# Derived Units Derived units are units of measurement formed by combining the seven base units of the International System of Units (SI). These combinations, created through multiplication or division, allow for the measurement of a vast array of physical quantities. Unlike base units, derived units are dependent on the definitions of these base units. ### Definition Derived units are combinations of the seven fundamental SI base units. They are used to measure quantities that are not base quantities, such as velocity, force, and energy. ### Formation Derived units are created by mathematically manipulating base units through multiplication and division. For instance, the unit for speed, meters per second (m/s), is derived from the base unit for length (meter) and time (second). ### Coherent vs. Non-Coherent Units **Coherent derived units** are products of powers of base units without any numerical factor other than one. An example is the unit of velocity, meters per second. **Non-coherent units** are formed when prefixes are used with coherent units, introducing a numerical factor. For example, kilometers per hour. ## Examples of Derived Units Many derived units have been given special names and symbols. The table below details some common examples: | Derived Quantity | Special Name | Symbol | Formula | Expression in SI Base Units | | :--- | :--- | :--- | :--- | :--- | | Frequency | Hertz | Hz | $1/\text{s}$ | $\text{s}^{-1}$ | | Force | Newton | N | mass $\times$ acceleration | $\text{kg}\cdot\text{m}\cdot\text{s}^{-2}$ | | Pressure | Pascal | Pa | force/area | $\text{kg}\cdot\text{m}^{-1}\cdot\text{s}^{-2}$ | | Energy, Work | Joule | J | force $\times$ distance | $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-2}$ | | Power | Watt | W | energy/time | $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}$ | | Electric Charge | Coulomb | C | current $\times$ time | $\text{s}\cdot\text{A}$ | | Electric Potential | Volt | V | power/current | $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-1}$ | | Electric Resistance | Ohm | $\Omega$ | voltage/current | $\text{kg}\cdot\text{m}^2\cdot\text{s}^{-3}\cdot\text{A}^{-2}$ | ## Possible Questions and Answers **Q:** How are derived units created from fundamental units? **A:** Derived units are formed by expressing a physical quantity in terms of fundamental quantities through mathematical relationships. For example, according to Newton's second law, Force = Mass $\times$ Acceleration. The fundamental unit for mass is the kilogram (kg), and acceleration has units of meters per second squared (m/s$^2$). Therefore, the unit for force is a combination of these: kg$\cdot$m/s$^2$, which is named the newton (N). **Q:** What are examples of derived units named after individuals? **A:** Examples include the newton (N) for force, joule (J) for energy, watt (W) for power, hertz (Hz) for frequency, and coulomb (C) for electric charge. - Derived units are created by combining the seven SI base units through multiplication or division. - They are essential for measuring a wide range of physical quantities like force, energy, and pressure. - Many derived units have special names, such as the newton (N) for force and the joule (J) for energy. - Understanding derived units is fundamental to applying physical principles and ensuring consistency in scientific measurements. - Derived units can be expressed in terms of of base quantities.