Physical Quantities

These are the properties of solids, liquids and gases that can be measured/quantified. Examples include the volume of a gas, temperature of a cup of tea, height of a building, etc.

Fundamental quantities

These are the physical quantities from which all other physical quantities are derived. According to the International System of Units (SI), there are seven (7) fundamental quantities:

Derived quantities

These are combinations of the fundamental quantities. The following are examples of derived quantities. Note the fundamental quantities which are involved in each. $$ \begin{equation}\begin{aligned} area=length\times width\\ \end{aligned}\end{equation} $$ Width is a length measurement and is essentially Length disguised under another word. This would mean that height, breadth, distance, displacement, perimeter and circumference are also length measurements (they fall under the fundamental quantity Length). $$ \begin{equation}\begin{aligned} volume&=length\times width\times height\\ OR&\\ volume&=area\times height\\ \end{aligned}\end{equation} $$ Notice that area is a derived quantity. Thus volume can be seen as the product of three fundamental quantities (length, width and height) or as the product of a derived quantity (area) and a fundamental quantity (height).

Let’s look at some other examples. $$ \begin{equation}\begin{aligned} speed&=\frac{distance}{time}\\ velocity&=\frac{displacement}{time}\\ acceleration&=\frac{velocity}{time}\\ force&=mass\times acceleration\\ pressure&=\frac{force}{area}\\ \end{aligned}\end{equation} $$

Magnitude and unit

Every physical quantity can be expressed as the product of a magnitude (how many) and unit (of what): $$ \begin{equation}\begin{aligned} measurement&\rightarrow magnitude\times unit\\ e.g.\ 3\ kg&\rightarrow 3\times kg\\ \end{aligned}\end{equation} $$

The exception to this rule are dimensionless quantities. These are physical quantities which do not have any units. They are often the ratio of two physical quantities with the same units.

Some examples of dimensionless quantities are:

• Relative density
• Refractive index
• Mechanical advantage
• Velocity ratio

We can use their formulae to determine the units: $$ \begin{equation}\begin{aligned} &MA=\frac{load}{effort}\rightarrow \frac{N}{N}\rightarrow no\ units (dimensionless)\\ &relative\ density=\frac{density\ of\ substance}{density\ of\ water}\rightarrow \frac{kgm^{-3}}{kgm^{-3}}\rightarrow no\ units\\ &refractive\ index=\frac{sin(\hat i)}{sin(\hat r)}\rightarrow \frac{no\ units}{no\ units}\rightarrow no\ units\\ \end{aligned}\end{equation} $$

Finding the units of derived quantities

We can find the units of derived quantities if we substitute the units of the other quantities which make it up: $$ \begin{equation}\begin{aligned} speed&=\frac{distance}{time}\\ &\rightarrow \frac{m}{s}\ OR\ ms^{-1}\\ \end{aligned}\end{equation} $$

Mission details

Find the units of the derived quantities:

• velocity
• acceleration
• force
• pressure

Some physical quantities have units which are placeholders for the fundamental units e.g. in measuring force, we use the Newton ($N$) acknowledging that this unit is just a placeholder for $kgms^{-2}$.

Research Questions

By the end of this section, you will be competent enough to answer the following:

• What is a fundamental/base quantity?
• What are the seven ($7$) fundamental quantities?
• What are the SI base units?
• What is a derived quantity?
• How can you determine the units of derived quantities given the units of other physical quantities and a formula?
• How can we express the units of derived quantities in terms of the SI base units?
• What do we mean when we say that a measurement of a physical quantity can be written as the product of a magnitude and unit?
• What are multiples and submultiples?
• What is the purpose of using multiples and submultiples in science?
• What is meant by a homogeneous equation?
• Do we need to use SI units in order to show an equation is homogeneous?