1.2 SI Units

SI Base Units

• There are 7 SI base quantities with SI base units.
• The SI base quantities are sometimes called dimensions.
• All other units can be derived using these base units.

Quantity	Unit	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Thermodynamic Temperature	Kelvin	K
Electric Current	Ampere	A
Amount of Substance	Mole	mol
Luminous Intensity	Candela	cd

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Expressing Derived Units as Base Units

• Derived units are obtained by multiplying or dividing SI base units.
• Equations such as $F=ma$ in the case of force can be used to express derived units into SI base units.

Example: Deriving SI base units for Force (Newtons)

• We start with a relevant equation which contains the physical quantity we are interested in.
• In this case, we will start with $F=ma$.
• We then add square brackets $[\dots ]$ to show that we are working with the dimensions. For example, $m$ represents mass, but $[m]$ represents the SI base quantities of mass, $kg$.
• Adding square brackets to $F=ma$ we get: $[F]=[m][a]$. This represents that the SI base quantities of force is equal to the product of the SI base quantities of mass and acceleration.
• We can now substitute the SI base quantities for mass and acceleration to get: $[F]=kg\times m{s}^{-2}$
• Therefore, Newton expressed in SI base units is $kgm{s}^{-2}$.

Workings:

F &= ma \ [F] &= [m][a] \ [F] &= kg \times ms^{-2} \ [F] &= kgms^{-2} \end{align} $$

Note: Base units of constants

• The dimensions/base units of a constant such as $1$, $23.4$ or $2\pi$ is 1.
• Taking the equation $\omega =\frac{2\pi }{T}$ for example:

\begin{align}

\omega &= \frac{2\pi}{T} \ [\omega] &= \frac{[2\pi]}{[T]} \ [\omega] &= \frac{1}{s} \ [\omega] &= s^{-1} \end{align}$$

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Homogeneity of Equations

• An equation is homogeneous when the SI base units on the Left-Hand Side are the same as on the Right-Hand Side.
• To determine whether a given equation is homogeneous, we need to express the units on both sides in SI base units and see whether the units on the LHS match the units on the RHS.

Example 1: A homogeneous equation

• We are given the equation $F=\frac{m{v}^2}{r}$, where F: Force, m: mass, v: speed, r: radius.
• To check whether this equation is homogeneous, we express the units on both sides as SI base units: RHS:

[F] &= [m][a] \ [F] &= kg \times ms^{-2} \ [F] &= kgms^{-2} \end{align}$$

Therefore, the base units on the LHS are $kgm{s}^{-2}$ LHS:

\frac{{[m][v^2]}}{[r]} &= \frac{{kg \times (ms^{-1})^2}}{m} \ &= \frac{{kgm^2s^{-2}}}{m} \ &= kgms^{-2} \end{align}$$

Therefore, the base units on the RHS are $kgm{s}^{-2}$. Since the base units are the same on both sides, the equation is homogeneous.

Example 2: A heterogeneous/non-homogeneous equation

• We are given the equation $T=2\pi \sqrt{\frac{g}{L}}$, where T: time period, g: gravity, L: length
• To check whether this equation is homogeneous, we express the units on both sides as SI base units: RHS:

[T]=s \end{align}$$

Therefore, the base units on the LHS are $s$ LHS:

[2\pi]\left[ \sqrt{ \frac{g}{L} } \right] &= [2\pi]{\frac{[\sqrt{ g }]}{[\sqrt{ L }]}} \ &= 1 \times {\frac{\sqrt{ ms^{-2} }}{\sqrt{ m }}} \ &= {\frac{\sqrt{ m }}{\sqrt{ m }}}\times \sqrt{ s^{-2} } \ &= \sqrt{ s^{-2} } \ &= s^{-1} \end{align}$$

Therefore, the base units on the RHS are $s^{-1}$. Since the base units are not the same on both sides of the equation, the equation is heterogeneous/not-homogeneous.

Validity of an equation

• For an equation to be valid, it must be homogeneous, that is, SI base units on both sides of the equation must be the same.
• However, an equation that is homogeneous is not necessarily valid.
• For example, if we take one of the equations of motion: $v^2=u^2+2as$, the equation is homogeneous as the SI base units on both sides are $m^2{s}^{-2}$.
• However, if we remove the term $2as$, we are left with $v^2=u^2$ which is not a valid equation. But this wrong equation is still homogeneous as the SI base units on both sides are $m^2{s}^{-2}$.

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Unit Prefixes

• The different numerical magnitudes to be measured in the Universe vary over a vast range.
• For example, the mass of an electron is $9.11\times 10^{-31}kg$ while the mass of the sun is $1.99\times 10^{30}kg$.
• To accommodate this range, we often use prefixes in front of SI units to indicate multiples or submultiples of those quantities.

Multiplication Factor	Prefix	Symbol	Example length
$10^{12}$	tera	T	Radius of Pluto’s orbit (5.9 Tm)
$10^9$	giga	G	Earth-Moon distance (0.4 Gm)
$10^6$	mega	M	Earth radius (6.37 Mm)
$10^3$	kilo	k	Height of Mount Everest (8.85 km)
$10^{-1}$	deci	d	Volume of large drink bottle (2.0 $d{m}^3$)
$10^{-2}$	centi	c	Width of A4 page (21 cm)
$10^{-3}$	milli	m	Microwave wavelength (1 mm)
$10^{-6}$	micro	$\mu$	Visible light wavelength (500 $\mu$m)
$10^{-9}$	nano	n	Atomic diameter (0.1 nm)
$10^{-12}$	pico	p	Gamma ray wavelength (1 pm)