Dimensional Analysis

Dimensional Analysis is a technique used in physics, chemistry, engineering, and science in general to track the dimensions of all physical quantities while performing calculations. The purpose of this practice is to ensure our final value is accurate in that it is dimensionally consistent — that the units make sense.

The dimensions of a quantity within the process of dimensional analysis are the Seven Base Quantities of physical reality, which are the seven base quantities of the SI Unit Metric System. They are, in no particular order:

1. Mass — measured in the kilogram (kg)
2. Time — measured in seconds (s)
3. Length — measured in the meter (m)
4. Luminous Intensity — measured in the candela (cd)
5. Thermodynamic Temperature — measured in the Kelvin (K) units
6. Electrical Current — measured in amperes or amps (A) equivalent to coulombs per second (C/s)
7. Amount of Substance — measured in the mole (mol)

All physical quantities that describe reality are derived from these base quantities. We take advantage of this fact when performing dimensional analysis by taking note of the “dimensions” of a unit defined by the physical quantities used in the formula that defines each unit. We specifically identify the exact combination of these seven quantities within every quantity we are using.

A dimension symbol has been given to each of the base quantities in addition to their SI Unit symbols. So when performing dimensional analysis we use their dimensional symbol: Mass (M), Time (T), Length (L), Current (I), Amount (N), Luminous Intensity (J), and Temperature (Θ).

(Θ is the Greek letter capital Theta, in case you don’t yet know your Greek Alphabet yet).

Every physical quantity can be expressed as products, quotients, or powers of these base quantities and can therefore be signified with these 7 dimensional symbols for the purpose of dimensional analysis.

Basic Dimensional Analysis

Now that we have the definition, let’s explore some examples. As you can see above, the base quantity of length (measured in meters within the SI Unit system) has a dimension of L, which in dimensional analysis is signified as L¹.

L¹ is the first dimension of the physical quantity we call length. The term “dimension” is used in this context in the same way that we refer to the three dimensions of space or the four dimensions of space-time (found in our discussion of the Cartesian Coordinate System).

Area is the next dimension of length because it is length in two directions — commonly referred to as width and length — literally in two spatial dimensions. Mathematically area is equal to length times width (A= l × w) which is written as meters squared m² in SI units, but in dimensional analysis as L². Volume then is the measurement of length in three spatial dimensions, length cubed, L³.

This is the last physical dimension.

More Complex Derived Quantities

When we are considering speed or velocity (distance per unit of time, such as Mi/h, m/s, or km/h) we are dealing with quantities of length and time. In dimensional analysis is written as either L/T or LT^-1 (since a negative power is the same as dividing by a positive power).

When we are calculating acceleration we are assessing the SI Unit of meters per second squared — meters per second per second — written mathematically as m/s². This is the rate of change in velocity per second. We therefore have length L divided by time squared T² giving us the dimensions of L/T² or LT^-2.

If we are measuring the density of a substance (technically the “volumetric mass density”) we are computing with the dimensions of mass and volume. We know that mass is measured in kilograms (kg) symbolized in dimensional analysis with the symbol M. We also established that area is length squared, and volume is length cubed, so the dimensional units are L³.

The units of density are $kg/m^3$. Written out dimensionally the units for density are M/L³ or ML^-3.

Writing Dimensions in Dimensional Analysis

We can write the dimensions of any quantity with these symbols. Every quantity in existence will be some combination of the dimensions of mass (M), length (L), time (T), electrical current (I), temperature (Θ), amount of substance (N), and luminous intensity (J):

The powers a, b, c, d, e, f, and g represent unknown exponent values placeholders for potential values depending on what we may choose to measure. They are arbitrary values.

In terms of the symbols for dimensions every physical quantity will be composed of these seven base quantities to some power. Even if one of the physical quantities is not present in a unit, that dimension can still be expressed as the dimensional unit to the power of zero. (Anything to the power of zero equals one and therefore doesn’t effect the result.) For example, writing length to the power of zero L⁰ is a mathematical way of saying that in that physical quantity, length is not a factor, because L⁰ = 1.

If all of the exponents were set to zero, then our quantity would be a dimensionless quantity because it doesn’t exist in any of the seven base dimensions of physical reality. In other words, that quantity is a pure number.

In physics we will often use square brackets around the symbol for a physical quantity to signify that we are expressing its dimension. If we are referring to the radius, r, of a sphere we might write [r] = L. If we are referring to acceleration, a, then we might write [a] = LT². If we are writing density ρ then we might write [ρ] = ML^-3 = M/L³, and so on.

Relevance of Dimensional Analysis

The importance of assessing dimension in the mathematical equations of physics and science is to ensure that our expressions are dimensionally consistent. In essence this prevents algebra mistakes, ensuring the logic in our conclusion. If our dimensional analysis differs from the expected units in the end, then we probably made a mistake somewhere.

The basic rules are that:

• Every physical quantity will always have the same dimensions. Volume will always be L³, regardless of what units you use. Likewise, density will always be ML^-3 regardless of whether we use kilograms per meter cubed, micrograms per nanometre cubed, or pounds per cubic inch. All that this means is that these dimensions are even more fundamental conceptually than the units we apply to a quantity.
• Dimensions are conserved in an expression. No matter how we algebraically rearrange an equation, no matter what algebra we perform on it, the same dimensions must be present in the appropriate relationships before and after. Furthermore, when one quantity equals another, the dimensions must be the same on each side of the equality. [1]
• The arguments of mathematical functions (such as trigonometric, exponential, or logarithmic functions) are dimensionless. These functions require pure numbers as inputs and give pure numbers as outputs.

If these rules are violated, then an equation is not dimensionally consistent, and our mathematics are making a false statement. Dimensional analysis helps us to verify our expressions, check for mistakes, and even remember laws of physics with logical analysis using the fundamental quantities of nature. Dimensional analysis can even suggest the form that potential new laws of physics might take. [1]

Resources

1. University Physics Volume I. Openstax.