Dimensional Analysis in Physics

Dimensional analysis allows you to verify the correctness of formulas used in physics.

It ensures that each equation is dimensionally consistent.

In essence, dimensional analysis involves checking that both sides of an equation have the same physical dimensions.

If this principle isn’t met, it’s a clear sign that there’s an error in either the calculations or the problem setup.

Dimensional analysis acts as an initial check for dimensional consistency in formulas, making sure each term in the equation makes physical sense. While this method doesn’t confirm numerical accuracy, it’s an essential first step. If an equation isn’t dimensionally consistent, even correct calculations would lack value, as the expression would be physically meaningless. Once dimensional consistency is confirmed, you can then ensure the mathematical calculations are correct.

Dimensional analysis relies on two key principles:

Adding and subtracting like quantities only
You can only add or subtract quantities with the same physical dimensions. For example, adding distance to time doesn’t make sense because they represent different physical quantities.
Dimensional balance on both sides of an equation
Every physical equation must be dimensionally balanced, meaning the physical dimensions on the left side must match those on the right. This balance is critical to ensure the equation has physical meaning.

Example of Dimensional Analysis in Action

Consider the formula for average velocity:

$$ v = \frac{d}{t} $$

Where:

$ v $ is velocity, with dimensions $[L]/[T]$,
$ d $ is distance, with dimensions $[L]$,
$ t $ is time, with dimensions $[T]$.

To check dimensional correctness, substitute each quantity with its physical dimension:

$$ [v] = \frac{[L]}{[T]} = [L]/[T] $$

The result matches the physical dimension of velocity, confirming the equation is dimensionally correct.

Why Use Dimensional Analysis?

Dimensional analysis helps identify errors when solving problems.

For instance, if you end up with a velocity dimension of $[L]^2/[T]$ instead of $[L]/[T]$, you’ll know something went wrong.

At that point, you’ll need to go back, review your steps, and correct the calculations.

Imagine you’re calculating the speed of a car that travels $ 120 \, \text{km} $ in $ 2 \, \text{hours} $. The correct formula for average speed is $ v = \frac{d}{t} $, where $ v $ represents velocity, $ d $ is distance, and $ t $ is time. $$ v = \frac{d}{t} $$ Now, let’s say you mistakenly use an incorrect formula, such as: $$ v = \frac{d \cdot t}{2} $$ This formula is incorrect, but dimensional analysis can help you spot the error right away, even before calculating. Substitute each variable in the incorrect formula with its physical dimension. Here, velocity has dimensions $[L]/[T]$, distance $ d $ has dimensions $[L]$, and time $ t $ has dimensions $[T]$. $$ \frac{[L]}{[T]} = \frac{[L] \cdot [T]}{2} $$ Dimensional analysis quickly reveals the formula is wrong because the resulting dimension isn’t that of velocity. The two sides of the equation don’t match. This allows you to catch and correct the mistake before moving on to the calculations, saving you from wasted effort and physically meaningless results.

In short, dimensional analysis is a straightforward yet effective way to check the consistency of a physical equation.

It’s especially valuable when working with complex formulas or mixed units, helping you keep your problem-solving on track.