Units and Dimensions – Dimensional Analysis, Formula, Applications

Introduction to Units and Dimensions

To measure a physical quantity like length, mass and time, we require a standard of measurement. Every measurement has two parts. The first is a magnitude or number (n), and the next is a unit (u).

Q = nu

For example, the length of an object = 75 cm.

The number expressing the magnitude of a physical quantity is inversely proportional to the unit selected.

If n₁ and n₂ are the numerical values of a physical quantity corresponding to the units u₁ and u₂, then

n₁u₁ = n₂u₂

For example, 3.8 m = 380 cm; 7.9 kg = 7900 g.

Fundamental and Derived Quantities

• The quantities that are independent of other quantities are called fundamental quantities. The units that are used to measure these fundamental quantities are called fundamental units. There are four systems of units, namely CGS, MKS, FPS and SI.
• The quantities that are derived using the fundamental quantities are called derived quantities. The units that are used to measure these derived quantities are called derived units.

Fundamental and supplementary physical quantities in the SI system

Fundamental Quantity	System of Units
	CGS	MKS	FPS
Length	centimeter	meter	foot
Mass	gram	kilogram	pound
Time	second	second	second

Physical Quantity	Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Intensity of light	candela	cd
Quantity of substance	mole	mol

Supplementary Quantities

Plane angle	Radian	rad
Solid angle	Steradian	sr

Most SI units are used in scientific research. SI is a coherent system of units.

A coherent system of units is one in which the units of derived quantities are obtained as multiples or submultiples of certain basic units. The SI system is a comprehensive, coherent and rationalised MKS. The ampere system (RMKSA system) was devised by Prof. Giorgi.

1. Meter: A meter is equal to 1650763.73 times the wavelength of the light emitted in a vacuum due to the electronic transition from 2p¹⁰ state to 5d⁵ state in Krypton-86. But in 1983, the 17^th General Assembly of Weights and Measures adopted a new definition for the meter in terms of the velocity of light. According to this definition, a meter is defined as the distance travelled by light in a vacuum during a time interval of 1/299, 792, 458 of a second.
2. Kilogram: The mass of a cylinder of platinum-iridium alloy kept in the International Bureau of Weights and Measures preserved at Serves near Paris is called one kilogram.
3. Second: The duration of 9192631770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atoms is called one second.
4. Ampere: The current which, when flowing in each of two parallel conductors of infinite length and negligible cross-section and placed one meter apart in vacuum, causes each conductor to experience a force of 2 × 10^-7 newtons per meter of length is known as one ampere.
5. Kelvin: The fraction of 1/273.16 of the thermodynamic temperature of the triple point of water is called Kelvin.
6. Candela: The luminous intensity in the perpendicular direction of a surface of a black body of area 1/600000 m² at the temperature of solidifying platinum under a pressure of 101325 Nm^-2 is known as one candela.
7. Mole: The amount of a substance of a system which contains as many elementary entities as there are atoms in 12 × 10^-3 kg of carbon-12 is known as one mole.
8. Radian: The angle made by an arc of the circle equivalent to its radius at the centre is known as a radian. 1 radian = 57^o17^l45^ll.
9. Steradian: The angle subtended at the centre by one square meter area of the surface of a sphere of radius one meter is known as steradian.

Some Important Conclusions

• Angstrom is the unit of length used to measure the wavelength of light. 1 Å ⁼ 10^-10 m.
• Fermi is the unit of length used to measure nuclear distances. 1 Fermi = 10^-15 meter.
• A light year is the unit of length for measuring astronomical distances.
• Light year = distance travelled by light in 1 year = 9.4605 × 10¹⁵ m.
• Astronomical unit = Mean distance between the sun and earth = 1.5 × 10¹¹ m.
• Parsec = 3.26 light years = 3.084×10¹⁶ m.
• Barn is the unit of area for measuring scattering cross-section of collisions. 1 barn = 10^-28 m².
• Chronometer and metronome are time-measuring instruments. The quantity having the same unit in all the systems of units is time.

MACRO Prefixes	MICRO Prefixes
Kilo (K) = 103 Mega (M) = 106 Giga (G) = 109 Tera (T) = 1012 Peta (P) = 1015 Exa (E) = 1018 Zetta (Z) = 1021 Yotta (y) = 1024	milli (m) = 10-3 micro (μ) = 10-6 nano (n) = 10-9 pico (p) = 10-12 femto (f) = 10-15 atto (a) = 10-18 zepto (z) = 10-21 yocto (y) = 10-24

Note: The following are not used in the SI system.

• deca 10¹ deci 10^-1
• hecta 10² centi 10^-2

How to Write Units of Physical Quantities?

1. Full names of the units, even when they are named after a scientist, should not be written with a capital letter. For example, newton, watt, ampere, meter

2. The unit should be written either in full or in agreed symbols only

3. Units do not take the plural form. For example, 10 kg but not 10 kgs, 20 w but not 20 ws

4. No full stop or punctuation mark should be used within or at the end of symbols for units. For example, 10 W but not 10 W.

What Are Dimensions?

Dimensions of a physical quantity are the powers to which the fundamental units are raised to obtain one unit of that quantity.

Dimensional Analysis

Dimensional analysis is the practice of checking relations between physical quantities by identifying the dimensions of the physical quantities. These dimensions are independent of the numerical multiples and constants, and all the quantities in the world can be expressed as a function of the fundamental dimensions.

Dimensional Formula

The expression showing the powers to which the fundamental units are to be raised to obtain one unit of a derived quantity is called the dimensional formula of that quantity.

If Q is the unit of a derived quantity represented by Q = M^aL^bT^c, then M^aL^bT^c is called the dimensional formula, and the exponents a, b, and c are called dimensions.

What Are Dimensional Constants?

The physical quantities with dimensions and a fixed value are called dimensional constants. For example, gravitational constant (G), Planck’s constant (h), universal gas constant (R), velocity of light in a vacuum (C), etc.

What Are Dimensionless Quantities? 

Dimensionless quantities are those which do not have dimensions but have a fixed value.

• Dimensionless quantities without units: Pure numbers, π, e, sin θ, cos θ, tan θ etc.
• Dimensionless quantities with units: Angular displacement – radian, Joule’s constant – joule/calorie, etc.

What Are Dimensional Variables?

Dimensional variables are those physical quantities which have dimensions and do not have a fixed value. For example, velocity, acceleration, force, work, power, etc.

What Are the Dimensionless Variables?

Dimensionless variables are those physical quantities which do not have dimensions and do not have a fixed value. For example, specific gravity, refractive index, the coefficient of friction, Poisson’s ratio, etc.

Law of Homogeneity of Dimensions

1. In any correct equation representing the relation between physical quantities, the dimensions of all the terms must be the same on both sides. Terms separated by ‘+’ or ‘–’ must have the same dimensions.
2. A physical quantity Q has dimensions a, b and c in length (L), mass (M) and time (T), respectively, and n₁ is its numerical value in a system in which the fundamental units are L₁, M₁ and T₁ and n₂ is the numerical value in another system in which the fundamental units are L₂, M₂ and T_2, respectively, then

\(\begin{array}{l}{{n}_{2}}={{n}_{1}}{{\left[ \frac{{{L}_{1}}}{L{}_{2}} \right]}^{a}}{{\left[ \frac{{{M}_{1}}}{{{M}_{2}}} \right]}^{b}}{{\left[ \frac{{{T}_{1}}}{{{T}_{2}}} \right]}^{c}}\end{array} \)

Limitations of Dimensional Analysis

1. Dimensionless quantities cannot be determined by this method. Also, the constant of proportionality cannot be determined by this method. They can be found either by experiment (or) by theory.
2. This method does not apply to trigonometric, logarithmic and exponential functions.
3. This method will be difficult in the case of physical quantities, which are dependent upon more than three physical quantities.
4. In some cases, the constant of proportionality also possesses dimensions. In such cases, we cannot use this system.
5. If one side of the equation contains the addition or subtraction of physical quantities, we cannot use this method to derive the expression.

Some Important Conversions

• 1 bar = 10⁶ dyne/cm² = 10⁵ Nm^-2 = 10⁵ pascal
• 76 cm of Hg = 1.013×10⁶ dyne/cm²  = 1.013×10⁵ pascal = 1.013 bar.
• 1 toricelli or torr = 1 mm of Hg = 1.333×10³ dyne/cm² = 1.333 millibar.
• 1 kmph = 5/18 ms^-1
• 1 dyne = 10^-5 N,
• 1 H.P = 746 watt
• 1 kilowatt hour = 36×10⁵ J
• 1 kgwt = g newton
• 1 calorie = 4.2 joule
• 1 electron volt = 1.602×10^-19 joule
• 1 erg = 10^-7 joule

Some Important Physical Constants

• Velocity of light in vacuum (c) = 3 × 10⁸ ms^-1
• Velocity of sound in air at STP = 331 ms^-1
• Acceleration due to gravity (g) = 9.81 ms^-2
• Avogadro number (N) = 6.023 × 10²³/mol
• Density of water at 4^oC = 1000 kgm^-3 or 1 g/cc.
• Absolute zero = -273.15^oC or 0 K
• Atomic mass unit = 1.66 × 10^-27 kg
• Quantum of charge (e) = 1.602 × 10^-19 C
• Stefan’s constant = 5.67 × 10^–8 W/m²/K⁴
• Boltzmann’s constant (K) = 1.381 × 10^-23 JK^-1
• One atmosphere = 76 cm Hg = 1.013 × 10⁵ Pa
• Mechanical equivalent of heat (J) = 4.186 J/cal
• Planck’s constant (h) = 6.626 × 10^-34 Js
• Universal gas constant (R) = 8.314 J/mol–K
• Permeability of free space (μ₀) = 4π × 10^-7 Hm^-1
• Permittivity of free space (ε₀) = 8.854 × 10^-12 Fm^-1
• The density of air at S.T.P. = 1.293 kg m^-3
• Universal gravitational constant = 6.67 × 10^-11 Nm²kg^-2

Derived SI units with Special Names

Physical Quantity	SI Unit	Symbol
Frequency	hertz	Hz
Energy	joule	J
Force	newton	N
Power	watt	W
Pressure	pascal	Pa
Electric charge or quantity of electricity	coulomb	C
Electric potential difference and emf	volt	V
Electric resistance	ohm	\(\begin{array}{l}\Omega\end{array} \)
Electric conductance	siemen	S
Electric capacitance	farad	F
Magnetic flux	weber	Wb
Inductance	henry	H
Magnetic flux density	tesla	T
Illumination	lux	Lx
Luminous flux	lumen	Lm

Dimensional Formulas for Physical Quantities

Physical Quantity	Unit	Dimensional Formula
Acceleration or acceleration due to gravity	ms–2	LT–2
Angle (arc/radius)	rad	MoLoTo
Angular displacement	rad	MoloTo
Angular frequency (angular displacement/time)	rads–1	T–1
Angular impulse (torque x time)	Nms	ML2T–1
Angular momentum (Iω)	kgm2s–1	ML2T–1
Angular velocity (angle/time)	rads–1	T–1
Area (length x breadth)	m2	L2
Boltzmann’s constant	JK–1	ML2T–2θ–1
Bulk modulus \(\begin{array}{l}\left(\Delta P.\frac{V}{\Delta V} \right)\end{array} \)	Nm–2, Pa	M1L–1T–2
Calorific value	Jkg–1	L2T–2
Coefficient of linear or areal or volume expansion	oC–1 or K–1	θ–1
Coefficient of surface tension (force/length)	Nm–1 or Jm–2	MT–2
Coefficient of thermal conductivity	Wm–1K–1	MLT–3θ–1
Coefficient of viscosity \(\begin{array}{l}\left(F=\eta A\frac{dv}{dx} \right)\end{array} \)	poise	ML–1T–1
Compressibility (1/bulk modulus)	Pa–1, m2N–2	M–1LT2
Density (mass/volume)	kgm–3	ML–3
Displacement, wavelength, focal length	m	L
Electric capacitance (charge/potential)	CV–1, farad	M–1L–2T4I2
Electric conductance (1/resistance)	Ohm–1 or mho or siemen	M–1L–2T3I2
Electric conductivity (1/resistivity)	siemen/metre or Sm–1	M–1L–3T3I2
Electric charge or quantity of electric charge (current x time)	coulomb	IT
Electric current	ampere	I
Electric dipole moment (charge x distance)	Cm	LTI
Electric field strength or intensity of electric field (force/charge)	NC–1, Vm–1	MLT–3I–1
Electric resistance \(\begin{array}{l}\left(\frac{potential\ difference}{current} \right)\end{array} \)	ohm	ML2T–3I–2
Emf (or) electric potential (work/charge)	volt	ML2T–3I–1
Energy (capacity to do work)	joule	ML2T–2
Energy density \(\begin{array}{l}\left(\frac{energy}{volume} \right)\end{array} \)	Jm–3	ML–1T–2
Entropy \(\begin{array}{l}\left(\Delta S=\Delta Q/T \right)\end{array} \)	Jθ–1	ML2T–2θ–1
Force (mass x acceleration)	newton (N)	MLT–2
Force constant or spring constant (force/extension)	Nm–1	MT–2
Frequency (1/period)	Hz	T–1
Gravitational potential (work/mass)	Jkg–1	L2T–2
Heat (energy)	J or calorie	ML2T–2
Illumination (Illuminance)	lux (lumen/metre2)	MT–3
Impulse (force x time)	Ns or kgms–1	MLT–1
Inductance (L) \(\begin{array}{l}\left(energy =\frac{1}{2}L{{I}^{2}} \right)\end{array} \) or coefficient of self-induction	henry (H)	ML2T–2I–2
Intensity of gravitational field (F/m)	Nkg–1	L1T–2
Intensity of magnetization (I)	Am–1	L–1I
Joule’s constant or mechanical equivalent of heat	Jcal–1	MoLoTo
Latent heat (Q = mL)	Jkg–1	MoL2T–2
Linear density (mass per unit length)	kgm–1	ML–1
Luminous flux	lumen or (Js–1)	ML2T–3
Magnetic dipole moment	Am2	L2I
Magnetic flux (magnetic induction x area)	weber (Wb)	ML2T–2I–1
Magnetic induction (F = Bil)	NI–1m–1 or T	MT–2I–1
Magnetic pole strength (unit: ampere–meter)	Am	LI
Modulus of elasticity (stress/strain)	Nm–2, Pa	ML–1T–2
Moment of inertia (mass x radius2)	kgm2	ML2
Momentum (mass x velocity)	kgms–1	MLT–1
Permeability of free space \(\begin{array}{l}\left(\mu_o = \frac{4\pi Fd^{2}}{m_1m_2} \right)\end{array} \)	Hm–1 or NA–2	MLT–2I–2
Permittivity of free space \(\begin{array}{l}\left({{\varepsilon }_{o}}=\frac{{{Q}_{1}}{{Q}_{2}}}{4\pi F{{d}^{2}}} \right)\end{array} \)	Fm–1 or C2N–1m–2	M–1L–3T4I2
Planck’s constant (energy/frequency)	Js	ML2T–1
Poisson’s ratio (lateral strain/longitudinal strain)	––	MoLoTo
Power (work/time)	Js–1 or watt (W)	ML2T–3
Pressure (force/area)	Nm–2 or Pa	ML–1T–2
Pressure coefficient or volume coefficient	oC–1 or θ–1	θ–1
Pressure head	m	MoLTo
Radioactivity	disintegrations per second	MoLoT–1
Ratio of specific heats	––	MoLoTo
Refractive index	––	MoLoTo
Resistivity or specific resistance	\(\begin{array}{l}\Omega\end{array} \) –m	ML3T–3I–2
Specific conductance or conductivity (1/specific resistance)	siemen/metre or Sm–1	M–1L–3T3I2
Specific entropy (1/entropy)	KJ–1	M–1L–2T2θ
Specific gravity (density of the substance/density of water)	––	MoLoTo
Specific heat (Q = mst)	Jkg–1θ–1	MoL2T–2θ–1
Specific volume (1/density)	m3kg–1	M–1L3
Speed (distance/time)	ms–1	LT–1
Stefan’s constant \(\begin{array}{l}\left( \frac{heat\ energy}{area\ x\ time\ x\ temperatur{{e}^{4}}} \right)\end{array} \) .	Wm–2θ–4	MLoT–3θ–4
Strain (change in dimension/original dimension)	––	MoLoTo
Stress (restoring force/area)	Nm–2 or Pa	ML–1T–2
Surface energy density (energy/area)	Jm–2	MT–2
Temperature	oC or θ	MoLoToθ
Temperature gradient \(\begin{array}{l}\left(\frac{change\text{ in temperature}}{\text{distance}} \right)\end{array} \)	oCm–1 or θm–1	MoL–1Toθ
Thermal capacity (mass x specific heat)	Jθ–1	ML2T–2θ–1
Time period	second	T
Torque or moment of force (force x distance)	Nm	ML2T–2
Universal gas constant (work/temperature)	Jmol–1θ–1	ML2T–2θ–1
Universal gravitational constant \(\begin{array}{l}\left(F = G.\frac{{{m}_{1}}{{m}_{2}}}{{{d}^{2}}} \right)\end{array} \)	Nm2kg–2	M–1L3T–2
Velocity (displacement/time)	ms–1	LT–1
Velocity gradient (dv/dx)	s–1	T–1
Volume (length x breadth x height)	m3	L3
Water equivalent	kg	MLoTo
Work (force x displacement)	J	ML2T–2

Quantities Having the Same Dimensional Formula

1. Impulse and momentum.
2. Work, torque, the moment of force, energy.
3. Angular momentum, Planck’s constant, rotational impulse.
4. Stress, pressure, modulus of elasticity, energy density.
5. Force constant, surface tension, surface energy.
6. Angular velocity, frequency, velocity gradient.
7. Gravitational potential, latent heat.
8. Thermal capacity, entropy, universal gas constant and Boltzmann’s constant.
9. Force, thrust.
10. Power, luminous flux.

Applications of Dimensional Analysis

Dimensional analysis is very important when dealing with physical quantities. In this section, we will learn about some applications of dimensional analysis.

Fourier laid down the foundations of dimensional analysis. The dimensional formulas are used to:

1. Verify the correctness of a physical equation.
2. Derive a relationship between physical quantities.
3. Converting the units of a physical quantity from one system to another system.

Checking the Dimensional Consistency

As we know, only similar physical quantities can be added or subtracted. Thus, two quantities having different dimensions cannot be added together. For example, we cannot add mass and force or electric potential and resistance.

For any given equation, the principle of homogeneity of dimensions is used to check the correctness and consistency of the equation. The dimensions of each component on either side of the sign of equality are checked, and if they are not the same, the equation is considered wrong.

Let us consider the equation given below,

The dimensions of the LHS and the RHS are calculated

As we can see, the dimensions of the LHS and the RHS are the same. Hence, the equation is consistent.

Deducing the Relation among Physical Quantities

Dimensional analysis is also used to deduce the relation between two or more physical quantities. If we know the degree of dependence of a physical quantity on another, that is, the degree to which one quantity changes with the change in another, we can use the principle of consistency of two expressions to find the equation relating to these two quantities. This can be understood more easily through the following illustration.

Example: Derive the formula for centripetal force F acting on a particle moving in a uniform circle.

As we know, the centripetal force acting on a particle moving in a uniform circle depends on its mass m, velocity v and the radius r of the circle. Hence, we can write

F = m^a v^b r^c

Writing the dimensions of these quantities,

As per the principle of homogeneity, we can write,

a = 1, b + c = 1 and b = 2

Solving the above three equations, we get, a = 1, b = 2 and c = -1.

Hence, the centripetal force F can be represented as,

Frequently Asked Questions (FAQs) on Units and Dimensions

Q1

What is the meaning of dimension in Physics?

It is an expression that relates derived quantities to fundamental quantities. But it is not related to the magnitude of the derived quantity.

Q2

What is the dimension of force?

We know that, F = ma —– (1)
Mass is a fundamental quantity, but acceleration is a derived quantity and can be represented in terms of fundamental quantities.
a = [LT⁻²] —– (2)
Using (1) and (2),
F = [MLT⁻²]
This is the dimension of force.

Q3

What is dimensional analysis?

Dimensional analysis is based on the principle that two quantities can be compared only if they have the same dimensions. For example, I can compare kinetic energy with potential energy and say they are equal, or one is greater than another because they have the same dimension. But I cannot compare kinetic energy with force or acceleration as their dimensions are different.

Q4

How do you demonstrate dimensional analysis with an example?

Suppose I have the following equation,
F = E^a.V^b. Tc
Where, F = Force; E = Energy; V = Velocity; M = Mass
We need to find the value of a, b and c.
Following are the dimensions of the given quantities,
F = [MLT⁻²], E = [ML²T⁻²], V = [LT⁻¹]
According to dimensional analysis, the dimension of RHS should be equal to LHS; hence,
[MLT⁻²] = [ML²T⁻²]^a . [LT⁻¹]^b. [T]^c
[MLT⁻²] = [M^a L^2a+b T^−2a−b+c]
Now, we have three equations,
a = 1
2a+b = 1
−2a − b + c = −2
Solving the three equations, we get,
a = 1, b = −1 and c = −1.

Q5

What is meant by a unit?

The standard quantity with which a physical quantity of the same kind is compared is called a unit.

Q6

Why are mass, length, and time chosen as fundamental or base quantities in mechanics?

This is because mass, length and time are independent of each other. All the other quantities in mechanics can be expressed in terms of mass, length and time.

Q7

Define significant figures.

Significant figures are those digits in a number known with certainty plus one more uncertain number.

Q8

Define dimensions.

The dimensions of a physical quantity are the powers to which the fundamental quantities are raised to represent that physical quantity.

Q9

Can a quantity have dimensions but no units associated with it?

No. If a quantity has dimensions, then it should have units associated with it.

Q10

What are the dimensions of a physical quantity?

Dimensions of a physical quantity refer to the nature of the quantity, which defines it in some measurable terms. The physical dimensions are expressed in terms of length, mass and time, represented by L, M and T, respectively.

Q11

Define a unit

Any given physical quantity is measured compared to a basic unit that is arbitrarily chosen and internationally accepted. The result of a measurement is expressed by a number which is accompanied by a unit.

Q12

What are supplementary units?

Supplementary units are dimensionless physical quantities that are used along with fundamental units.

Q13

Give an example of derived units.

Velocity is a physical quantity derived from fundamental quantities of length and time.
The unit of velocity is ms^-1.