SI Derived Units

The International System of Units (SI) is built upon seven base units — the metre, kilogram, second, ampere, kelvin, mole, and candela. Every other measurable physical quantity is described by a combination of these base units called a derived unit. For example, speed is measured in metres per second (m s⁻¹), which is simply the base unit of length divided by the base unit of time. This logical structure means that any equation in physics remains dimensionally consistent no matter where in the universe it is applied. SI derived units are formed by multiplying or dividing base units according to the mathematical relationship that defines the physical quantity. Some derived units are given special names and symbols to make them easier to use. The newton (N) for force, the joule (J) for energy, the watt (W) for power, the pascal (Pa) for pressure, the hertz (Hz) for frequency, the coulomb (C) for electric charge, and the volt (V) for electric potential are among the most important named derived units encountered in high school physics. Understanding derived units is essential for dimensional analysis — a powerful technique used to check whether equations are physically valid, convert between unit systems, and even derive new relationships from fundamental principles. When you see a complex formula, expressing every quantity in its base-unit form lets you verify that both sides of the equation have the same dimensions, providing a crucial check on algebraic work and experimental calculations.

Key takeaways

Start with a Defining Equation
Assign a Special Name When Convenient
Build Further Derived Units on Top
Express Any Unit in Base Form
Dimensionless Derived Units

Start with a Defining Equation

Every derived unit originates from a defining equation that relates the new quantity to quantities already understood. For force, Newton's second law $F = ma$ states that force equals mass multiplied by acceleration. Since mass is measured in kilograms (kg) and acceleration in metres per second squared (m s⁻²), force must be measured in kg m s⁻².

Assign a Special Name When Convenient

The combination kg m s⁻² is cumbersome to write repeatedly, so the SI system assigns it the special name 'newton' with symbol N. This is purely a notational shortcut: $1\,\text{N} = 1\,\text{kg m s}^{-2}$ . The physical meaning is unchanged — a newton is still just a kilogram-metre per second squared.

Build Further Derived Units on Top

Derived units can themselves be combined to form even higher-level derived units. Energy (the joule) is defined as $W = Fd$ , where force F is in newtons and distance d is in metres, giving $1\,\text{J} = 1\,\text{N m} = 1\,\text{kg m}^2\,\text{s}^{-2}$ . Power (the watt) is energy per unit time: $P = W/t$ , so $1\,\text{W} = 1\,\text{J s}^{-1} = 1\,\text{kg m}^2\,\text{s}^{-3}$ .

Express Any Unit in Base Form

Any SI derived unit, no matter how complex, can be reduced entirely to powers of the seven base units. The volt, defined through $V = W/Q$ (energy per unit charge), becomes $1\,\text{V} = 1\,\text{kg m}^2\,\text{s}^{-3}\,\text{A}^{-1}$ . This reduction is the foundation of dimensional analysis and ensures all physical equations are internally consistent.

Dimensionless Derived Units

Some derived quantities are ratios of identical units, making them dimensionless — they have no base-unit expression. The radian (rad) for plane angle is defined as arc length divided by radius (m/m = 1), and the steradian (sr) for solid angle is area divided by radius squared (m²/m² = 1). These are still officially recognised SI derived units, they simply carry no dimensional signature.

The Key Equation

$$[F] = \text{kg}\cdot\text{m}\cdot\text{s}^{-2} \equiv \text{N}

- **F** — Force (newton (N)): The physical quantity being described; force causes acceleration in a mass. - **\text{kg}** — Kilogram (kg): The SI base unit of mass. - **\text{m}** — Metre (m): The SI base unit of length. - **\text{s}^{-2}** — Per second squared (s⁻²): The inverse square of the SI base unit of time, arising from the definition of acceleration as $\Delta v / \Delta t$. ## Worked Example **Problem.** Express the SI unit of pressure — the pascal (Pa) — entirely in terms of SI base units, and verify your answer using the definition of pressure as force per unit area.

1\,\text{Pa} = 1\,\text{kg}\,\text{m}^{-1}\,\text{s}^{-2}

**Answer.** The pascal in base SI units is kg m⁻¹ s⁻².

Figure

Diagram

Origins · 1 minGiovanni Giorgi and how this was discovered

Giovanni Giorgi

1871–1950 · Italian

In 1901, Giovanni Giorgi proposed the first fully coherent metric system of electrical and mechanical units, which became the intellectual foundation for the modern SI. He showed that by choosing the metre, kilogram, and second as mechanical base units and adding a single electrical base unit (later the ampere), all other units — including those for electromagnetism — could be derived without awkward conversion factors. His system, refined over subsequent decades, was formally adopted as the SI in 1960. Giorgi was also a distinguished civil engineer who helped design the water supply system for Rome, demonstrating that rigorous measurement and dimensional thinking have practical consequences far beyond the laboratory.

SI Base Units — Derived units are constructed entirely from base units; you must understand the seven base units before derived units make sense.
Dimensional Analysis — Dimensional analysis uses the base-unit expressions of derived units to check equations and convert between unit systems.
Unit Conversion and Prefixes — SI prefixes (kilo-, mega-, milli-, etc.) are applied to derived units just as to base units, so understanding conversions is the natural next step.
Newton's Second Law — The newton — the most commonly used named derived unit — is defined directly by $F = ma$ , making this law the prototype for understanding how derived units arise.
Work, Energy and Power — The joule and the watt are derived units built on top of the newton, illustrating how the hierarchy of derived units reflects the hierarchy of physical concepts.