KAHIBARO
Table of Contents
Introduction
In physics we describe the world using measurable quantities such as length, mass, and time. These are expressed in base units, like meter, kilogram, and second in the International System of Units (SI). Many physical quantities, however, cannot be described by a single base unit. They are combinations of base units, built by multiplying or dividing them. These are called derived units.
This chapter explains what derived units are, how they are built from base units, and why standard names and symbols are given to some of them. The focus here is on the idea of constructing units, not on the detailed physical meaning of each quantity, which will be studied in later parts of the course.
What Are Derived Units?
A derived unit is any unit that can be written as a product or quotient of powers of base units. The base SI units are those for length, mass, time, electric current, temperature, amount of substance, and luminous intensity, written respectively as meter $(\text{m})$, kilogram $(\text{kg})$, second $(\text{s})$, ampere $(\text{A})$, kelvin $(\text{K})$, mole $(\text{mol})$, and candela $(\text{cd})$.
When you define a new physical quantity using a formula that involves base quantities, the unit of that new quantity is determined by combining the units on the right-hand side of the formula. This combination is the derived unit.
A derived unit is any unit that can be expressed as a product of powers of SI base units, for example
$$
\text{m}^2,\quad \text{m}\,\text{s}^{-1},\quad \text{kg}\,\text{m}\,\text{s}^{-2}.
$$
For example, if a quantity is defined as distance divided by time, its unit is meter divided by second, written $\text{m}/\text{s}$ or $\text{m s}^{-1}$. Both notations are standard: $\text{m/s}$ and $\text{m s}^{-1}$ represent the same unit.
Combining Base Units: Products and Powers
To construct derived units, you use multiplication, division, and powers of base units. The rules follow the usual laws of exponents from algebra.
If a quantity $Q$ is defined by some formula involving other quantities, say
$$
Q = A \cdot B,
$$
then the unit of $Q$ is the product of the units of $A$ and $B$. If
$$
Q = \frac{A}{B},
$$
then the unit of $Q$ is the unit of $A$ divided by the unit of $B$.
When variables and their units are raised to powers, the same power applies to the unit. So if a length $L$ is in meters and the formula involves $L^2$, then the unit becomes $\text{m}^2$. Similarly, $L^3$ is in $\text{m}^3$.
The following table shows the algebraic rules for unit manipulation.
| Operation on quantity | Example with variables | Example with units |
|---|---|---|
| Product | $C = A B$ | $\text{unit}_C = \text{unit}_A \cdot \text{unit}_B$ |
| Quotient | $C = A / B$ | $\text{unit}_C = \text{unit}_A / \text{unit}_B$ |
| Power | $C = A^n$ | $\text{unit}_C = (\text{unit}_A)^n$ |
These rules allow you to build any derived unit once you know how a physical quantity is defined in terms of base quantities.
Derived Units Without Special Names
Many derived units are left written directly in terms of base units. For example, the unit of volume is often simply written as $\text{m}^3$.
Some important examples:
- Area: If area $A$ is length times length,
$$
A = L_1 \cdot L_2,
$$
and each length is in meters, then
$$
\text{unit of area} = \text{m} \times \text{m} = \text{m}^2.
$$
So the SI unit of area is the square meter, $\text{m}^2$. - Volume: If volume $V$ is length times width times height,
$$
V = L \cdot W \cdot H,
$$
with each dimension in meters, then
$$
\text{unit of volume} = \text{m} \times \text{m} \times \text{m} = \text{m}^3.
$$
So the SI unit of volume is the cubic meter, $\text{m}^3$. - Number density: If number density is defined as number of particles divided by volume, $n = N/V$, and $N$ is a pure number and $V$ is in $\text{m}^3$, then the unit becomes $\text{m}^{-3}$.
In all such cases you do not need a new name for the unit. You can simply use the expression in base units.
Named SI Derived Units
For some frequently used derived units, the SI system introduces special names and symbols. These names make formulas and numerical work shorter and more readable. The named units are still exactly equivalent to certain combinations of base units.
A named SI derived unit is a derived unit that has been given a special name and symbol, but it is still exactly equal to a product of powers of SI base units.
Below is a table of several important named SI derived units and their expressions in base units.
| Quantity | SI unit name | Symbol | In base units |
|---|---|---|---|
| Force | newton | N | $\text{N} = \text{kg m s}^{-2}$ |
| Pressure | pascal | Pa | $\text{Pa} = \text{N m}^{-2} = \text{kg m}^{-1}\text{s}^{-2}$ |
| Energy | joule | J | $\text{J} = \text{N m} = \text{kg m}^2\text{s}^{-2}$ |
| Power | watt | W | $\text{W} = \text{J s}^{-1} = \text{kg m}^2\text{s}^{-3}$ |
| Electric charge | coulomb | C | $\text{C} = \text{A s}$ |
| Electric potential difference | volt | V | $\text{V} = \text{W A}^{-1} = \text{kg m}^2\text{s}^{-3}\text{A}^{-1}$ |
| Electric resistance | ohm | $\Omega$ | $\Omega = \text{V A}^{-1} = \text{kg m}^2\text{s}^{-3}\text{A}^{-2}$ |
| Electric capacitance | farad | F | $\text{F} = \text{C V}^{-1} = \text{kg}^{-1}\text{m}^{-2}\text{s}^4\text{A}^2$ |
| Magnetic flux | weber | Wb | $\text{Wb} = \text{V s} = \text{kg m}^2\text{s}^{-2}\text{A}^{-1}$ |
| Magnetic flux density | tesla | T | $\text{T} = \text{Wb m}^{-2} = \text{kg s}^{-2}\text{A}^{-1}$ |
| Frequency | hertz | Hz | $\text{Hz} = \text{s}^{-1}$ |
| Luminous flux | lumen | lm | $\text{lm} = \text{cd sr}$ |
| Illuminance | lux | lx | $\text{lx} = \text{lm m}^{-2} = \text{cd sr m}^{-2}$ |
The last two examples involve the radian and steradian, which are special cases of derived units for angles and solid angles.
Example: Building the Newton from Base Units
To see how a named unit is built, consider force, denoted by $F$, which in classical mechanics is related to mass $m$ and acceleration $a$ through an equation that you will meet in detail later:
$$
F = m a.
$$
Acceleration is velocity change per time, and velocity itself is length per time. So acceleration has units
$$
\text{m s}^{-2}.
$$
Combining this with mass in kilograms,
$$
\text{unit of force} = \text{kg} \cdot \text{m s}^{-2} = \text{kg m s}^{-2}.
$$
This combined unit is given the special name newton.
The newton is defined as
$$
1\,\text{N} = 1\,\text{kg m s}^{-2}.
$$
When you see newtons in later chapters, you should always be able to remember that they are kilograms meters per second squared in terms of base units.
Example: Joule and Watt
Energy and power are also given special names. Consider energy $E$ that can be described as the product of force and distance:
$$
E = F \cdot d.
$$
The unit of $E$ is newton meter. Since $1\,\text{N} = 1\,\text{kg m s}^{-2}$, we get
$$
\text{unit of energy} = \text{N m} = \text{kg m s}^{-2} \cdot \text{m} = \text{kg m}^2\text{s}^{-2}.
$$
This unit is named joule.
The joule is defined as
$$
1\,\text{J} = 1\,\text{N m} = 1\,\text{kg m}^2\text{s}^{-2}.
$$
Power $P$ describes how fast energy is transferred or converted, so it is energy per unit time:
$$
P = \frac{E}{t}.
$$
The unit of $P$ is joule per second, which simplifies to
$$
\text{J s}^{-1} = \text{kg m}^2\text{s}^{-3}.
$$
This unit is called the watt.
The watt is defined as
$$
1\,\text{W} = 1\,\text{J s}^{-1} = 1\,\text{kg m}^2\text{s}^{-3}.
$$
Named units like joule and watt help keep expressions short, but you can always expand them back into base units if needed.
Angles as Derived Units: Radian and Steradian
Angles are dimensionless in the SI system, but their units are still treated as derived units. The plane angle is defined as the ratio of arc length to radius. If an arc of length $s$ is cut from a circle of radius $r$, the angle $\theta$ in radians is
$$
\theta = \frac{s}{r}.
$$
Both $s$ and $r$ are lengths, so their units are meters. The ratio of two lengths has no physical dimension, but we still introduce the special name radian with symbol $\text{rad}$ to show clearly that we are talking about an angle.
Similarly, the steradian $\text{sr}$ is used for solid angles and is also considered dimensionless in base units. Although they are dimensionless, $\text{rad}$ and $\text{sr}$ behave like derived units built from the concept of length.
Using Derived Units Consistently
Being consistent with units is important in physics. Derived units provide a way to check the consistency of equations. If you combine quantities in a formula, the resulting units must match the unit of the quantity on the left-hand side. If they do not, this usually indicates an error in the formula or in the way it is used.
For instance, suppose a quantity $X$ is defined by
$$
X = \frac{E}{V},
$$
where $E$ is energy in joules and $V$ is volume in cubic meters. The unit of $X$ is
$$
\frac{\text{J}}{\text{m}^3} = \frac{\text{kg m}^2\text{s}^{-2}}{\text{m}^3} = \text{kg m}^{-1}\text{s}^{-2}.
$$
This is a derived unit that might correspond to an energy density. Even if it has no special name, the combination of base units completely defines it.
In more advanced work, you often rewrite named units like newton or joule back into base units to check dimensions or to simplify expressions. This is closely related to dimensional analysis, which will be addressed later in this section of the course.
Visualizing Relationships Between Units
It can be useful to imagine the relations between some common named derived units and the base units as a network. The following schematic drawing indicates how a few derived units connect back to the SI base units.
Each arrow here represents a relation in a defining equation. For example, $F = m a$ gives the connection from kilogram, meter, and second to newton, and $E = F d$ connects newton and meter to joule.
Summary
Derived units arise whenever a physical quantity is built from other quantities through multiplication, division, or powers. They are expressed as products of powers of SI base units, and some of them are given special names and symbols, such as newton, joule, and watt. Understanding how to construct and interpret derived units is essential for working with equations in physics and for checking that formulas are used consistently with respect to units.