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.

$$n_1 u_1 = n_2 u_2

Dimensions and Dimensional Formula

The dimensions of a physical quantity may be defined as the powers to which the fundamental units of mass, length, and time must be raised to represent the physical quantity.$\\$E.g$\\$

$\text{velocity} = \dfrac{\text{displacement}}{\text{time}} = \dfrac{L}{T} = [M^0 L^1 T^{-1}]$

The dimension of velocity is 0 in mass, 1 in length, and -1 in time.

$\text{acceleration} = \dfrac{\text{velocity}}{\text{time}} = \dfrac{LT^{-1}}{T} = [M^0 L^1 T^{-2}]$

The dimension of velocity is 0 in mass, 1 in length, and -2 in time.

$\text{Force} = \text{mass} \times \text{acceleration}$

=$M \times LT^{-2}$

= $[MLT^{-2}]$

The dimension of force is 1 in mass, 1 in length, and -2 in time.

Different Types of Variables and Constants

$\to$ Dimensional Variable: Quantities having dimension and variable values. eg. Acceleration, Volume, Force, etc.

$\to$ Dimensional Constant: Quantities having dimension but constant values. eg. gravitational constant, Planck's constant, Stefan's constant, etc.

$\to$ Non-dimensional Variable: Quantities not having dimension with variable values. eg. strain, angle, relative density, etc.

$\to$ Non-dimensional Constant: Quantities not having dimension with constant values. eg. $\pi$, $e$, $1$ etc.

Non-Dimensional Quantities

$\to$ Relative density

$\to$ Angle and solid angle

$\to$ Strain

$\to$ Poisson’s ratio

$\to$ Refractive Index

$\to$ Mechanical Equivalent of Heat

$\to$ Emissivity

$\to$ Magnetic Susceptibility

$\to$ Electric Susceptibility

$\to$ Relative Permittivity

$\to$ Relative Permeability

$\to$ Coefficient of Friction

$\to$ Loudness ( Decibel is the Unit of Intensity Level)

$\to$ Dielectric Constant

Physical Quantities	Dimension	SI unit
Area (A = $L^2$)	[$L^2$]	m$^2$
Volume (V = $L^3$)	[$L^3$]	m$^3$
Density ($\rho = \dfrac{m}{v}$)	[$ML^{-3}$]	kg/m$^{3}$
Acceleration ($a =\dfrac{\text{v}}{\text{t}}$)	[$LT^{-2}$]	m/s$^2$
Speed ($v =\dfrac{\text{d}}{\text{t}}$)	$[LT^{-1}]$	m/s
Momentum (p = mv )	$MLT^{-1}$	kgm/s
Acceleration ($a =\dfrac{\text{v}}{\text{t}}$)	[$LT^{-2}$]	m/s$^2$
Force ($F =ma$)	[$MLT^{-2}$]	N
Impulse ($I =Ft$)	[$MLT^{-1}$]	Ns
Work ($W =Fd$)	[$ML^2T^{-2}$]	J
Power($P=W/t$)	[$ML^2T^{-3}$]	W
Pressure($P =F/A$)	[$ML^{-1}T^{-2}$]	N/m$^2$

Use of Dimensional Formula

Conversion of one system of the unit to another.

If $M_1, L_1, T_1$ are the fundamental units on one system of units, $M_2, L_2, T_2$ are the fundamental units on the second system of units.

and, the dimensional formula of the physical quantity is $M^a L^b T^c$ then,

$n_2 = n_1 \bigg (\dfrac{M_1}{M_2} \bigg )^a \bigg (\dfrac{L_1}{L_2}\bigg )^b \bigg (\dfrac{T_1}{T_2} \bigg )^c$

Where $n_1$ and $n_2$ are the numerical values in each system.

$\to$ Example: Convert 1 Joule into erg.

Joule and erg are the units of energy. So, dimensional formula = $[ML^2 T^{-2}]$

and Joule is the unit of the SI system and erg is the unit of the CGS system.

$n_2 = n_1 \bigg (\dfrac{M_1}{M_2} \bigg )^a \bigg (\dfrac{L_1}{L_2}\bigg )^b \bigg (\dfrac{T_1}{T_2} \bigg )^c$

$n_2 = 1 \bigg (\dfrac{1 kg}{1 gm} \bigg )^1 \bigg (\dfrac{1 m}{1 cm}\bigg )^2 \bigg (\dfrac{1 s}{1 s} \bigg )^{-2}$

$n_2 = 1 \bigg (\dfrac{1000 gm}{1 gm} \bigg )^1 \bigg (\dfrac{100 cm}{1 cm}\bigg )^2 \bigg (\dfrac{1 s}{1 s} \bigg )^{-2}$

$n_2 = 10^7$

Hence, $1 J = 10^7 erg$

Checking the consistency of the equation

The principle of homogeneity states that "dimensions of each of the terms of a dimensional equation on both sides should be the same. "

$\to$ Example: check the consistency of the equation $v^2 = u^2 + 2as$

Dimensional formula of $v = [M^0L^1T^{-1}]$

Dimensional formula of $u = [M^0L^1T^{-1}]$

Dimensional formula of $2 = [M^0L^0T^0]$

Dimensional formula of $a = [M^0L^1T^{-2}]$

Dimensional formula of $s = [M^0L^1T^0]$

Hence,

Dimension of $v^2 = [LT^{-1}]^2 = [L^2 T^{-2}]$

Dimension of $u^2 = [LT^{-1}]^2 = [L^2 T^{-2}]$

Dimension of $2as =[M^0L^0T^0] [M^0L^1T^{-2}] [M^0L^1T^0] = [L^2 T^{-2}]$

Hence, the given equation is dimensionally consistent.

Note: The dimensionally consistent equations are not necessarily correct physical equations. But, dimensionally inconsistent equations are always incorrect physical equations.

Deriving the equation

$\to$ Example: The viscous force acting on a sphere is directly proportional to the following parameters:

the radius of the sphere(r)

coefficient of viscosity($\eta$)

the terminal velocity of the object $v$

Find the expression for the viscous force on the sphere if the proportional constant is $6\pi$

Solution:

$F \propto \eta^a r^b v^c$

Using homogeneity of dimension:

$[MLT^{-2}] = [ML^{-1}T^{-1}]^a [L]^b [LT^{-1}]^c$

$[MLT^{-2}] = [M^{a} L^{-a+b+c} T^{-a-c}]$

Then,

$a=1$

$-a+b+c = 1$

$-a-c = -2$

solving we get, $a=b=c=1$

$F = 6 \pi \eta r v$

Dimensional Formula of Important Physical Quantities

Physical Quantities	Dimension	SI unit
Angular displacement($\theta =l/r$)	[$M^0L^{0}T^{0}$]	rad
Angular velocity($\omega = \theta/t$)	[$M^0L^{0}T^{-1}$]	rad/s
Frequency($\eta = 1/T$)	[$M^0L^{0}T^{-1}$]	Hz
Angular acceleration($\alpha = \omega/t$)	[$M^0L^{0}T^{-2}$]	rad/s^$2$
Moment of Inertia($I = mr^2$)	[$ML^{2}$]	$kgm^2$
Torque($\tau = I \alpha$)	[$ML^2T^{-2}$]	Nm
Angular Momentum($J = I \omega$)	[$ML^2T^{-1}$]	Js
Force constant($k = F/x$)	[$MT^{-2}$]	Nm$^{-1}$
Surface tension($k = F/l$)	[$MT^{-2}$]	Nm$^{-1}$
Gravitational Constant($G = \dfrac{Fr^2}{m_1m_2})$	[$M^{-1}L^3T^{-2}$]	$Nm^2kg^{-2}$
velocity gradient($\dfrac{\delta v}{\delta x})$	[$T^{-1}$]	$s^{-1}$
Coefficient of viscosity ($\eta = \dfrac{F}{A \times dv/dx})$	[$ML^{-1}T^{-1}$]	$kgm^{-1}s^{-1}$
Gravitational Potential ($= \dfrac{W}{m})$	[$L^2T^{-2}$]	$m^2kg^{-2}$
Planck's Constant ($h=\dfrac{E}{\nu})$	[$ML^2T^{-1}$]	$Js$
Gravitational Intensity($= \dfrac{F}{m})$	[$LT^{-2}$]	$Nkg^{-1}$
Charge($Q = It)$	[$AT$]	$As$
Electric Potential ($=\dfrac{W}{Q})$	[$ML^2T^{-3}A^{-1}$]	$JC^{-1}$
Electric Dipole ($=q \times 2l$	[$LT A$]	$Cm$
Current Density ($J=\dfrac{I}{A})$	[$AL^{-2}$]	$Am^{-2}$
Capacitance ($C=\dfrac{Q}{V})$	[$M^{-1}L^{-2}T^{4}A^{2}$]	$Farad$
Permittivity ($=\dfrac{C d}{A})$	[$M^{-1}L^{-3}T^{4}A^{2}$]	$Farad/m$
Resistance ($R=\dfrac{V}{I} = \dfrac{W}{QI})$	[$ML^{2}T^{-3}A^{-2}$]	$Ohm$
Conductance ($=\dfrac{1}{R}$	[$M^{-1}L^{-2}T^{3}A^{2}$]	$mho$

Combined Dimension:

The dimension of $\dfrac{E}{B}$ is the same as that of velocity. (E stands for Electric Field Intensity and B stands for magnetic field intensity.)

$RC= \dfrac{L}{R} = \sqrt{LC}$ has the dimension of Time.

The dimensions of solar constant are$\dfrac{\text{energy}}{(\text{area} \times \text{time})}$

Physical Quantities having the same dimensions

$\to$ Inertia and mass have the same dimensional formula.

$\to$ acceleration, gravitational field strength, and acceleration due to gravity. $[M^0LT^{-2}] \\$

$\to$ Linear momentum and impulse.$[MLT^{-1}] \\$

$\to$ Work, energy, torque, couple, moment, internal energy, heat, work function$[ML^2T^{-}2]$.$\\$

$\to$ Angular velocity, frequency, velocity gradient, radioactive decay constant $[M^0L^0T^{-1}]$.$\\$

$\to$ Spring constant, force gradient, surface energy. $[ML^0T^{-2}] \\$

$\to$ Pressure, stress, modulus of elasticity, energy density . $[ML^{-1}T^{-2}] \\$

$\to$ Gravitational Potential, specific latent heat. $[M^0L^2T^{-2}] \\$

$\to$ Electric potential, EMF, and electric potential difference. $[ML^2T^{-3}A^{-1|}] \\$

$\to$ Intensity of radiation and solar constant. $[ML^0T^{-3}] \\$

$\to$ Angular momentum and Planck’s Constant.$[ML^2T^{-1}A] \\$

$\to$ Magnetising force and intensity of magnetization. $[M^0L^{-1}T^0A] \\$

$\to$ Rydberg’s constant and wavenumber. $[M^0L^{-1}T^0]$.$\\$

$\textbf P.S.$ Never memorize the dimensional formula of any physical quantity. Instead, memorize the formula which contains that physical quantity and try to obtain the dimensional formula of the designated physical quantity.$\\$

For eg; For calculating the dimensional formula of permittivity, we first have to know the Formula which uses the Permittivity. The simplest one which I come across is the following relation obtained from Coulomb's Law:$\\$

For vacuum $F_v = \dfrac{1}{4\pi \epsilon_0 } \dfrac {q_1q_2}{r^2} \\$

For medium$F_m = \dfrac{1}{4\pi \epsilon_0 \epsilon_r } \dfrac {q_1q_2}{r^2} \\$

$\therefore \dfrac{F_m}{F_v} = \epsilon_r \\$From the above relation, we know the dimensional formula of Force(F), Charge(q), and distance(r). $\Pi$ is a dimensionless quantity.$\\$

Through simplification, we can easily obtain the dimensional formula of $\epsilon_0$.
Where $\epsilon_0$ Is the permittivity of free space.

Non-Standard Bases of Dimensional Formula

Taking force, length, and time to be the fundamental quantities. Find the density, momentum, and energy.

Density

$[ML^{-3}] =[F]^a[L]^b[T]^c$

$[ML^{-3}] =[MLT^{-2}]^a[L]^b[T]^c$

$[ML^{-3}] =[M^aL^{a+b}T^{-2a +c}]$

From homogeneity of dimension:

$a=1, a+b =-3, -2a +c =0$

Solving, we get, $a=1, b=-4, c =2$

$[\text{Density}]= [F^1 L^{-4}T^2]$

Momentum

$[MLT^{-1}] =[F]^a[L]^b[T]^c$

$[MLT^{-1}] =[MLT^{-2}]^a[L]^b[T]^c$

$[MLT^{-1}] =[M^aL^{a+b}T^{-2a +c}]$

From homogeneity of dimension:

$a=1, a+b =1, -2a +c =-1$

Solving, we get, $a=1, b=0, c =1$

$[\text{Density}]= [F^1 L^{0}T^1]$

Energy

$[ML^{2}T^{-2}] =[F]^a[L]^b[T]^c$

$[ML^{2}T^{-2}] =[MLT^{-2}]^a[L]^b[T]^c$

$[ML^{2}T^{-2}] =[M^aL^{a+b}T^{-2a +c}]$

From homogeneity of dimension:

$a=1, a+b =2, -2a +c =-2$

Solving, we get, $a=1, b=1, c =0$

$[\text{Density}]= [F^1 L^{1}T^0]$

Percentage Change

For percentage change $\leq 10 \%$

Equation type $A= kx^n$

Taking log in both sides,

$\log A = log k + n \log x$

differentiating,

$\dfrac{\Delta A}{A} = n \times \dfrac{\Delta x}{x}$

$\dfrac{\Delta A}{A} \times 100 \% = n \times \dfrac{\Delta x}{x} \times 100 \%$

Due to properties of logarithm:

The power is changed into multiplication.

$y = x^n$

$\log y = n \log x$

The multiplication is changed into addition and

$y= ab$

$\log y = \log a + \log b$

The division is changed into subtraction.

$y= \dfrac{a}{b}$

$\log y = \log a - \log b$

For percentage change $> 10 \%$

Equation type $A= kx^n$

$\to$ If the percentage increase in x is $a \%$, $\% \text{change in A} = \Bigg[\Bigg(\dfrac{100+a}{100}\Bigg)^n-1\Bigg] \times 100 \%$

$\to$ If the percentage decrease in x is $a \%$, $\% \text{change in A} = \Bigg[\Bigg(\dfrac{100-a}{100}\Bigg)^n-1\Bigg] \times 100 \%$

Errors in Measurement

There is a certain difference between the measured value and the actual value during measurement. It is called an error in measurement.

Error in Equation type $f= kx^n$

Taking log in both sides,

$\log f = log k + n \log x$

differentiating,

$\dfrac{\Delta f}{f} = n \times \dfrac{\Delta x}{x}$

$\% \text{error in f} = \dfrac{\Delta f}{f} \times 100 \%= n \times \dfrac{\Delta x}{x} \times 100 \%$

Error in Equation type $f= k \dfrac{x^a y^b}{z^c}$

Relative Error: $\dfrac{\Delta f}{f} = a \times \dfrac{\Delta x}{x} + b \times \dfrac{\Delta y}{y} + c \times \dfrac{\Delta z}{z}$

$\text{Percentage Error}: \dfrac{\Delta f}{f} \times 100 \%= \Bigg(a \times \dfrac{\Delta x}{x} + b \times \dfrac{\Delta y}{y} + c \times \dfrac{\Delta z}{z}\Bigg) \times 100 \%$

There is no error of measurement in constant value.

The power is changed into multiplication.

The multiplication and division are changed into addition.