Dimensions of Physical Quantities

In physics and engineering, physical quantities are composed of a set of numbers with fundamental quantities. Every other physical unit of measurement or quantity can be mentioned in terms of the fundamental quantities. In other words, dimension is an expression of the character of a derived quantity about fundamental quantities, without regard for its numerical value.

The list of fundamental quantities along with their representations as dimension are listed below:

  1. Mass [M]
  2. Length [L]
  3. Time [T]
  4. Current [A]
  5. Temperature [T]
  6. Amount of Substance [mol]
  7. Luminous Intensity [cd]

How do we express the Physical Quantities?

Every physical quantity used in the world can be expressed as a combination of these fundamental quantities. The square brackets are added to the letters to show that the quantity used is a dimensional quantity. The dimensions of any other quantity will involve one or more of these basic dimensions.

For instance, a measurement of the volume of an object will involve the product of three lengths and the dimensions of volume are therefore [L]3. In the same way, a measurement of velocity requires a length divided by a time, and so the dimensions of velocity are [L][T]-1.

In order to understand the technique to write dimensions of a derived quantity, we consider the case of force. Force is defined as:

F = m ∗ a

F= [M][a]

The dimension of acceleration, represented as [a] is in fact a derived dimensional quantity which is ratio of velocity to time. In turn, velocity is also a derived quantity, being ratio of length and time.

F = [M][a] = [M] [vT−1]

F = [M][LT−1T−1] = [MLT−2]

The table below shows the dimensions of the various common quantities in mechanics.

Quantity Formula Dimensions
Area Length x Length [L]2
Volume Length x Length x Length [L]3
Velocity Length/ Time [L][T]-1
Acceleration Velocity/ Time [L][T]-2
Force Mass x Acceleration [M][L][T]-2
Energy Force x Length [M][L]2[T]-2
Power Energy/ Time [M][L]2[T]-3
Pressure Force/ Area [M][L]-1[T]-2
Momentum Mass x Velocity [M][L][T]-1

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.

The dimension of any physical quantity is the combination of the basic physical dimensions that composes it. Dimensional analysis is based on the fact that physical law must be independent of the units used to measure the physical variables. It can be used to check the plausibility of derived equations, computations, and hypothesis.

The dimensions of the quantities of each side of an equation must match: those on the left-hand side must equal those on the right. Both sides of the equation must equate to the same dimensions. For example;

s = ut + ½ at2

Dimensional representation gives us:

[L] = [L][T]–1[T] + [L][T]–2[T]2

Therefore,

[L] = [L] + [L]

This proves that the equation is dimensionally correct.

If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not to be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation is definitely wrong.