Dimensional Analysis

Dimensional Analysis

Dimensions of a physical quantity are the powers to which the fundamental quantities must be raised.

We know that velocity   = displacement / time

=  [L] / [T] = [MoL1Tâˆ'1]

where [M], [L] and [T] are the dimensions of the fundamental quantities mass, length and time respectively.

Therefore velocity has zero dimension in mass, one dimension in length and âˆ'-dimension in time. Thus the dimensional formula for velocity is [MoL1Tâˆ'1] or simply [LTâˆ'1].The dimensions of fundamental quantities are given in Table  and the dimensions of some derived quantities are given in List.

Dimensions of fundamental quantities

Fundamental quantity    Dimension

Length        L

Mass M

Time  T

Temperature         K

Electric current     A

Luminous intensity        cd

Amount of subtance      mol

Dimensional formulae of some derived quantities

Physical      quantity     Expression Dimensional formula

Area :                   length x breadth   [L2]

Density :               mass / volume      [MLâˆ'3]

Acceleration :       velocity / time       [LTâˆ'2 ]

Momentum :         mass x velocity    [MLTâˆ'1]

Force :                  mass x acceleration        [MLTâˆ'2 ]

Work :                  force x distance    [ML2Tâˆ'2 ]

Power :                 work / time [ML2Tâˆ'3 ]

Energy :                Work [ML2Tâˆ'2 ]

Impulse :              force  x  time        [MLTâˆ'1 ]

Radius of gyration :       Distance      [L]

Pressure :             force / area  [MLâˆ'1Tâˆ'2 ]

Surface       tension :      force / length        [MTâˆ'2 ]

Frequency :          1 / time period      [Tâˆ'1]

Tension :              force  [MLTâˆ'2 ]

Moment of force (or torque) : force x distance    [ML2Tâˆ'2 ]

Angular      velocity :     angular displacement / time     [Tâˆ'1]

Stress :                 force / area  [MLâˆ'1Tâˆ'2]

Heat :                   energy         [ML2Tâˆ'2 ]

Heat capacity :     heat energy/ temperature         [ML2T-2K-1]

Charge :                current x time       [AT]

Faraday constant :         Avogadro constant x     elementary charge [AT mol-1]

Magnetic induction :      force / (current x length)          [MT-2 A-1]

Dimensional quantities

Constants which possess dimensions are called dimensional constants. Planck's constant, universal gravitational constant are dimensional constants.

Dimensional variables are those physical quantities which possess dimensions but do not have a fixed value. Example âˆ' velocity, force, etc.

Dimensionless quantities

There are certain quantities which do not possess dimensions. They are called dimensionless quantities. Examples are strain, angle, specific gravity, etc. They are dimensionless as they are the ratio of two quantities having the same dimensional formula.

Principle of homogeneity of dimensions

An equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same. This is called the principle of homogeneity of dimensions. This principle is based on the fact that two quantities of the same dimension only can be added up, the resulting quantity also possessing the same dimension.

The equation A + B = C is valid only if the dimensions of A, B and C are the same.