How are the base unit and derived units related?

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Do you ever wonder how tall Mount Everest is? Or how far is San Francisco from New Delhi? Well, Mount Everest is 8,849 meters tall, and San Francisco is 12,349 kilometers away from New Delhi. But to derive a precise answer to such a question, we need to find a way to measure physical quantities.

But before understanding the measurement of physical quantities, let us define physical quantities, fundamental quantities and derived quantities. 

Physical Quantities

‘Quantities which can be measured either directly or indirectly’ are defined as physical quantities. We can express laws of physics in terms of physical quantities. Some examples of physical quantities includes length, mass, speed, force, current, temperature etc.

Further, physical quantities are divided into fundamental quantities and derived quantities.

Fundamental Quantities

The fundamental quantities are defined as physical quantities that don’t depend upon other quantities for measurement. The fundamental quantities are length, mass, time, current, temperature, luminous intensity and amount of substance.

Derived quantities

The derived quantities are physical quantities that depend upon fundamental quantities for measurement. E.g. Velocity = distance/time, where distance(length) and time are fundamental quantities.

How to measure physical quantities?

The measurement of a physical quantity is the process of comparing this measured quantity with a standard amount of physical quantity of the same kind called its unit. In ancient times humans measured physical quantities in various units such as Cubit, Fathom and Hand-Span.

Presently, the following four systems are in use:-

  1. Foot-Pound-Second system (FPS)
  2. Centimeter-Gram-Second system (CGS)
  3. Meter-Kilogram-Second (MKS)
  4. International system of units (SI)

SI System (Système International)

The SI system of measurement was published in 1960 based on the MKS system. It was devised to be used in technical & scientific research to avoid confusion. The SI system is based on seven basic units which can be used to generate an unlimited number of additional units called derived units.

What Base units?

Units that measure fundamental quantities and are mutually independent are known as Base units. These base or fundamental units can neither be derived from one another, nor they can be further resolved into more simpler units.

There are seven base units as mentioned in the table given below.

Table 1. Examples of SI base units

Base Quantity Base UnitSymbol
Lengthmeterm
Masskilogramkg
Timeseconds
Electric CurrentampereA
Thermodynamic TemperaturekelvinK
Amount of Substancemolemol
LuminositycandelaCd

 Apart from these 7 base units we also have two supplementary units namely radial and steradian used for plane angle and solid angle respectively.

Know more about What are the units of measurement in the metric system?

What are Derived Units?

Units that measure derived quantities and are dependent on the base units for their measurement are known as derived units. So, these are the physical units which can be expressed in terms of fundamental or base units.

Table 2. Examples of SI derived units

Derived QuantityBasic Unit NameSymbol
AreaSquare meter $[m^2]$
VolumeCubic meter $[m^3]$
SpeedMeter per second $[m^1s^{-1}]$
AccelerationMeter per second squared $[m^1s^{-2}]$
Wave NumberReciprocal of meter $[m^{-1}]$
Mass DensityKilogram per cubic meter $[Kg^1m^{-3}]$
Specific VolumeCubic meter per kilogram$[m^3Kg^{-1}]$
ForceKilogram meter per second squared$[Kg^1m^1s^{-1}]$
PressureKilogram per meter second squared$[Kg^1m^{-1}s^{-2}]$
EnergyKilogram meter squared per second squared$[Kg^1m^2s^{-2}]$
PowerKilogram meter squared per second cube$[Kg^1m^2s^{-3}]$

How are base unit & derived unit related?

From our previous discussion we know that every derived unit can be expressed in terms of its base units.

For example, one newton is defined as the force that is required to accelerate a mass of 1kg by 1m.s-2 in the direction of applied force

 Therefore, $1N = 1 kg \times 1 m / s^2$

$1N = m^1Kg^1s^{-2}$

 Similarly, 1 pascal is defined as the 1 Newton force applied on an area of 1 meter square.

Therefore, $1Pa=1N/1m^2$

$1Pa=1Kg\times \frac{1m}{s^2} \times 1m^{-2}$

$1Pa=1Kg/(s^2\times m^1)$

$1Pa= [Kg^1 s^{-2}m^{-1}]$

Thus, in this way, every derived unit is related to the base unit.

Conclusion

A measurement method is needed in any system of units to define the units for these physical quantities. The algebraic combinations of the base quantities can then be used to represent all other physical quantities. All of these physical quantities is then referred to as a derived quantity, with each unit being referred to as a derived unit.

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