day	= 24 hr = 86,400 s	Convenience
1 year	1 yr	1= 365.25 day = 3.156 $\times 10^7$ s	Convenience
1 shake	1 Shake	1 $\times$ 10 $^{-8}$ s	Used in nuclear physics

Other Conversion

Quantities	Non SI values	Value in SI unit
Pressure	1 Pascal	1 Nm $^{-2}$
Pressure	1 atm = 760 mmHg	1.01 $\times 10^{5}$ Nm $^{-2}$
Pressure	1 bar= 1 atm	1.01 $\times 10^{5}$ Nm $^{-2}$
Pressure	1 torr= 1 mmHg	133.322 Nm $^{-2}$
Volume	1 liter = 1 cm $^3$	$1 \times 10^{-3} m^3$
Energy	1 eV	1.6 $\times 10^{-19}$ J
Energy	1 erg	1 $\times 10^{-7}$ J
Energy	1 KwHr	3.6 $\times 10^{6}$ J
Power	1 Hp	746 W
Power of lens	1 D	1 m $^{-1}$
Plane Angle	1 Degree	$\dfrac{\pi}{180} rad$

Measurement

The comparison of the amount of physical quantity in terms of standard reference of the same quantity is called measurement.

$\to$ The standard reference amount is a unit (u).

$\to$ The scale of the amount with reference to the unit is the numerical value (n).

Hence,

Measured Quantity(Q) = Numerical value (n) $\times$ unit (u)

$\to$ Eg. 10 Kg where, numerical value = 10 and unit = 1 Kg.

For a given measurement, the measured quantity is constant. i.e. Q = nu = constant.