Dimensions of Physical Quantities

The seven SI base dimensions are length [L], mass [M], time [T], electric current [I], thermodynamic temperature [Θ], amount of substance [N], and luminous intensity [J]. Square brackets around a symbol denote 'the dimensions of'. Purely numerical quantities (like angles, which are ratios of lengths) are dimensionless, written as [1] or M⁰L⁰T⁰.

Every derived quantity is defined by an equation connecting it to simpler quantities. Start from the most fundamental definition you know. For example, velocity is defined as displacement divided by time: $v = \Delta x / \Delta t$. Since displacement has dimensions of length [L] and time has dimensions [T], the dimensions of velocity are $[v] = \text{L}\,\text{T}^{-1}$. Always work step by step from the definition rather than trying to recall the formula from memory.

When quantities are multiplied, their dimensions multiply; when quantities are divided, their dimensions divide. Exponents behave algebraically: $[A^n] = [A]^n$. For example, Newton's second law gives force as $F = ma$, so $[F] = [m][a] = \text{M} \cdot \text{L}\,\text{T}^{-2} = \text{M}\,\text{L}\,\text{T}^{-2}$. This result is the dimension of the SI unit of force, the newton (N).

The Principle of Dimensional Homogeneity states that only quantities with identical dimensions may be added or subtracted. In the kinematic equation $s = ut + \tfrac{1}{2}at^2$, each term must have dimensions of length: $[ut] = (\text{L}\,\text{T}^{-1})(\text{T}) = \text{L}$ and $[at^2] = (\text{L}\,\text{T}^{-2})(\text{T}^2) = \text{L}$. Both check out. If any term had a different dimensional formula, the equation would be physically impossible.

Arguments of transcendental functions such as $\sin$, $\cos$, $\exp$, and $\ln$ must always be dimensionless. If you see $\sin(\omega t)$ in a wave equation, you can immediately conclude that angular frequency $\omega$ has dimensions $[\omega] = \text{T}^{-1}$ so that $\omega t$ is dimensionless. Ratios of like quantities — such as strain (extension/original length) or the coefficient of restitution — are also dimensionless.

With practice, you will memorise the dimensional formulas for common quantities: energy $[E] = \text{M}\,\text{L}^2\,\text{T}^{-2}$; pressure $[P] = \text{M}\,\text{L}^{-1}\,\text{T}^{-2}$; electric charge $[Q] = \text{I}\,\text{T}$; electric potential $[V] = \text{M}\,\text{L}^2\,\text{T}^{-3}\,\text{I}^{-1}$; magnetic flux density $[B] = \text{M}\,\text{T}^{-2}\,\text{I}^{-1}$. Knowing these allows much faster checking of electromagnetic and thermodynamic equations.

\left[2\pi\sqrt{\frac{\ell}{g}}\right] = \left(\frac{\text{L}}{\text{L}\,\text{T}^{-2}}\right)^{1/2} = (\text{T}^2)^{1/2} = \text{T} = [T_{\text{period}}] \checkmark

• Dimensional Homogeneity and Equation Checking — Once you can write dimensional formulas, the next step is using the principle that every term in a valid physical equation must share the same dimensions — the primary application of what you have learned here.
• Deriving Relationships by Dimensional Analysis — Dimensional formulas can be used to deduce how one quantity depends on others (up to a dimensionless constant) without solving the full physics — a powerful technique called the Buckingham Pi theorem.
• SI Base Units and Prefixes — Dimensions are abstract; SI base units are their concrete measurements. Understanding the distinction between a dimension (e.g., length) and its unit (e.g., metre) is essential for fluent use of dimensional analysis.
• Unit Conversion and Dimensional Consistency — Converting between unit systems (e.g., CGS to SI) relies directly on dimensional formulas: if you know the dimensions of a quantity, you know exactly how its numerical value scales when base units change.
• Newton's Second Law — Newton's second law F = ma is the most commonly used equation for deriving the dimensional formula of force and is a gateway to finding dimensions of energy, momentum, and pressure.