Dimensions of Physical Quantities

So far you have worked with units, significant figures, and measurement uncertainties. But there is a deeper way to look at any physical quantity: instead of asking “what unit is it measured in?”, you can ask “what kind of quantity is it made of?” That question leads you to the idea of dimensions.

What Are Dimensions?

Every derived physical quantity, no matter how complicated, is built from a combination of a few fundamental quantities. These fundamental quantities are the same seven base quantities of the SI system you already know. When you express a derived quantity in terms of these base quantities, the powers (exponents) that appear on each base quantity are called the dimensions of that derived quantity.

There is a standard notation to keep dimensions separate from units or numerical values. You write the base quantity symbol inside square brackets:

• Length: $[L]$
• Mass: $[M]$
• Time: $[T]$
• Electric current: $[A]$
• Thermodynamic temperature: $[K]$
• Luminous intensity: $[cd]$
• Amount of substance: $[mol]$

The square brackets serve a specific purpose: they signal that you are talking about the dimension (the type) of the quantity, not its numerical value or its unit.

The Three Dimensions of Mechanics

Although there are seven base dimensions in total, a large portion of physics, specifically mechanics (the study of motion, forces, and energy), uses only three of them:

• $[M]$ for mass
• $[L]$ for length
• $[T]$ for time

Every mechanical quantity, from velocity to pressure to energy, can be broken down into some combination of these three. The remaining four base dimensions ($[A]$, $[K]$, $[cd]$, $[mol]$) come into play only when you move into areas like electromagnetism, thermodynamics, optics, or chemistry.

Building Dimensions from Scratch

The best way to understand dimensions is to build them yourself for a few familiar quantities.

Volume

Volume is the space occupied by an object. You calculate it by multiplying three lengths: length, breadth, and height.

$\text{Volume} = \text{length} \times \text{breadth} \times \text{height}$

Each of these is a length, so in terms of dimensions:

$[L] \times [L] \times [L] = [L^3]$

Volume has three dimensions in length. It does not involve mass or time at all, so both of those carry a zero exponent:

$\text{Dimensions of volume} = [M^0 \, L^3 \, T^0]$

Writing $[M^0]$ and $[T^0]$ simply means “this quantity has nothing to do with mass or time.” Any base quantity raised to the power zero equals 1, contributing nothing to the expression.

Force

Force is a more interesting case because it involves all three mechanical base quantities. Start from the defining relationship:

$\text{Force} = \text{mass} \times \text{acceleration}$

Acceleration itself is a derived quantity. It tells you how quickly velocity changes, and velocity is length per time, so:

$\text{Acceleration} = \frac{\text{length}}{\text{time}^2}$

In dimensional terms:

$[\text{Acceleration}] = \frac{[L]}{[T]^2} = [L \, T^{-2}]$

Now multiply by mass:

$[\text{Force}] = [M] \times [L \, T^{-2}] = [M \, L \, T^{-2}]$

This tells you that force has one dimension in mass, one dimension in length, and minus two dimensions in time. The negative exponent on time reflects the “per second squared” in acceleration. All other base dimensions (electric current, temperature, luminous intensity, amount of substance) are zero for force.

Dimensions Describe the Nature, Not the Number

Here is a crucial point that separates dimensions from units. When you write the dimensional formula of a quantity, you strip away all numerical information. No coefficients, no multiplying constants, no specific values appear. Dimensions capture purely the qualitative type of the quantity.

This has a powerful consequence. Consider these five quantities:

• Initial velocity
• Final velocity
• Average velocity
• Change in velocity
• Speed

They describe different physical situations: one might be the speed of a car at the start of a journey, another might be how much the speed changed. But as far as their nature goes, every single one of them is a length divided by a time. Their dimensional formula is identical:

$[L \, T^{-1}]$

It does not matter whether you call it “speed” or “change in velocity” or “average velocity.” Dimensionally, they are all the same kind of quantity. This idea will become extremely useful when you learn to check equations and derive relationships using dimensions in the topics that follow.