Dimensional Analysis

2.3. Dimensional Analysis#

Learning Objectives#

By the end of this section you will be able to:

  • Analyze a mathematical relationship for dimensional consistency

  • Determine dimensions necessary to make a relationship dimensionally consistent

Dimensional Analysis#

Dimensional analysis is a method of checking an equation or a solution to a problem by analyzing the dimensions in which it is expressed.

Dimensional analysis is useful for establishing the form, but not the numerical coefficients of an empirical relationship.

Dimensional Consistency#

Any mathematical equation or mathematical relation denotes not only a numerical equivalency but also a dimensional equivalency. Thus, any equation must be dimensionally consistent.

Meaning – if the two sides of an equation do not have the same dimensions the equation is wrong or inconsistent.

If two sides of a mathematical equation or relationship do have the same dimensions, the equation may still be wrong, but the error is likely to be in the arithmetic rather than the method of solution.

Rules for Dimensional Consistency#

  • For terms to be combined (added or subtracted), the terms must be of the same dimension

  • You cannot add or subtract quantities unless they have the same dimension.

  • For example, you cannot add apples to oranges

  • The argument of a function must be dimensionless. Functions such as trigonometric functions (such as sine and cosine), logarithms, or exponential functions that appear in the equation must be dimensionless. These functions require pure numbers as inputs and give pure numbers as outputs.

    • For example what is cos(3 feet)? Toes?

  • Constants are not considered, treat them as dimensionless

    • They have no dimensions and are used for evaluating the arithmetic

  • The rules of algebra apply to the combination or rearrangement of terms

Example of Dimensional Consistency#

Consider the equation below for the vertical position of a particle undergoing constant acceleration:

\[ y = y_0 + v_{0}t - \frac{1}{2}gt^2 \]

\(y\) = height of particle at time \(t\) [unit of length \(L\)]

\(y_0\) = initial height of particle (at time \(t\) = 0) [unit of length \(L\)]

\(v_0\) = initial velocity of particle (at time \(t\) = 0) [unit of velocity \(L/t\)]

\(t\) = time [unit of time \(t\)]

\(g\) = gravitational acceleration [unit of acceleration \(L/t^2\)]

We can rewrite the equation substituting the dimensions for the variables. Note the dimensions are enclosed in brackets to indicate that they represent dimensions and not variables or values.

\[ [L] \approx [L] + [\frac{L}{t}][t] + [\frac{L}{t^2}][t^2] \]

If we simplify the right-hand side of the equation, by canceling out the \([t]\) and \([t^2]\) dimensions, we end up with the equation below:

\[ [L] \approx [L] + [L] + [L] \]

Since units of length added together (or subtracted) produce units of length, the equation is dimensionally consistent.

\[ [L] \approx [L] \]
Worked Example

GIVEN:

\[ y = A \sin{(\omega t + \phi)} + m \]

where \(y\) is a unit of length, \(A\) is a unit of length, \(m\) is a unit of length and \(t\) is a unit of time.

FIND:

What dimensions must \(\omega\) and \(\phi\) have to make this relationship dimensionally consistent?

SOLUTION:

Rewrite the equation in known base dimensions. Base dimensions are enclosed with square brackets [ ].

\[ [L] \approx [L] \sin{(\omega [t] + \phi)} + [L] \]

Note that terms are of the same dimension, but need the argument of the function to be dimensionless thus \(\omega t + \phi\) must be dimensionless.

\[ \omega = \left[\frac{L}{Lt}\right] \]
\[ \phi = \left[\frac{L}{L}\right] \]

Section Summary#

This section was concerned with dimensional analysis and dimensional consistency.

To evaluate the dimensional consistency of a mathematical equation:

  • Remove all numerical constants and coefficients

  • Convert variables and values that have units into their base dimensions, typically square brackets [ ] are used

  • Adding or subtracting dimensions produces the same dimension

  • Algebra rules apply for multiplication and division

  • Function arguments must be dimensionless

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