Base Quantity

A base quantity (or basic quantity) is chosen and arbitrarily defined, rather than being derived from a combination of other physical quantities.

The 7 base quantities are:

Physical quantity	Base SI unit
Mass (m)	Kilogram (Kg)
Length ($l$)	Metre (m)
Time (t)	Second (s)
Current ($\text{I}$)	Ampere (A)
Temperature (T)	Kelvin (K)
Amount of sub. (n)	Molar (mol)
Luminous Intensity (L)	Candela (cd)

SI units are used as standardised units in all measurements in the world. SI is the short form for “International System of Units”.

Derived Quantity

A derived quantity is defined based on a combination of base quantities and has a derived unit that is the exponent, product or quotient of these base units.

Note:

• Units, such as the joule, newton, volt and ohm, are SI units, but they are not base SI units.
• Dimensional analysis: The main idea of “deriving” a derived unit is to treat units like algebraic terms, and manipulate them accordingly to get the right derived unit for the quantity. (See example 3 for a walkthrough)

Examples on Derived Quantities

Derived Quantity	Unit Name	Unit Symbol	Base Units
Force	Newton	$\text{N}$	$\text{kg m s}^{-2}$
Energy	Joule	$\text{J}$	$\text{kg m}^{2}\text{ s}^{-2}$
Power	Watt	$\text{W}$	$\text{kg m}^{2}\text{ s}^{-3}$
Frequency	Hertz	$\text{Hz}$	$\text{s}^{-1}$
Charge	Coulomb	$\text{C}$	$\text{A s}$
Voltage	Volt	$\text{V}$	$\text{kg m}^{2}\text{s}^{-3}\text{ A}^{-1}$
Resistance	Ohm	$\Omega$	$\text{kg m}^{2}\text{s}^{-3}\text{ A}^{-2}$

Practice on Derived Quantity

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Example 1

An example of derived quantity is energy which has a derived unit of Joules which is $\text{kg} \, \text{m}^{2} \, \text{s}^{-2}$ OR $\text{kg} \, \text{m}^{2} / \text{s}^{2}$in base SI units.

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Example 2

Another example of derived quantity is density which has a derived quantity of $\text{kg} \, \text{m}^{-3}$ or $\text{kg}/\text{m}^{3}$ in base SI units.

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If you are unclear how to express derived quantity in terms of the base SI units, please study the example below:

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Example 3

The equation for density is:

$$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$$

Now, you know that the unit for mass is $\text{kg}$, while the unit for volume is $\text{m}^{3}$.

Hence,

$$\begin{aligned} \text{Unit for density} &= \frac{\text{kg}}{\text{m}^{3}} \\ &= \text{kg} / \text{m}^{3} \\ &= \text{kg} \, \text{m}^{-3} \end{aligned}$$

Homogeneous Equation

When each of the terms in a physical equation has the same base units, the equation is said to be homogeneous. This means that the units on both sides of the equation must be the same.

An equation which is not homogeneous must be wrong. However, when a physical equation is homogeneous, it does not necessarily imply that the equation is correct.

Example of A Homogeneous Equation

$$\begin{aligned} s &= ut + \frac{1}{2}at^{2} \\ \text{Units: m} &= \, \text{m s}^{-1} \times \text{s} \,\,\, \text{m s}^{-2} \times \text{s}^{2} \\ \end{aligned}$$

Notice that the unit for all the terms within the equation above is “m”! The equation is a homogeneous equation.