Derived and Base SI Units
A derived unit is a type of measurement unit within the International System of Units (SI) that is formed by combining one or more of the seven fundamental base units. These units may either be dimensionless or can be obtained by multiplying the base units together.
Derived Unit Names and Symbols
Derived units are commonly expressed in lowercase letters. Although most of them are formed by combining base units, there are 22 units with distinct names. If a unit is named after a person, its symbol typically starts with an uppercase letter.
Examples of Derived Units
Some examples of derived units that are named after individuals are the watt, hertz, and coulomb, which are abbreviated as W, Hz, and C, respectively. On the other hand, other examples of derived units include cubic meters (m3), meters per second (m/s), and joule per kelvin (J/K).
How Many Derived Units Are There?
Among the derived units, there are 22 units that have unique names, which include the dimensionless units called the radian (rad) and steradian (sr). However, there are more than 100 other derived units that are expressed in relation to the base units.
D E R I V E D U N I T S
| Quantity | Name | Symbol | Base Units |
|---|---|---|---|
| Area | square meter | A | m² |
| Volume | cubic meter | V | m³ |
| Density | Kilogram per cubic meter | p | kg/m³ |
| Speed | Meters per second | v | m/s |
| Acceleration | Meters per second squared | a | m/s² |
| Pressure | pascal | Pa | kg mˉ¹sˉ² |
| Force | newton | N | kg msˉ² |
| Energy | joule | J | kg m²sˉ² |
| Frequency | hertz | Hz | sˉ¹ |
| Power | watt | W | kg m²sˉ³ |
| Voltage | volt | V | kg m²sˉ³Aˉ¹ |
| Charge | coulomb | C | A s |
Derived Unit List
Here are the 22 derived units with names:
| Name | Symbol | Quantity | SI Base Units |
| hertz | Hz | frequency | s-1 |
| radian | rad | angle | 1 |
| steradian | sr | solid angle | 1 |
| newton | N | force | kg⋅m⋅s−2 |
| pascal | Pa | pressure | kg⋅m−1⋅s−2 |
| joule | J | energy | kg⋅m2⋅s−2 |
| watt | W | power | kg⋅m2⋅s−3 |
| coulomb | C | electric charge | s⋅A |
| volt | V | voltage or potential difference | kg⋅m2⋅s−3⋅A−1 |
| farad | F | electrical capacitance | kg−1⋅m−2⋅s4⋅A2 |
| ohm | Ω | electrical resistance | kg⋅m2⋅s−3⋅A−2 |
| siemens | S | electrical conductance | kg−1⋅m−2⋅s3⋅A2 |
| weber | Wb | magnetic flux | kg⋅m2⋅s−2⋅A−1 |
| tesla | T | magnetic flux density | kg⋅s−2⋅A−1 |
| henry | H | electrical inductance | kg⋅m2⋅s−2⋅A−2 |
| degree Celsius | °C | temperature relative to 273.15 K | K |
| lumen | lm | luminous flux | cd |
| lux | lx | illuminance | cd⋅m−2 |
| becquerel | Bq | radioactive decays per unit time | s−1 |
| gray | Gy | absorbed dose of ionizing radiation | m2⋅s−2 |
| sievert | Sv | equivalent dose of ionizing radiation | s−1⋅mol |
| katal | kat | catalytic activity | s−1⋅mol |
Derived Units and Dimensional Analysis
There are numerous derived units that can be obtained through mathematical combinations of SI base units, but they do not have unique names. The following are some examples:
| Quantity | Symbol | Unit | Abbreviation | Derivation |
| area | A | square meter | m2 | length x width |
| volume | V | cubic meter | m3 | length x width x height |
| density | ρ | kilograms per cubic meter | kg/m3 | mass / volume |
| concentration (molarity) | c or M | moles per liter | mol/L | amount / volume |
| speed | v | meters per second | m/s | length / time |
| acceleration | a | meters per second per second | m/s2 | speed / time |
| angular velocity | rad/s | radians per second | rad/s | angle / time |
| current density | A/m2 | ampere per square meter | A/m2 | current / time |
| wave number | m-1 | reciprocal meter | m-1 | 1 / length |
| specific volume | ν | cubic meter per kilogram | m3/kg | volume / mass |
It’s important to note that several derived units do not have unique symbols or abbreviations.
Let’s consider the following example of converting cubic meters to cubic centimeters:
By using the conversion factor of 100 cm/1 m, we can obtain the equivalent of 100 cm³/1 m³. Simplifying further, this can be expressed as 1,000,000 cm³/1 m³. It’s important to avoid the common mistake of assuming that there are only 100 cm³ per 1 m³ simply because there are 100 cm in 1 m.
Who Can Make a Derived Unit?
Although it may seem like anyone could create a derived unit by using the base units, a unit can only be recognized as a valid measurement if it is included in The International System of Units (SI). The CGPM (General Conference on Weights and Measures) oversees the management of the SI (or metric system) and provides recommendations to the CIPM (International Committee for Weights and Measures). The BIPM (International Bureau of Weights and Measures) is responsible for periodically updating the list and definitions of units.
Non-SI Units
In addition to base and derived units, the metric system also includes several other units that are neither base units nor derived units. These units are included in the metric system either because they are multiples or fractions of SI units, or because they serve practical purposes. Some of the non-SI units that are allowed in the metric system are:
| Name | Symbol | Quantity |
| minute | min | time |
| hour | h | time |
| day | d | time |
| astronomical unit | au | length |
| degree | ° | plane angle |
| minute | ′ | plane angle |
| second | ″ | plane angle |
| hectare | ha | area |
| liter or litre | l | volume |
| tonne | t | mass |
| dalton | Da | mass |
| electronvolt | eV | energy |
| neper | Np | logarithmic ratio |
| bel, decibel | B, dB | logarithmic ratio |
The 9th SI brochure does not include some of the units that were permitted in the 8th brochure. Examples of these units are the bar (pressure), mmHg (pressure), ångström (length), and gauss (magnetic flux density).
- The term “Systeme International” or SI Units refers to a standardized system of measurement that is based on internationally recognized definitions.
- The SI Units consist of 7 base units, which serve as the foundation for all other derived units.
- Metre (m) – Length
- Kilogram (Kg) – Mass
- Second (s) – Time
- Kelvin (°K) – temperature
- Ampere (A) – Current
- Candela (Cd) – Luminous Intensity
- Mole (mol) – Amount of a substance
- Each of the seven base units in the SI system has a standard definition. For instance, the meter is defined as the distance traveled by light in a vacuum over a period of
seconds.
- Derived units are created by combining the base units of the SI system. Some examples of these derived units include:
- Velocity (ms-1)
- Acceleration (ms-2)
- Density (Kgm-2)
- In addition to the base units, some derived units have their own unique names. Here are a few examples:
- Coulomb (C) – (As) Charge
- Pascal (Pa) – (Kgm-1s-2) Pressue
Ohms (Ω) – (Kgm2s-3A-2) Resistence
Prefixes
- To facilitate working with numbers that are very large or very small, the SI system uses prefixes that involve multiplying the unit by a specific power of ten.
- Yotta (Y) – 1024
- Zetta (Z) – 1021
- Exa (E) – 1018
- Peta (P) – 1015
- Tera (T) – 1012
- Giga (G) – 109
- Mega (M) – 106
- Kilo (k) – 103
- Milli (m) – 10-3
- Micro (μ) – 10-6
- Nano (n) – 10-9
- Pico (p) – 10-12
- Femto (f) – 10-15
- Atto (a) – 10-18
- Zepto (z) – 10-21
- Yocto (y) – 10-24
- In addition, these prefixes are also used:
- Hecto (h) – 102
- Deca (da) – 101
- Deci (d) – 10--1
- Centi (c) – 10-2
- A unit expressed in scientific notation with a large or small exponent can be written more conveniently using SI prefixes. For instance, 0.005 A can be written as 5 mA, while 6900000000000000000000000 m can be expressed as 6.9 Ym.
The seven base SI units are the meter (m), kilogram (kg), second (s), ampere (A), Kelvin (K), mole (mol), and candela (cd). These units are used to define all other SI units.
A derived unit is a unit that is created by combining two or more base units. For example, the unit for speed is meters per second (m/s), which is a combination of the meter (length) and second (time) base units.
Scalar quantities have only a magnitude (size) and no direction, while vector quantities have both a magnitude and a direction. Examples of scalar quantities include temperature and mass, while examples of vector quantities include velocity and force.
To convert between units, you need to use conversion factors. For example, to convert 1 meter to centimeters, you would use the conversion factor 1 meter = 100 centimeters. You can then multiply or divide by the conversion factor to get the desired unit.
Precision refers to the degree of exactness or reproducibility of a measurement, while accuracy refers to how close a measurement is to the true value. A measurement can be precise but not accurate, accurate but not precise, or both precise and accurate.
Dimensional analysis is a method of checking the dimensional consistency of equations by ensuring that the units on both sides of the equation are the same. It is a powerful tool for verifying the correctness of equations and can help identify errors in calculations.
Some common derived units in physics include meters per second (m/s) for speed, meters per second squared (m/s^2) for acceleration, newtons (N) for force, joules (J) for energy, and watts (W) for power.
Using SI units ensures that measurements are standardized and can be easily compared and communicated across different countries and disciplines. It also simplifies calculations and reduces the risk of errors due to unit conversion.
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