Dimensional Analysis

In ancient times, the length of a foot, the width of a finger, and the distance of a step were commonly used as different units of measurements. In 1790, the French created a standard unit of measurement called the metric system.  For the sake of uniformity, scientists all over the world have accepted a set of standard units of measurement. The system of units now used is known as the International System of Units (SI units).

Fundamental and Derived Units

In mechanics, there are three physical quantities namely mass, length and time which are called fundamental physical quantities and the units of these physical quantities are called fundamental units. These units are called fundamental because they are all independent of each other.

But while discussing other branches of Physics, we come across other physical quantities like temperature, luminous intensity, electric current and the amount of substance which are also called fundamental physical quantities and their units kelvin (K), candela (Cd), ampere (A), mole (mol) respectively are called fundamental units.

In addition, there are two supplementary fundamental units - radian for plane angle and steradian for solid angle.

The units which are not independent of each other and are obtained from the fundamental units, are called the derived units. For example, the units of area, velocity, acceleration, volume, force, etc. are the derived units.

S.I. Units or International System of Units

In this system, there are seven fundamental units and two supplementary units. Seven fundamental units are:

  1. metre (for length)
  2. kilogram (for mass)
  3. second (for time)
  4. kelvin (for temperature)
  5. ampere (for current)
  6. candela (for luminous intensity
  7. mole (for amount of substance)

Two supplementary units are:

  1. radian (for plane angle)
  2. steradian (for solid angle)

Dimensions

The dimensions of a physical quantity are the powers to which the fundamental units are raised in order to obtain the units of that quantity. The fundamental quantities mass, length, time, temperature, luminous intensity and current are respectively represented as M, L, T, θ, cd and A.

If the dimensions of a physical quantity are a in length, b in mass, c in time and d in temperature, then the dimensions of that physical quantity are written as:

[LaMbTcθd]

This is the dimensional formula of that quantity.

Important Dimensional Formulae

  • Density [ML-3]
  • Velocity [LT-1]
  • Angular velocity [T-1]
  • Acceleration [LT-2]
  • Angular acceleration [T-2]
  • Impulse [MLT-1]
  • Torque [ML2T-2]
  • Force [MLT-2]
  • Moment of inertia [ML-2]
  • Work [ML2T-2]
  • Power [ML2T-3]
  • Pressure [ML-1T-2]
  • Stress [ML-1T-2]
  • Strain (dimensionless)
  • Modulus of elasticity [ML-1T-2]
  • Surface tension [MT-2]
  • Gravitational constant [M-1L3T-2]
  • Co-efficient of viscosity [ML-1T-1]