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Dimensions and Dimensional formula of physical quantities

This article mainly focuses on the dimensions of physical quantities, dimensional formulas and dimensional analysis. Before going through the dimensions and dimensional formulas, first you require a piece of basic knowledge about the basics of measurement and for that consider reading my previous article about the measurement and different system of units.

Now, let’s start with the dimensions. All physical quantities can be expressed in terms of seven fundamental (base) quantities such as mass, length, time, temperature, electric current, luminous intensity and amount of substance. These seven quantities are called the seven dimensions of the physical world. We can use symbols instead of the names of the base quantities. M, L and T are used to represent the dimensions of the three mechanical quantities mass, length and time respectively. They can also be denoted by using the brackets [M], [L] and [T]. Other dimensions are denoted by K (for temperature), I (for electric current), cd (for luminous intensity) and mol (for the amount of substance). The dimensions of a physical quantity and the dimensions of its unit are the same. The letters [M], [L], [T] etc. specify only the nature of the unit and not its magnitude.

What do you mean by dimensions of physical quantity?

Each derived quantity requires proper power for fundamental quantities so as to represent it. The powers of fundamental quantities, through which they are to be raised to represent unit derived quantity, are called dimensions. In other words, the dimensions of a physical quantity are the powers to which the base quantities (fundamental quantities) are raised to represent that quantity.

For eg:

The area is the product of two lengths.
Area = Length X breadth = [L] x [L] = [L²]
Therefore, [A] = [L²] That is, the dimension of area is 2 dimension in length and zero dimension in mass and time.
Or [A] = [M⁰L²T⁰]
Similarly, the volume is the product of three lengths.
Volume = Length X breadth X height = [L] x [L] x [L] = [L³]
Therefore, [V] = [L³] That is, the dimension of volume is 3 dimension in length and zero in mass and time.
Or [V] = [M⁰L³T⁰]
Similarly, acceleration is the rate of change of velocity per unit of time.
$large begin{aligned} Acceleration=frac{Velocity}{Time}=frac{Displacement}{Timetimes Time} = frac{L}{T^{2}} end{aligned}$
Therefore, [a] = [L¹T^-2] That is, the dimension of acceleration is 1 dimension in length, -2 dimension in time and zero dimension in mass.
Or [a] = [M⁰L¹T^-2]

Thus, the dimensions of a physical quantity are the powers(or exponents) to which the fundamental units of length, mass, time etc. must be raised to represent it or the dimension of the units of a derived physical quantity is defined as the number of times the fundamental units of length, mass, time etc appear in the physical quantity.

What is dimensional formula and dimensional equation?

The dimensional formula is a compound expression showing how and which of the fundamental quantities are involved in making that physical quantity.
The dimensional equation of a physical quantity is an equation, equating the physical quantity with its dimensional formula. That is, the dimensional equations are the equations, which represent the dimensions of a physical quantity in terms of the base quantities.

For example, the above expressions like[M⁰L²T⁰], [M⁰L³T⁰], [M⁰L¹T^-2] etc are known as dimensional formulae and the equations such as [A] = [M⁰L²T⁰], [V] = [M⁰L³T⁰], [a] = [M⁰L¹T^-2] etc. are known as dimensional equations.

Dimensional formula of physical quantities

The dimensional formula and SI units for more than 100 physical quantities are given in the table below.

Sl. No	Physical Quantity	Formula	Dimensional Formula	S.I Unit
1	Area (A)	Length x Breadth	[M⁰L²T⁰]	m²
2	Volume (V)	Length x Breadth x Height	[M⁰L³T⁰]	m³
3	Density (d)	Mass / Volume	[M¹L^-3T⁰]	kgm^-3
4	Speed (s)	Distance / Time	[M⁰L¹T^-1]	ms^-1
5	Velocity (v)	Displacement / Time	[M⁰L¹T^-1]	ms^-1
6	Acceleration (a)	Change in velocity / Time	[M⁰L¹T^-2]	ms^-2
7	Acceleration due to gravity (g)	Change in velocity / Time	[M⁰L¹T^-2]	ms^-2
8	Specific gravity	Density of body/density of water at 4^oC	No dimensions [M⁰L⁰T^-0]	No units
9	Linear momentum (p)	Mass x Velocity	[M¹L¹T^-1]	kgms^-1
10	Force (F)	Mass x Acceleration	[M¹L¹T^-2]	N
11	Work (W)	Force x Distance	[M¹L²T^-2]	J (Joule)
12	Energy (E)	Work	[M¹L²T^-2]	J
13	Impulse (I)	Force x Time	[M¹L¹T^-1]	Ns
14	Pressure (P)	Force / Area	[M¹L^-1T^-2]	Nm^-2
15	Power (P)	Work / Time	[M¹L²T^-3]	W
16	Universal constant of gravitation (G)	$largebegin{aligned}frac{Force times (distance)^{2}}{(mass)^{2}}end{aligned}$	[M^-1L³T^-2]	Nm²kg^-2
17	Moment of inertia (I)	Mass x (distance)²	[M¹L²T⁰]	kgm²
18	Moment of force, moment of couple	Force x distance	[M¹L²T^-2]	Nm
19	Surface tension (T)	Force / Length	[M¹L⁰T^-2]	Nm^-1
20	Surface energy (E)	Energy / unit area	[M¹L⁰T^-2]	Nm^-1
21	Force constant (x)	Force / Displacement	[M¹L⁰T^-2]	Nm^-1
22	Coefficient of viscosity ( η )	$largebegin{aligned}frac{Force}{areatimes velocity;gradient}end{aligned}$	[M¹L^-1T^-1]	Nsm^-2
23	Thrust (F)	Force	[M¹L¹T^-2]	N
24	Tension (T)	Force	[M¹L¹T^-2]	N
25	Stress	Force / Area	[M¹L^-1T^-2]	Nm^-2
26	Strain	Change in dimension / Original dimension	No dimensions [M⁰L⁰T^-0]	No unit
27	Modulus of Elasticity (E)	Stress / strain	[M¹L^-1T^-2]	Nm^-2
28	Radius of gyration (k)	Distance	[M⁰L¹T⁰]	m
29	Angle ( θ), Angular displacement	Arc length / Radius	No dimensions [M⁰L⁰T^-0]	rad
30	Trigonometric ratio ( sin θ, cos θ, tan θ, etc)	Length / length	No dimensions [M⁰L⁰T^-0	No unit
31	Angular velocity( ω )	Angle / Time	[M⁰L⁰T^-1]	rad s^-1
32	Angular acceleration( α )	Angular velocity / Time	[M⁰L⁰T^-2]	rad s^-2
33	Angular momentum (J)	Moment of inertia x Angular velocity	[M¹L²T^-1]	kgm²s^-1
34	Torque (𝞽)	Moment of inertia x Angular acceleration	[M¹L²T^-2]	Nm
35	Velocity gradient $large begin{aligned} left ( frac{dv}{dx} right ) end{aligned}$	Velocity / Distance	[M⁰L⁰T^-1]	s^-1
36	Rate flow	Volume / Time	[M⁰L³T^-1]	m³s^-1
37	Wavelength( 𝛌 )	Length of a wavelet	[M⁰L¹T⁰]	m
38	Frequency( $large begin{aligned} nu end{aligned}$ )	Number of vibrations/second or 1/time period	[M⁰L⁰T^-1]	Hz or s^-1
39	Angular frequency (ω)	2π x frequency	[M⁰L⁰T^-1]
40	Planck’s constant (h)	Energy / Frequency	[M¹L²T^-1]	Js
41	Buoyant force	Force	[M¹L¹T^-2]	N
42	Relative density	Density of substance / density of water at 4^oC	No dimensions [M⁰L⁰T^-0]	No unit
43	Pressure gradient	Pressure / Dstance	[M¹L^-2T^-2]	Nm^-3
44	Pressure energy	Pressure x Volume	[M¹L²T^-2]	J
45	Temperature	——	[M⁰L⁰T⁰K¹]	K
46	Heat (Q)	Energy	[M¹L²T^-2]	J
47	Latent heat (L)	Heat / Mass	[M⁰L²T^-2]	Jkg^-1
48	Specific heat (S)	$begin{aligned} frac{Heat}{Masstimes temperature}end{aligned}$	[M⁰L²T^-2K^-1]	Jkg^-1K^-1
49	Thermal expansion coefficient or thermal expansivity	$begin{aligned}frac{Change;in;dimension}{original;dimensiontimes temperature}end{aligned}$	[M⁰L⁰T⁰K^-1]	K^-1
50	Thermal conductivity	$begin{aligned}frac{Heat;energytimes thickness}{Areatimes temperaturetimes time}end{aligned}$	[M¹L¹T^-3K^-1]	Wm^-1K^-1
51	Bulk modulus or (compressibility)^-1	$begin{aligned}frac{Volumetimes (change;in;pressure)}{(change;in;volume)}end{aligned}$	[M¹L^-1T^-2]	Nm^-2 or Pascals
52	Centripetal acceleration	$begin{aligned}frac{(velocity)^2}{radius}end{aligned}$	[M⁰L¹T^-2]
53	Stefan constant (σ)	$begin{aligned}frac{(Energy / area times time)}{(temperature)^4}end{aligned}$	[M¹L⁰T^-3K^-4]	Wm⁻²K⁻⁴
54	Wien constant	Wavelength X temperature	[M⁰L¹T⁰K¹]	mK
55	Gas constant (R)	$begin{aligned} frac{Pressure times Volume}{Temperature}end{aligned}$	[M¹L²T^-2K^-1]	JK^-1
56	Boltzmann constant (K)	Energy / temperature	[M¹L²T^-2K^-1]	JK^-1
57	Charge (q)	Current x time	[M⁰L⁰T¹A¹]	C
58	Current density	Current / area	[M⁰L^-2T⁰A¹]	A m⁻²
59	Electric potential (V), voltage, electromotive force	Work / Charge	[M¹L²T^–3A^-1]	V
60	Resistance (R)	Potential difference / Current	[M¹L²T^–3A^-2]	ohms (Ω)
61	Capacitance	Charge / potential difference	[M^–¹L^–²T⁴A²]	F (Farad)
62	Electrical resistivity or (electrical conductivity)^-1	$begin{aligned}frac{Resistance times area}{length}end{aligned}$	[M¹L³T^-3A^–²]	Ωm ( resistivity)
63	Electric field (E)	Force / Charge	[M¹L¹T^-3A^-1]	NC^-1
64	Electric flux	Electric field X area	[M¹L³T^–³A^-1]	Nm²C^-1
65	Electric dipole moment	Torque / electric field	[M⁰L¹T¹A¹]	C m
66	Electric field strength or electric intensity	Potential difference / distance	[M¹L¹T^-3A^-1]	NC^-1
67	Magnetic field (B), magnetic flux density, magnetic induction	$begin{aligned}frac{Force}{current times length}end{aligned}$	[M¹L⁰T^-2A^-1]	T (Tesla)
68	Magnetic flux (Φ)	Magnetic field X area	[M¹L²T^-2A^-1]	Wb (Weber)
69	Inductance	Magnetic flux / current	[M¹L²T^-2A^-2]	H (Henry)
70	Magnetic dipole moment	Torque /field or current X area	[M⁰L²T⁰A¹]	Am²
71	Magnetic field strength (H), magnetic intensity or magnetic moment density	Magnetic moment / volume	[M⁰L^-1T⁰A¹]	Am^-1
72	Hubble constant	Recession speed / distance	[M⁰L⁰T^-1]	s^-1
73	Intensity of wave	(Energy/time)/area	[M¹L⁰T^-3]	Wm^-2
74	Radiation pressure	Intensity of wave / speed of light	[M¹L^–¹T^-2]
75	Energy density	Energy / volume	[M¹L^-1T^-2]	Jm^-3
76	Critical velocity	$begin{aligned}frac{Reynold's;number times coefficient;of;viscocity}{Mass; density times radius}end{aligned}$	[M⁰L¹T^-1]	ms^-1
77	Escape velocity	$begin{aligned}(2 times acceleration;due;to;gravitytimes earth's;radius)^{1/2}end{aligned}$	[M⁰L¹T^-1]	ms^-1
78	Heat energy, internal energy	Work ( = Force X distance)	[M¹L²T^-2]	J
79	Kinetic energy	$begin{aligned}frac{1}{2}mass times (velocity)^{2}end{aligned}$	[M¹L²T^-2]	J
80	Potential energy	Mass X acceleration due to gravity X height	[M¹L²T^-2]	J
81	Rotational kinetic energy	$begin{aligned}frac{1}{2}times moment;of;inertiatimes (angular;velocity)^{2}end{aligned}$	[M¹L²T^-2]	J
82	Efficiency	$begin{aligned}frac{output;work;or;energy}{input;work;or;energy}end{aligned}$	No dimensions [M⁰L⁰T⁰]	No unit
83	Angular impulse	Torque X time	[M¹L²T^-1]	Js (Joule second)
84	Permitivity constant (of free space)	$begin{aligned}frac{Charge times charge}{4pitimes electric;forcetimes (distance)^{2} }end{aligned}$	[M^-1L^-3T⁴A²]	F m^-1
85	Permeability constant (of free space)	$begin{aligned}frac{2pi times forcetimes distance}{currenttimes currenttimes length}end{aligned}$	[M¹L¹T^-2A^-2]	NA^-2
86	Refractive index	$begin{aligned}frac{Speed;of;light;in;vacuum}{Speed;of;light;in;medium}end{aligned}$	No dimensions [M⁰L⁰T⁰]	No unit
87	Faraday constant (F)	Avogadro constant X elementary charge	[M⁰L⁰T¹A¹mol^-1]	C mol^-1
88	Wave number	$begin{aligned}frac{2pi }{wavelength}end{aligned}$	[M⁰L^-1T⁰]
89	Radiant flux, Radiant power	Energy emitted / time	[M¹L²T^-3]	W(Watt)
90	Luminosity of radiant flux or radiant intensity	$begin{aligned}frac{Radiant;power;or;radiant;flux;of;source}{Solid;angle}end{aligned}$	[M¹L²T^-3]	W sr^-1(Watt/steradian)
91	Luminous power or luminous flux of source	$begin{aligned}frac{Luminous;energy;emitted}{time}end{aligned}$	[M¹L²T^-3]	lm (lumen)
92	Luminous intensity or illuminating power of source	Luminous flux / Solid angle	[M¹L²T^-3]	cd (candela)
93	Intensity of illumination or luminance (`L`_v)	$begin{aligned}frac{Luminous;intensity}{(distance)^{2}}end{aligned}$	[M¹L⁰T^-3]	cd m^-2
94	Relative luminosity	Luminous flux of a source of given wavelength / luminous flux of peak sensitivity wavelength(555 nm) source of the same power	No dimensions [M⁰L⁰T⁰]	No unit
95	Luminous efficiency	Total luminous flux / Total radiant flux	No dimensions [M⁰L⁰T⁰]	No unit
96	Illuminance or illumination	Luminous flux incident / Area	[M¹L⁰T^-3]	lx (lux)
97	Mass defect	(Sum of masses of nucleons) – (mass of the nucleus)	[M¹L⁰T⁰]
98	Binding energy of nucleus	$begin{aligned}Mass;defecttimes (Speed;of;light;in;vacuum)^{2}end{aligned}$	[M¹L²T^-2]
99	Decay constant	0.693 / half-life	[M⁰L⁰T^-1]
100	Resonant frequency	$begin{aligned}(Inductancetimes capacitance)^{-1/2}end{aligned}$	[M⁰L⁰T^-1A⁰]
101	Quality factor or Q-factor of coil	$begin{aligned}frac{Resonant;frequency times inductance}{Resistance}end{aligned}$	No dimensions [M⁰L⁰T⁰]	No unit
102	Power of lens	$begin{aligned}frac{1}{focal;length}end{aligned}$	[M⁰L^-1T⁰]	D (dioptre)
103	Magnification	Image distance / Object distance	No dimensions [M⁰L⁰T⁰]	No unit
104	Fluid flow rate	$begin{aligned}frac{(pi /8)(pressure)times (radius)^{4}}{Viscosity;coefficienttimes length}end{aligned}$	[M⁰L³T^-1]	m³s^-1
105	Capacitive reactance (X_c)	(Angular frequency X capacitance)^-1	[M¹L²T^-3A^-2]	ohms (Ω)
106	Inductive reactance (X_L)	(Angular frequency X inductance)	[M¹L²T^-3A^-2]	ohms (Ω)

Table 1: Dimensional formula of physical quantities

Also, the values of some important physical constants and their symbols used to represent them are already written in my previous article about the different charts used in the measurement.

Physical quantities with same dimensional formula

Given below is the list of some important physical quantities having the identical dimensional formula.

Sl. No	Physical Quantity	Dimensional Formula
1	Momentum and impulse	[M¹L¹T^-1]
2	Angular momentum and Planck’s constant	[M¹L²T^-1]
3	Work, Energy, Moment of a force, Torque and couple	[M¹L²T^-2]
4	Frequency, Angular Frequency, Angular velocity and Velocity gradient	[M⁰L⁰T^-1]
5	Pressure, Stress, Elastic constant and Energy density	[M¹L^-1T^-2]
6	Force constant(spring), Surface Tension and surface energy	[M¹L⁰T^-2]
7	Radius of gyration, light year and wavelength	[M⁰L¹T⁰]

Table 2: List of physical quantities with same dimensional formula

Distinguish between dimension variable, dimensionless variables, dimensional constants and dimensionless constants

Depending upon the dimensional formula, the various physical quantities can be divided into four categories.

Dimensional variables

Physical quantities which have dimensions and do not have a constant value are called dimensional variables.
eg: velocity, work, power.
Dimensionless variables

Physical quantities which have no dimensions but are variables, are called dimensionless (non-dimensional) variables.
eg: strain, plane angle
Dimensional constants

Physical quantities that have constant values but still have dimensions, are called dimensional constants.
eg: Planck’s constant (h), Universal gravitational constant (G)
Dimensionless constants

Pure numbers like, 1,2,3, π etc. are called dimensionless (non-dimensional) constants.

What is dimensional analysis

The dimensions of base quantities and combination of these dimensions describe the nature of physical quantities. Dimensional analysis can be used to check the dimensional consistency of equations, deducing relation among the various physical quantities, etc. A dimensionally consistent equation need not be actually an exact or correct equation, but a dimensionally wrong or inconsistent equation must be wrong.

Applications of Dimensional Analysis

The important applications of dimensional analysis are

To convert the value of a physical quantity from one system to another.
To check the correctness of a given relation.
To derive a relation between various physical quantities.

To convert the value of a physical quantity from one system to another.

We can convert the value of a physical quantity from one system to another by using dimensional analysis and by lying the principle of homogeneity.
Let physical quantity as represented in system one = n₁[M₁^xL₁^YT₁^Z], where x, y and z are the dimensions of the given physical quantity.
Similarly, physical quantity as represented in system two = n₂[M₂^xL₂^YT₂^Z]n₁ and n₂ are the numerical values in the two systems.
Since the quantity is same in both the systems,
n₁[M₁^xL₁^YT₁^Z] = n₂[M₂^xL₂^YT₂^Z] $n_{2} = n_{1} left [ frac{M_{1}}{M_{2}} right ]^{x} left [ frac{L_{1}}{L_{2}} right ]^{y} left [ frac{T_{1}}{T_{2}} right ]^{z}$
To check the correctness of a given relation.

we can check the correctness of the given relation by finding out the dimensional formula of every term on either side of the relation. If the dimensions are identical, the relation is correct. (Principle of homogeneity)
eg: Check the corectness of the equation $S = ut + frac{1}{2}at^{2}$
Dimensional formula of S = [ L ]
Dimensional formula of ut = [ L T^-1] x [ T ] = [ L ]
Dimensional formula of $frac{1}{2}at^{2}$ = [ L T^-2] x [ T² ] = [ L ] $
Here 1/2 is a constant and has no dimensions.
Since all the dimensions in the three terms are the same, the equation is correct.
To derive a relation between various physical quantities.

When one physical quantity depends on several physical quantities, then the relationship between the quantities can be derived using the dimensional method.
eg: Deduce an expression for the time period of a simple pendulum.
The factors on which the time period (T) may possibly depend on are:
a. mass of the bob (m)
b. length of the pendulum (l)
c. acceleration due to gravity (g)
d. the angle of swing of the pendulum (θ)
That is, $T propto m^al^bg^cTheta ^d$
∴ $T = Km^al^bg^cTheta ^d$ , where K is a dimensionless constant of proportionality.
Taking dimesnional formula for each quantity.
[M⁰L⁰T¹] = [ML⁰T⁰]^a[M⁰LT⁰]^b[M⁰LT^-2]^c [M⁰L⁰T¹] = [M^aL^b+c T^–2c], angle θ is a dimensionless quantity. ∴ d=0.
Equating the indices of corresponding dimensions on the two sides
a = 0, b + c = 0, c = –b
1 = –2c , c = –1/2, therefore, b = 1/2
∴ substituting in equation for T,
$T = Kl^frac{1}{2}g^frac{-1}{2} = Ksqrt{frac{l}{g}}$
From experiments we find K to be equal to 2
∴ $T = 2pi sqrt{frac{l}{g}}$

Limitations of Dimensional Analysis

The main limitations of dimensional analysis are listed below.

This method doesn’t tell anything about the dimensionless constants.
It is not possible to derive relations that contain more than one term like $S = ut + frac{1}{2}at^{2}$ .
This method fails if the relation contains more than three unknown quantities.
This method can’t be applied if the relation consists of trigonometric functions or logarithmic or complex or exponential functions.
Since there are many physical quantities with the same dimensions, it is very difficult to identify them by knowing dimensions alone.

So that’s all about the topic of dimensions, dimensional formula and dimensional analysis of physical quantities. Also, there may be errors arise while measuring a quantity and you can go through my next article to learn more about the errors and different types of errors that may occur in measurement.

I would love to hear your feedback about this post as well as your thoughts on dimensions of physical quantities via the comments section given below.

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Science

Dimensions and Dimensional formula of physical quantities

Dimensions and Dimensional formula of physical quantities

What do you mean by dimensions of physical quantity?

What is dimensional formula and dimensional equation?

Dimensional formula of physical quantities

Physical quantities with same dimensional formula

Distinguish between dimension variable, dimensionless variables, dimensional constants and dimensionless constants

Dimensional variables

Dimensionless variables

Dimensional constants

Dimensionless constants

What is dimensional analysis

Applications of Dimensional Analysis

To convert the value of a physical quantity from one system to another.

To check the correctness of a given relation.

To derive a relation between various physical quantities.

Limitations of Dimensional Analysis

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