Dimensions and Units

Dimensions and Units#

Base Quantities#

All physical quantities are expressed in terms of base quantities and standard international (SI) units:

Base Quantity	Symbol	SI Unit	SI Unit Symbol
time	T	second	s
length	L	meter	m
mass	M	kilogram	kg
temperature	\(\Theta\)	kelvin	K
electric current	I	ampere	A
amount of substance	N	mole	mol
luminous intensity	J	candela	cd

Dimensions of Physical Quantities#

All other physical quantities are called derived quantities because they are expressed in terms of base quantities. The dimensions of a physical quantity refers to its expression in terms of the base quantities. For example, common derived quantities are:

Physical Quantity	Dimensions	Base SI Units
velocity	L T^-1	m s^-1
acceleration	L T^-2	m s^-2
area	L²	m²
volume	L³	m³
density	M L^-3	kg m^-3

Some derived quantities have their own SI units:

Physical Quantity	Dimensions	SI Unit	SI Unit Symbol	Base SI Units
force	M L T^-2	newton	N	kg m s^-2
pressure	M L^-1 T^-2	pascal	Pa	kg m^-1 s^-2
energy	M L² T^-2	joule	J	kg m² s^-2
molar concentration	N L^-3	molar	M	N m^-3
power	M L² T^-3	watt	W	kg m² s^-3
frequency	T^-1	hertz	Hz	s^-1
angle	1	radian	rad	1

Note that an angle is a ratio of lengths, arc length over radius, therefore its dimensions are L L^-1 = 1. This is an example of a dimensionless quantity.

Dimensional Homogeneity#

An equation is dimensionally homogeneous if the expressions on both sides of the equation have the same dimensions. During the mathematical modelling process we should always check that the equations in our models are dimensionally homogeneous.

For example, consider the mass-spring-damper equation derived from Newton’s second law of motion:

\[ m x'' + c x' + k x = F(t) \]

Each term in the equation is a force therefore each side of the equation has dimensions M L T^-2 and the equation is dimensionally homogeneous.

Let’s find the dimensions of the variables and parameters in the mass-spring-damper equation:

Variable/Parameter	Symbol	Dimensions
position	\(x\)	L
velocity	\(x'\)	L T^-1
acceleration	\(x''\)	L T^-2
mass	\(m\)	M
damping coefficient	\(c\)	M T^-1
spring coefficient	\(k\)	M T^-2
external force	\(F(t)\)	M L T^-2