Dimensional Analysis

What is Dimensional Analysis?

Dimensional analysis (or dimensional check) involves breaking down a physical quantity into its fundamental quantities that compose it.

Every physical law represents a mathematical relationship between fundamental or derived physical quantities.

To perform dimensional analysis, replace each physical quantity with its corresponding dimension.

What is the Dimension of a Physical Quantity?

The dimension of a physical quantity expresses it in terms of fundamental quantities.

Fundamental Quantity	Quantity Symbol	Unit of Measurement	Unit Symbol
Length	l	Meter	m
Mass	m	Kilogram	kg
Time	t	Second	s
Electric Current	i	Ampere	A
Temperature	T	Kelvin	K
Amount of Substance	n	Mole	mol
Luminous Intensity	iv	Candela	cd

To indicate a dimension, the symbol for the quantity should be enclosed in square brackets [ ].

For example, the dimension of length is written as

$$ [l] $$

The Difference Between Dimension and Unit of Measurement. It's essential not to confuse these two concepts. The dimension refers to the physical nature of the quantity (e.g., time rather than length or mass), while the unit of measurement specifies the unit used to measure it (e.g., seconds, hours, days, etc.).

Length, mass, time, electric current, amount of substance, and luminous intensity are known as fundamental quantities because they have basic dimensions that serve as the building blocks for defining the dimensions of all other physical quantities.

Physical quantities expressed using fundamental quantities are called derived quantities.

For example, velocity is a derived quantity obtained by dividing a length (l) by time (t).

$$ [v] = \frac{[l]}{[t]} $$

Note. In dimensional analysis, dimensionless quantities—those with no dimension (e.g., pure numbers)—are not considered. A dimensionless quantity is purely a numeric value without a unit of measurement, and therefore it cannot be substituted with a physical quantity symbol.

What is the Purpose of Dimensional Analysis?

Dimensional analysis is a valuable tool for verifying the physical correctness of calculations.

Physical quantities with the same dimension can be simplified using algebraic operations such as addition, subtraction, multiplication, and division.

A calculation is physically correct if it has the same dimension as the physical law it corresponds to.

Note. Dimensional correctness does not necessarily mean that the calculation is mathematically accurate. There may still be mathematical errors. Dimensional analysis serves as a preliminary check to help avoid logical errors, making it a necessary but not sufficient condition for the correctness of a calculation.

How to Perform Dimensional Analysis

Follow these steps to perform dimensional analysis on a physical quantity:

1. Analyze the equation that defines the quantity.
2. Substitute each quantity in the equation with its dimension (e.g., t for time, l for lengths, l/t for velocity, etc.).
3. Enclose the symbols of the physical quantities in square brackets.
4. The final result is the dimensional equation.

A Practical Example

Let’s analyze the dimension of velocity.

Velocity (v) is a physical quantity derived by dividing the distance traveled in (m)eters by the time in (s)econds.

$$ v = \frac{m}{s} $$

Substitute each physical quantity in the equation with its dimension.

In this case, there are two fundamental quantities, making the process straightforward.

Replace meters with the symbol for length [l] and seconds with the symbol for time [t].

$$ [v] = \frac{[l]}{t} $$

or, alternatively,

$$ [v] = [l] \cdot [t]^{-1} $$

This provides the physical dimension of velocity: velocity is length divided by time.

Note. You can also use units of measurement instead of physical quantity symbols. Knowing that the unit of measurement for length is the meter [m], and for time is the second [s], the dimensional equation for velocity can also be written as: $$ [v] = \frac{[m]}{[s]} = [m] \cdot [s]^{-1} $$

Any calculation involving velocity must align with the dimension of the physical law of velocity.

For instance, if a car travels 100 meters in 5 seconds, the velocity of the car is:

$$ v = \frac{100 \ m}{5 \ s } = 20 \ m/s $$

To verify the logical correctness of this calculation, I perform dimensional analysis on both sides of the equation.

On the left side, I replace velocity (v) with the dimension of velocity [v]=[l]/[t]

$$ \frac{[l]}{[t]} = \frac{100 \ m}{5 \ s } = 20 \ m/s $$

Now, I carry out dimensional analysis on the calculations on the right side of the equation.

On the right side, I replace meters with [l] and seconds with [t].

$$ \frac{[l]}{[t]} = \frac{[l]}{[t]} $$

The matching dimensions confirm that the logic is sound.

Of course, this doesn’t eliminate the possibility of mathematical errors in the calculations.

If I had written 100m / 5s = 30 m/s instead of 20 m/s, the dimensional analysis would still be correct, but the result would be incorrect.

Note. To further clarify dimensional analysis, it helps to examine a logical error. If the calculation were $$ v = \frac{100 \ m}{5 \ s^2 } = 20 \ m/s^2 $$ The dimensional analysis would be $$ \frac{[l]}{[t]} = \frac{[l]}{[t^2]} $$ In this case, the dimensions do not match because I am comparing velocity [l]/[t] with acceleration [l]/[t²]. This discrepancy indicates that the wrong formula, the one for acceleration, was used. At this point, it is unnecessary to check the mathematical correctness of the calculation. There is a mismatch between the physical quantities, and the calculation must be redone from scratch, following the correct formula for velocity.

And so on.