Units and Dimensions - Dimensional Formula

Introduction to Units and Dimensions

Every measurement has two parts. The first is a number (n), and the next is a unit (u). Q = nu. For example, the length of an object = 40 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, 2.8 m = 280 cm; 6.2 kg = 6200 g.

Table of Contents

• Units
• Dimensions
• What Is the Dimensional Formula?
• Limitations
• Physical Constants
• Dimensional Formulas for Physical Quantities
• Quantities with the Same Dimensional Formula
• Applications
• FAQs

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

Supplementary Quantities

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.

⇒ Also Read: List of All SI Units

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.

Read More: Dimensional Analysis

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

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

Dimensional Formulas for Physical Quantities

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,

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