Understanding Units and Dimensions of Physical Quantities

A physical measurement consists of two parts: a numeric value that shows the quantity and a measurement unit that defines the measurement standard. The physical quantity is defined by these components, which give the expression Q = n × u.

For instance, the length of an object can be expressed as 40″ cm” because 40 serves as the numerical value, while cm (centimetres) functions as the unit. The product of a physical quantity’s value and its unit system remains unchanged when the quantity is measured using two distinct unit systems. The equivalences 45″ min” = 2700″ s” and 3.5 L = 3500 mL serve as examples.

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Table of Contents

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Define Units

A unit is a standard measure of a physical quantity. It tells us in what terms the measurement is expressed.

Unit Examples

The unit of length is the meter, which has the abbreviation (m). The unit of mass is the kilogram, which has the abbreviation (kg). The unit of time is the second, which has the abbreviation (s).

Define Dimensions

Dimensions define the characteristics of a physical quantity using basic units, such as length (L), mass (M), and time (T). The dimension of velocity shows its representation through [LT^(-1)] while the dimension of force shows its representation through [MLT^(-2) ].

Dimensional Analysis and Related Concepts

Dimensional analysis involves studying how physical quantities relate to one another by identifying their dimensions.

Dimensional Formula

The dimensional formula shows how a derived quantity relates to fundamental units through the specific powers of those units. The dimensional formula for the derived quantity Q is Q = M^a L^b T^c; the term M^a L^b T^c is its dimensional formula, where the exponents a, b, and c are called the dimensions of the quantity.

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Dimensional Constants

Physical quantities that have fixed values and possess dimensions are called dimensional constants. Examples include the gravitational constant G, Planck’s constant h, the universal gas constant R, and the speed of light in a vacuum c.

Dimensionless Quantities

Dimensionless quantities refer to physical, chemical, or mathematical values that lack measurement units and possess a net dimension of one.

Without Units: These quantities have no dimensions and no units; they are pure numbers, such as π, e, and trigonometric functions like sin⁡θ, cos⁡θ, and tan⁡θ.
With Units: Some dimensionless quantities are expressed with special units, for example, angular displacement measured in radians or Joule’s constant expressed as joules per calorie.

Dimensional Variables

Quantities with dimensions whose values can vary are called dimensional variables. Common examples include velocity, acceleration, force, work, and power.

Dimensionless Variables

Dimensionless variables are physical quantities that have no dimensions and whose values can vary. Examples include specific gravity, refractive index, coefficient of friction, and Poisson’s ratio.

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Law of Homogeneity of Dimensions

For any physically meaningful equation relating physical quantities, the dimensions of every term on both sides must be identical.

Conversion of Numerical Values Between Unit Systems

Consider a physical quantity with dimensions a, b, and c corresponding to length (L), mass (M), and time (T), respectively. If n_1is the numerical value of Qin a system with fundamental units L_1, M_1, and T_1, and n_2 is its numerical value in another system with fundamental units L_2, M_2, and T_2, then the values are related by:

Useful Unit Conversions in Physics

Pressure Conversions:

1 bar = 10⁶ dyne/cm² = 10⁵ N/m² = 10⁵ Pa
76 cm of Hg ≈ 1.013 × 10⁶ dyne/cm² = 1.013 × 10⁵ Pa ≈ 1.013 bar
1 torr (1 mm Hg) = 1.333 × 10³ dyne/cm² ≈ 1.333 mbar

Force and Mass:

1 dyne = 10⁻⁵ N
1 kgf (kilogram-force) = 9.81 N

Energy and Work:

1 calorie (cal) ≈ 4.184 joule (J)
1 erg = 10⁻⁷ J
1 electron volt (eV) = 1.602 × 10⁻¹⁹ J
1 kilowatt-hour (kWh) = 3.6 × 10⁶ J

Power:

1 horsepower (H.P.) = 746 W

Speed:

1 km/h = 5/18 m/s ≈ 0.2778 m/s

Essential Physical Constants

Fundamental Speeds:

Velocity of light in vacuum: c = 3 × 10⁸ m/s
Velocity of sound in air at STP: v = 331 m/s

Gravitational & Mechanical Constants:

Acceleration due to gravity: g = 9.81 m/s²
Universal gravitational constant: G = 6.67 × 10⁻¹¹ N·m²/kg²
Mechanical equivalent of heat: J = 4.186 J/cal

Atomic & Molecular Constants:

Avogadro number: N_A = 6.022 × 10²³ mol⁻¹
Atomic mass unit: 1 u = 1.66 × 10⁻²⁷ kg
Quantum of charge: e = 1.602 × 10⁻¹⁹ C

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Thermal & Radiation Constants:

Boltzmann constant: k = 1.381 × 10⁻²³ J/K
Stefan-Boltzmann constant: σ = 5.67 × 10⁻⁸ W/m²·K⁴
Absolute zero: 0 K = −273.15 °C

Gas & Pressure Constants:

Universal gas constant: R = 8.314 J/mol·K
One atmosphere: 1 atm = 76 cm Hg = 1.013 × 10⁵ Pa

Electromagnetic Constants:

Permittivity of free space: ε₀ = 8.854 × 10⁻¹²
Permeability of free space: μ₀ = 4π × 10⁻⁷ H/m
Planck’s constant: h = 6.626 × 10⁻³⁴ J·s

Density Constants:

Density of water at 4 °C: ρ = 1000 kg/m³ = 1 g/cm³
Density of air at STP: ρ = 1.293 kg/m³

Derived SI Units with Special Names

The following are commonly used derived SI units that have been assigned special names:

Physical Quantity	SI Unit	Symbol
Frequency	hertz	Hz
Energy	joule	J
Force	newton	N
Power	watt	W
Pressure	pascal	Pa
Electric charge (quantity of electricity)	coulomb	C
Electric potential difference / EMF	volt	V
Electric resistance	ohm	Ω
Electric conductance	siemens	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 of Common Physical Quantities

The dimensional formula of a physical quantity expresses it in terms of the fundamental quantities: Mass (M), Length (L), Time (T), Electric current (I), Temperature (θ), Luminous intensity (J), and Amount of substance (N). Derived quantities are written as products of these fundamentals raised to specific powers.

Physical Quantity	Unit	Dimensional Formula
Acceleration (or gravity)	m/s²	LT⁻²
Angle / Angular displacement	rad	MoLoTo
Angular frequency	rad/s	T⁻¹
Angular impulse (torque × time)	N·m·s	ML²T⁻¹
Angular momentum (Iω)	kg·m²/s	ML²T⁻¹
Angular velocity	rad/s	T⁻¹
Area	m²	L²
Boltzmann constant	J/K	ML²T⁻²θ⁻¹
Bulk modulus	Pa	ML⁻¹T⁻²
Calorific value	J/kg	L²T⁻²
Coefficient of linear/areal/volume expansion	°C⁻¹ or K⁻¹	θ⁻¹
Surface tension (force/length)	N/m	MT⁻²
Thermal conductivity	W/m·K	MLT⁻³θ⁻¹
Viscosity	poise	ML⁻¹T⁻¹
Compressibility (1/bulk modulus)	Pa⁻¹	M⁻¹LT²
Density	kg/m³	ML⁻³
Displacement / Wavelength / Focal length	m	L
Electric capacitance (charge/potential)	F	M⁻¹L⁻²T⁴I²
Electric conductance	S	M⁻¹L⁻²T³I²
Electric conductivity	S/m	M⁻¹L⁻³T³I²
Electric charge	C	IT
Electric current	A	I
Electric dipole moment	C·m	LTI
Electric field strength (force/charge)	N/C	MLT⁻³I⁻¹
Electric resistance	Ω	ML²T⁻³I⁻²
EMF / Electric potential	V	ML²T⁻³I⁻¹
Energy	J	ML²T⁻²
Energy density	J/m³	ML⁻¹T⁻²
Entropy	J/θ	ML²T⁻²θ⁻¹
Force	N	MLT⁻²
Force constant / Spring constant	N/m	MT⁻²
Frequency	Hz	T⁻¹
Gravitational potential	J/kg	L²T⁻²
Heat / Energy	J	ML²T⁻²
Illumination / Illuminance	lux	MT⁻³
Impulse	N·s	MLT⁻¹
Inductance	H	ML²T⁻²I⁻²
Intensity of gravitational field	N/kg	LT⁻²
Intensity of magnetisation	A/m	L⁻¹I
Joule’s constant	J/cal	MoLoTo
Latent heat	J/kg	L²T⁻²
Linear density	kg/m	ML⁻¹
Luminous flux	lumen	ML²T⁻³
Magnetic dipole moment	A·m²	L²I
Magnetic flux	Wb	ML²T⁻²I⁻¹
Magnetic induction	T	MT⁻²I⁻¹
Magnetic pole strength	A·m	LI
Modulus of elasticity	Pa	ML⁻¹T⁻²
Moment of inertia	kg·m²	ML²
Momentum	kg·m/s	MLT⁻¹
Permeability of free space	H/m	MLT⁻²I⁻²
Permittivity of free space	F/m	M⁻¹L⁻³T⁴I²
Planck’s constant	J·s	ML²T⁻¹
Poisson’s ratio	–	MoLoTo
Power	W	ML²T⁻³
Pressure	Pa	ML⁻¹T⁻²
Pressure coefficient / Volume coefficient	°C⁻¹	θ⁻¹
Pressure head	m	MoLTo
Radioactivity	disintegrations/s	MoLoT⁻¹
Ratio of specific heats	–	MoLoTo
Refractive index	–	MoLoTo
Resistivity	Ω·m	ML³T⁻³I⁻²
Specific conductance	S/m	M⁻¹L⁻³T³I²
Specific entropy	1/K	M⁻¹L⁻²T²θ
Specific gravity	–	MoLoTo
Specific heat	J/kg·θ	L²T⁻²θ⁻¹
Specific volume	m³/kg	M⁻¹L³
Speed	m/s	LT⁻¹
Stefan’s constant	W/m²·θ⁴	MLT⁻³θ⁻⁴
Strain	–	MoLoTo
Stress	Pa	ML⁻¹T⁻²
Surface energy density	J/m²	MT⁻²
Temperature	°C or θ	MoLoToθ
Temperature gradient	°C/m	MoL⁻¹Toθ
Thermal capacity	J/θ	ML²T⁻²θ⁻¹
Time period	s	T
Torque / Moment of force	N·m	ML²T⁻²
Universal gas constant	J/mol·θ	ML²T⁻²θ⁻¹
Universal gravitational constant	N·m²/kg²	M⁻¹L³T⁻²
Velocity	m/s	LT⁻¹
Velocity gradient	s⁻¹	T⁻¹
Volume	m³	L³
Water equivalent	kg	MoLoTo
Work	J	ML²T⁻²

Physical Quantities Sharing the Same Dimensional Formula

Impulse and momentum
Force and thrust
Work, energy, torque, and moment of force
Angular momentum, Planck’s constant, and rotational impulse
Stress, pressure, modulus of elasticity, and energy density
Force constant, surface tension, and surface energy
Angular velocity, frequency, and velocity gradient
Gravitational potential and latent heat
Thermal capacity, entropy, universal gas constant, and Boltzmann constant
Power and luminous flux

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Applications of Dimensional Analysis

Dimensional analysis is a powerful tool in physics for understanding and working with physical quantities. It helps simplify complex problems and ensure consistency in equations. Fourier established the foundations of dimensional analysis, and it has several important applications:

Checking Equation Consistency:
Dimensional formulas can be used to verify whether a physical equation is dimensionally correct.
Deriving Relationships:
It allows us to find possible relationships between different physical quantities, even without detailed experimental data.

Unit Conversion:
Dimensional analysis provides a systematic way to convert a physical quantity from one system of units to another.

How to Check Dimensional Consistency?

Identify all quantities in the equation and write down their dimensional formulas.
Calculate the dimensions of each term on the left-hand side (LHS) and the right-hand side (RHS).
Compare LHS and RHS:
- If the dimensions match, the equation is dimensionally consistent.
- If they differ, the equation is incorrect.
Check additions and Subtractions: Only quantities with the same dimensions can be added or subtracted.

Example: Dimensional Analysis of equation: F=m⋅a Step 1: Left-Hand Side (LHS)

Force, F

Dimensional formula: [F]=MLT−2

Step 2: Right-Hand Side (RHS)

Mass × Acceleration: m⋅a
Mass: [m]=M
Acceleration: [a]=LT^−2
Multiply: [m] [a] = M⋅ LT−2=MLT−2

Step 3: Compare [F]_LHS=[F]_RHS = MLT^-2 Since LHS = RHS, the equation is dimensionally consistent.

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FAQs?

What does the term dimension mean in physics?

In physics, a dimension describes the nature of a physical quantity in terms of fundamental units such as mass (M), length (L), and time (T). It shows how a derived quantity relates to these basics, but it does not indicate the magnitude of the quantity.

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How can we determine the dimension of a force?

Force is defined as F = ma, where m is mass and a is acceleration. Mass is a fundamental quantity [M], and acceleration has the dimension [LT^(-2) ]. Therefore, the dimension of force is:

[F]=[M][LT^(-2)]=[MLT^(-2) ]

What is the purpose of dimensional analysis in physics?

Dimensional analysis is a method for checking equations or relationships among physical quantities. It is based on the principle that only quantities with the same dimensions can be compared, added, or equated. For example, kinetic energy can be compared with potential energy, but not with force or acceleration.

How do you use dimensional analysis to find unknown exponents?

Write all quantities in terms of M, L, T. Express the RHS with unknown powers, equate the dimensions of both sides, and solve the resulting system of equations to find the exponents.