Dimensional Consistency

Any physical quantity’s dimension indicates its dependency on the base quantities as a product of symbols (or power of symbols) representing the basic quantities. A measurement of length, for example, is said to have dimension L, a measurement of mass has dimension M, and a measurement of time has dimension T.

Dimensions, like units, follow algebraic laws.

Because area is the product of two lengths, it has the dimension L2  or Length squared.

In general, every physical quantity’s dimension may be expressed as:

LaMbTcIdΘeNfJg

for some powers a, b, c, d, e, f, a, b, c, d, e, f, and g.

The dimensions of a length can be written in this way, with a = 1a= 1 and the following six powers all set to zero:

L1 = L1M0T0I0Θ0N0J0

Dimensionless refers to any quantity having a dimension that may be represented so that all seven powers are 0. Dimensionless quantities are referred to as pure numbers by physicists.

Dimensionally consistent

The meaning of dimensionally consistent is that equality or equation is represented by equal signs, requiring not only that the value is the same but also that the units on both sides of the equation be the same.

Dimensionally consistent equations are those in which all of the terms have the same dimensions and all of the arguments of any mathematical functions that appear in the equation are dimensionless.

Checks for dimensional consistency in equations

A sequence of actions may be taken to execute the algorithm check for dimensional consistency in equations. They are as follows:

1. Constructing the constraint tree for each equation

2. Using known naming standards, determine the probable dimensions of the variables and consequently the dimensions of the constraint tree’s lead nodes.

3. Propagating values in the constraint tree and determining each tree’s consistency

4. Imposing consistency requirements on the dimensions of all leaf nodes corresponding to a single variable, and propagating the constraints across the resultant overall graph

5. Providing feedback to the user

Dimensional Consistency Rule

The idea of dimension is important because every mathematical equation linking physical quantities must be dimensionally consistent.

The rules to check dimensional consistency are as follows:

· Every term in an equation must have the same dimension; adding or subtracting amounts of different dimensions makes no sense. In an equation, the expressions on either side of the equation must have the same dimensions.

· The arguments of any typical mathematical functions that appear in the equation, such as trigonometric functions, logarithms, or exponential functions, must be dimensionless. These functions accept only pure numbers as inputs and return only pure numbers as outputs.

If the dimensional consistency rules are broken, an equation is no longer dimensionally consistent and cannot be used to express physical law. This fact may be used to check for typos and algebraic errors, to assist recall the numerous laws of physics, and even to suggest new laws of physics.

Dimensionally consistent examples

Consider the physical quantities s, v, a, and t, which have dimensions, [s]= L, [v]= LT-1, [a] = LT-1 and [t]= T, where L stands for length and T for time. Determine the dimensional consistency of each of the following equations:

a.   s= vt + 0.5at2

b.   s= vt2 + 0.5at

c.   v= sin (at2 /s)

Strategy

Dimensional consistency is defined as ensuring that each term in a given equation has the same dimension as the other terms in that equation and that the arguments of any standard mathematical functions are dimensionless.

Solutions:

a.  Because there are no trigonometric, logarithmic, or exponential functions to consider in this equation, we just need to consider the dimensions of each term. There are three words, one in the left expression and two in the right expression, therefore we’ll go over them one by one:

[s] = L

[vt] = [v]. [t] = LT-1.T = LT0 =L

[0.5at2] = [a]. [t]2 = LT-2.T2 = LT0 = L

b. Because there are no trigonometric, logarithmic, or exponential functions, we can only look at the dimensions of the three components in the equation:

[s] = L

[vt] = [v]. [t]2 = LT-1. T2 = LT

[at] = [a].[t] = LT-2 .T= LT-1

Because none of the three words have the same dimension as the others, this is as far from being dimensionally consistent as you can go. The technical name for such an equation is gibberish.

c. Because this equation contains a trigonometric function, we must first ensure that the sine function’s input is dimensionless:

[at2/s] = ([a].[t]2)/[s] = (LT-2 .T2)/L = L/L = 1

The debate has no dimensions. We must now examine the size of each of the two terms in the equation:

[v]= LT-1

[sin(at2/s)] = 1

Because the two words have distinct dimensions, it is not dimensionally consistent.

Conclusion

We now know that the dimension of a physical quantity is just an expression of the basic quantities from which it is generated, after understanding the dimensional consistency meaning, few-dimensional consistency examples, and the dimensional consistency equation. Dimensional consistency is required for all equations representing physical laws or principles. This fact may be used to help recall physical rules, examine whether stated correlations between physical quantities are conceivable, and may even generate new physical laws.