1.1 Derived Units | Measurements | Physics 11

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, metres per second ( $m \cdot s^{- 1}$ ), is derived from the base unit for length (metre) 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, metres per second ( $m \cdot s^{- 1}$ ).

Non-coherent units are formed when prefixes are used with coherent units, introducing a numerical factor. For example, kilometres per hour ( $km \cdot h^{- 1}$ ) introduces a factor of $\frac{1000}{3600}$ .

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	$1/ s$	$s^{- 1}$
Force	Newton	N	mass $\times$ acceleration	$kg \cdot m \cdot s^{- 2}$
Pressure	Pascal	Pa	force/area	$kg \cdot m^{- 1} \cdot s^{- 2}$
Energy, Work	Joule	J	force $\times$ distance	$kg \cdot m^{2} \cdot s^{- 2}$
Power	Watt	W	energy/time	$kg \cdot m^{2} \cdot s^{- 3}$
Electric Charge	Coulomb	C	current $\times$ time	$s \cdot A$
Electric Potential	Volt	V	power/current	$kg \cdot m^{2} \cdot s^{- 3} \cdot A^{- 1}$
Electric Resistance	Ohm	$Ω$	voltage/current	$kg \cdot m^{2} \cdot s^{- 3} \cdot A^{- 2}$

Worked Example: Deriving the Unit of Force

Q: How is the unit of force (newton) derived from SI base units?

A: According to Newton's second law, $F = ma$ . The SI unit of mass is the kilogram (kg) and the unit of acceleration is metres per second squared ( $m \cdot s^{- 2}$ ). Therefore: $1 N = 1 kg \cdot m \cdot s^{- 2}$

Worked Example: Deriving the Unit of Pressure

Q: How is the unit of pressure (pascal) expressed in SI base units?

A: Pressure = Force / Area. Force has units $kg \cdot m \cdot s^{- 2}$ and area has units $m^{2}$ . Therefore: $1 Pa = \frac{kg \cdot m \cdot s ^{- 2}}{m ^{2}} = kg \cdot m^{- 1} \cdot s^{- 2}$

Worked Example: Deriving the Unit of Energy

Q: Express the joule (J) in terms of SI base units.

A: Work = Force $\times$ Distance. Force = $kg \cdot m \cdot s^{- 2}$ ; Distance = m. Therefore: $1 J = kg \cdot m \cdot s^{- 2} \times m = kg \cdot m^{2} \cdot s^{- 2}$

Key Points

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 Dimensions→ of base quantities.

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	$1/ s$	$s^{- 1}$
Force	Newton	N	mass $\times$ acceleration	$kg \cdot m \cdot s^{- 2}$
Pressure	Pascal	Pa	force/area	$kg \cdot m^{- 1} \cdot s^{- 2}$
Energy, Work	Joule	J	force $\times$ distance	$kg \cdot m^{2} \cdot s^{- 2}$
Power	Watt	W	energy/time	$kg \cdot m^{2} \cdot s^{- 3}$
Electric Charge	Coulomb	C	current $\times$ time	$s \cdot A$
Electric Potential	Volt	V	power/current	$kg \cdot m^{2} \cdot s^{- 3} \cdot A^{- 1}$
Electric Resistance	Ohm	$Ω$	voltage/current	$kg \cdot m^{2} \cdot s^{- 3} \cdot A^{- 2}$

Key Points

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 Dimensions→ of base quantities.