Dimensional Analysis in Research

We have witnessed above that speed can be expressed in different units; the same speed can be expressed, if we wish, in miles per second, meters per day, inches per year, or in any other units we choose. In each case, the number will be different, combined with the corresponding unit. But there is something common among all these quantities: they are all obtained by dividing a quantity of length by a quantity of time. Symbolically we may represent this fact as

Speed = L/T

where L is length and Tis time. Relative to such representation, it is said that “L/T” or LT ^-1 is the “dimension” of speed. The dimensions of some of the other simple quantities are shown below:

• Area = length x length = L x L = L²
• Volume = area x length = L² x L = L³
• Density = mass + volume = M ÷ L³= ML ^-3
• Specific gravity = density + density = ML ^-3 ÷ ML ^-3 = 1, meaning, it is “dimensionless”
• Acceleration = speed + time = [L+T] ÷ T = L÷T² = LT^-2
• Force = mass x acceleration = M x LT^-2  = MLT~²

A close study of the dimensions of the quantities shown above reveals some significant features:

1. Dimensions are free of units. Whatever the units used to express the quantity, the dimensions of that quantity remain unaltered.
2. Dimensions of a derived physical property are obtained by representing with symbols (with required exponents) the various fundamental physical properties conforming to the definition of the derived property.
3. Dimensions have no numerical significance. Stripped of numbers, they reveal the structure, rather than the value, of the quantity. High speed or low speed, for instance, have the same dimen­sions (LIT, or LT’¹).
4. Dimensions of the product of two kinds of quanti­ties are the same as the product of the dimensions of those quantities.
5. Dimensions of the ratio of two kinds of quantities are the same as the ratio of the dimensions of those quantities.
6. The ratio of two quantities having the same units is “dimensionless”; it is a pure number. Both the cir­cumference and diameter of a circle, for instance, are units of length; their ratio is only a number (n); they have no dimensions. Similarly, the specific gravity of a substance, which is a ratio of its density and the density of water, each having the same units, is a pure number; it has no dimensions.

When quantitative relations between or among parameters are attempted, it is often necessary to state such relations in the form of equations. The most significant application of dimensional analysis is to check that such equations are “balanced,” as explained following.

Firstly, a comparison of two quantities is meaningful only when their dimensions are the same; 5 centimeters can be com­pared to 7 miles because both have the same dimensions, (L). But comparing 5 grams to 7 miles makes no sense. Similarly, we can add two quantities when their dimensions are the same. We can add 3 milligrams to 45 tons, but adding 4 inches to 6 grams is nonsense.

In the course of formulating the relation among four quanti­ties, let us suppose that we form an equation:

S = X (Y² + Z)             (3.1)

To see that the quantities of the same units may be added in this equation, it is necessary that Y² and Z have the same units. And to see that the quantities of the same units are compared, X (Y + Z) should have the same units as 5. Both these conditions depend on what the actual units of 5, X, Y, and Z are. To make this issue concrete, let us suppose that the quantities have the fol­lowing units:

S: mile

X: mph

Y: hour

Z: hour²

Substituting only the units—ignoring the numerical values for the moment—we have

Mile = (mile ÷ hour) x (hour² + hour²)                                (3.2)

The first condition, namely, that Y² and Z of (3.1) have the same units, is fulfilled above.

Then, on further simplification of (3.2), we have

Mile = mile x hour                                                                  (3.3)

which is obviously wrong. If X had units mile/hour² instead, and all other quantities had the same units as before, the reader might find that the equation could have been balanced.

Such balancing would have been much easier if we used dimensions instead of units, for then the structure of the equa­tion, free of quantities, would be revealed. This is done below, incorporating the changed units of X.

S: L

X: L/T² = LT^-2

Y: T

Z: T²

Now, (3.1), in terms of dimensions, is reduced to

L = LT^-2 x (T² + T²)                                                        (3.4)

L = L

The reader should have noticed in (3.4) that (T² + T²) is taken to be T², not 2T², if we simply follow the rules of algebra. In fact, a similar jump should also have been noticed in (3.2), wherein (hour² + hour²) is taken to be simply hour². Thus, yet another feature of dimensional analysis worth adding to the list of other features given previously may be stated as follows:

Whereas in multiplication and division, alphabetical terms with exponents used in dimensional analysis may be treated sim­ply as algebraic terms, in addition and subtraction, special treat­ment is necessary.

Two or more similar terms, when added (or subtracted), are subsumed into the same term.

Source: Srinagesh K (2005), The Principles of Experimental Research, Butterworth-Heinemann; 1st edition.