Dimensional Formula

Beyond Units: What is a Quantity Made Of?

While units can change (e.g., meters, feet, miles), the fundamental nature of a quantity does not. This fundamental nature is its dimension. A dimension tells us what 'ingredients' a quantity is made of, using the base quantities as a recipe.

For example, any measurement of length, whether it's in meters or light-years, has the dimension of Length. We represent the fundamental dimensions with symbols, typically enclosed in square brackets:

• [Mass] = M
• [Length] = L
• [Time] = T
• [Electric Current] = A
• [Temperature] = K

(In mechanics, we primarily use M, L, and T.)

Dimensional Formula & Equation

The dimensional formula is the expression that shows the complete 'recipe' for a quantity. It is written as the base dimensions raised to certain powers. For example, the dimensional formula for Force is $[M^1 L^1 T^{-2}]$.

A dimensional equation is simply an equation formed by setting a quantity equal to its dimensional formula. For example: $[F] = [M L T^{-2}]$.

How to Find the Dimensional Formula

1. Start with the definition or a formula for the quantity (e.g., Speed = Distance / Time).
2. Replace each quantity in the formula with its dimension (e.g., [Speed] = [L] / [T]).
3. Simplify the expression using algebra. (e.g., [Speed] = $[L T^{-1}]$).

Dimensionless Quantities

Some quantities are just pure numbers, representing a ratio of two similar quantities. These have no units and no dimensions. Their dimensional formula is $[M^0 L^0 T^0]$. Examples include strain (change in length / original length) and refractive index.

Reference Table: Dimensional Formulas of Common Quantities

The table below collects the dimensional formulas of quantities you will meet across mechanics, oscillations, and heat, so you have a single place to check or derive them from. Work through a few derivations yourself (e.g. Pressure = Force/Area) to confirm the pattern before relying on the table.

A note on the table: a few of these entries are easy to get wrong by pattern-matching alone — always re-derive them from their defining formula if in doubt. For instance, Young's modulus is stress/strain = (force/area)/(dimensionless) = $[MLT^{-2}]/[L^2] = [ML^{-1}T^{-2}]$, not simply $[T]$. Likewise, angular momentum is $I\omega$, giving $[ML^2]\times[T^{-1}] = [ML^2T^{-1}]$ — moment of inertia itself carries no time dependence, only mass and length squared ($I = \sum mr^2$). And wave intensity is power per unit area, $[ML^2T^{-3}]/[L^2] = [MT^{-3}]$, not $[ML^{-1}T^{-3}]$.

The Dimensional 'Lie Detector'

The first and most common use of dimensional analysis is as a 'lie detector' for equations. Based on the Principle of Homogeneity, we can quickly test if a formula is dimensionally plausible. If it fails the test, it is guaranteed to be incorrect. If it passes, it is dimensionally consistent, which is a good sign, but not a complete proof of its correctness.

How to Check an Equation

We test the formula for the period of a pendulum: $T = 2\pi \sqrt{l/g}$, where T is the time period, l is the length, and g is the acceleration due to gravity.

Step 1: Isolate and Analyze the Left-Hand Side (LHS)

The LHS is just the time period, T.
[LHS] = [T]

Step 2: Isolate and Analyze the Right-Hand Side (RHS)

The RHS is $2\pi \sqrt{l/g}$.
First, ignore any dimensionless constants (like $2\pi$).
Now, substitute the dimensions for the remaining variables:
[RHS] = $\sqrt{[l]/[g]} = \sqrt{\frac{[L]}{[LT^{-2}]}}$

Step 3: Simplify the RHS and Compare

Simplify the expression for the RHS dimensions using algebra.
[RHS] = $\sqrt{\frac{[L]}{[L][T^{-2}]}} = \sqrt{\frac{1}{[T^{-2}]}} = \sqrt{[T^2]} = [T]$
Now, compare: [LHS] = [T] and [RHS] = [T].

Step 4: Conclude

Since [LHS] = [RHS], the equation is dimensionally consistent.

'You Can't Add Apples and Oranges'

The entire field of dimensional analysis is built on one simple, powerful idea: The Principle of Homogeneity. This principle states that for any physical equation to be valid, the dimensions of every term in the equation must be the same.

This is the mathematical equivalent of the saying, 'You can't add apples and oranges.' It is meaningless to add a length to a time, or subtract a mass from a velocity. A valid physical equation must balance not just in terms of numerical value, but also in its fundamental nature.

The Rules of the Game:

1. Equality: If $A = B$, then $[A]$ must be equal to $[B]$.
2. Addition/Subtraction: If $A = B + C$, then $[A]$ must equal $[B]$ and $[C]$.

Checking an Equation for Consistency

We can use this principle to check if a physics equation is plausible. We test the kinematic equation: $s = ut + \frac{1}{2}at^2$.

• Term 1: Displacement (s)
  The dimension is simply $[L]$.
• Term 2: Initial Velocity × Time (ut)
  The dimensions are $[u][t] = [LT^{-1}] \times [T] = [L T^{-1+1}] = [L]$.
• Term 3: ½ × Acceleration × Time² (½at²)
  (Note: Pure numbers like ½ are dimensionless)
  The dimensions are $[a][t^2] = [LT^{-2}] \times [T^2] = [L T^{-2+2}] = [L]$.

Since every term in the equation has the dimension of Length $[L]$, the equation is dimensionally consistent. This does not guarantee the equation is correct (for example, the factor of ½ could be wrong), but it is a necessary first check.