Derived Units: The Building Blocks of Measurement
From Base to Derived: The Logic of Combination
Imagine you only have a few basic Lego blocks. You can build simple walls, but to create a car or a spaceship, you need to combine those basic blocks in specific ways. The International System of Units, or SI, works in a similar way. It starts with seven base units that measure fundamental quantities. These are the building blocks for all other measurements.
A derived unit is then defined as a unit of measurement for a derived quantity. A derived quantity is one that is defined in terms of the seven base quantities via a system of equations. In simple terms, it's a measurement unit we get by multiplying, dividing, or in other ways combining the base units.
| Base Quantity | SI Base Unit Name | Unit Symbol |
|---|---|---|
| Length | meter | m |
| Mass | kilogram | kg |
| Time | second | s |
| Electric Current | ampere | A |
| Thermodynamic Temperature | kelvin | K |
| Amount of Substance | mole | mol |
| Luminous Intensity | candela | cd |
A Catalog of Common Derived Units
Let's look at some of the most common derived units you will encounter in science class and everyday life. Each one has a special name, but it can always be broken down into its base unit components.
| Derived Quantity | SI Derived Unit Name | Unit Symbol | Expression in SI Base Units |
|---|---|---|---|
| Area | square meter | m$^2$ | m$^2$ |
| Volume | cubic meter | m$^3$ | m$^3$ |
| Speed, Velocity | meter per second | m/s | m $\cdot$ s$^{-1}$ |
| Acceleration | meter per second squared | m/s$^2$ | m $\cdot$ s$^{-2}$ |
| Force | newton | N | kg $\cdot$ m $\cdot$ s$^{-2}$ |
| Pressure, Stress | pascal | Pa | kg $\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$ |
| Energy, Work, Heat | joule | J | kg $\cdot$ m$^2$ $\cdot$ s$^{-2}$ |
| Power, Radiant Flux | watt | W | kg $\cdot$ m$^2$ $\cdot$ s$^{-3}$ |
| Electric Charge | coulomb | C | s $\cdot$ A |
| Electric Potential | volt | V | kg $\cdot$ m$^2$ $\cdot$ s$^{-3}$ $\cdot$ A$^{-1}$ |
Force is defined by Isaac Newton's second law of motion: $F = m \cdot a$, where $F$ is force, $m$ is mass, and $a$ is acceleration. Mass is measured in kilograms (kg) and acceleration in meters per second squared (m/s$^2$). Therefore, the unit of force is kg $\cdot$ (m/s$^2$), which is simplified to kg $\cdot$ m $\cdot$ s$^{-2}$ and given the special name Newton.
The Power of Dimensional Analysis
Dimensional analysis is a powerful problem-solving technique that uses the units (or "dimensions") of quantities to check the validity of equations and to convert from one unit to another. The core idea is simple: you can only add, subtract, or compare quantities that have the same units. Furthermore, the units on both sides of an equation must be identical.
Let's say you can't remember the formula for speed. You know it involves distance and time. Using dimensional analysis, you can figure it out. Speed must have units of length divided by time (e.g., m/s). If you mistakenly thought speed was distance multiplied by time, the units would be m $\cdot$ s, which is not a unit for speed. This quick check would tell you your formula is wrong.
Example: Converting km/h to m/s
You want to convert a speed of 72 km/h to m/s. We use conversion factors as fractions that equal 1.
$72 \frac{km}{h} \times \frac{1000 m}{1 km} \times \frac{1 h}{3600 s}$
Notice how the units "km" and "h" cancel out, leaving only "m" and "s".
$72 \times \frac{1000}{3600} \frac{m}{s} = 20 \frac{m}{s}$
So, 72 km/h is equal to 20 m/s.
Derived Units in Action: From Classrooms to Cities
Derived units are not just abstract concepts; they are used everywhere. Understanding them helps us make sense of the world.
In Sports: A baseball pitcher throws a fastball. The speed of the ball is measured in meters per second (m/s) or miles per hour (mph), a derived unit for velocity. The force with which the batter hits the ball can be described in newtons (N).
In Weather Reports: The atmospheric pressure that determines if it will be sunny or stormy is reported in hectopascals (hPa), which is a multiple of the pascal (Pa).
In Your Home: The electricity you use is billed in kilowatt-hours (kWh). A kilowatt (kW) is 1000 watts, a unit of power. A kilowatt-hour is the amount of energy used when a 1000-watt appliance runs for one hour. So, 1 kWh = 1 kW $\times$ 1 h = 1000 W $\times$ 3600 s = 3,600,000 J. This shows how a common utility unit is directly related to the joule.
In Medicine: Blood pressure is measured in millimeters of mercury (mmHg), which is a unit of pressure related to the pascal. Doctors use this derived unit to assess cardiovascular health.
Common Mistakes and Important Questions
Q: Is liter a base unit or a derived unit?
A: A liter is a derived unit, even though it's common. It is defined as a special name for a cubic decimeter. 1 L = 1 dm$^3$ = 0.001 m$^3$. Since it comes from the meter (length) cubed, it is a derived unit for volume.
Q: Why do some derived units have special names (like Newton) and others don't (like m/s)?
A: It's a matter of convenience and honoring scientists. Units that are used very frequently, like force, pressure, and energy, are given simple names to make communication easier. It's simpler to say "10 newtons" than "10 kilogram-meters per second squared." For less common or simpler combinations, we just use the base unit names.
Q: Can a derived unit be made from other derived units?
A: Yes, but it will always simplify to base units. For example, pressure in pascals (Pa) is defined as force (N) per unit area (m$^2$). So, Pa = N / m$^2$. Since N = kg $\cdot$ m $\cdot$ s$^{-2}$, then Pa = (kg $\cdot$ m $\cdot$ s$^{-2}$) / m$^2$ = kg $\cdot$ m$^{-1}$ $\cdot$ s$^{-2}$. The final expression is always in terms of the seven base units.
Footnote
[1] SI system: Abbreviation for "Systeme International d'Unites" (International System of Units). It is the modern form of the metric system and the most widely used system of measurement in the world.