Physical quantities are used to classify physical objects and events in terms of numbers. Physical quantities are not all the same.
Base physical quantities, also called
primary physical quantities, are defined entirely in terms of physical operations. For such quantities, equality and addition are defined in physical terms.
Length is a familiar physical quantity that is primary. Two sticks are of equal length if they cover each other perfectly when one is put on top of the other. Physical operations corresponding to addition of two lengths,
A and
B, are defined in the familiar way.
Observe that there are no numbers involved in the equality and addition operations. They are defined entirely in physical terms. In order for a physical quantity to be primary, the operation of equality and addition must satisfy the following familiar laws:
\begin{align*}
A = B \quad \& \quad B = C \quad \Longrightarrow \quad A &= C ,
\\
A + B &= B + A ,
\\
A + ( B+C ) &= (A +B ) + C ,
\end{align*}
and so others others natural relationships.
Other recognizable base quantities are mass, time, area, volume, velocity, and force. A primary quantity that most likely you know is cardinality, which is a measure of the number of elements of the set.
In order to measure physical quantities, a unit is needed. This allows us to provide numerical values to objects. However, a numerical value to be assigned depends on units in use. For example, my height is
\begin{align*}
h &= 1.73\,\mbox{m},
\\
h &= 173\,\mbox{cm},
\\
h &= 17300000\,\mu\mbox{m},
\\
h &= 5'-08'' \quad (\mbox{5 feet and some inches}).
\end{align*}
There are varying reasons for choosing a particular unit in a given situation. Obviously
light-years (= 9.96… × 10
15 m) or
microns (= 10
−6 m) are not a suitable unit for my height.
Another motivation for choosing units is to simplify algebraic manipulations by getting rid of constant appearing in mathematical formulas. For example, if we choose units for length. mass, time, charge, and temperature to be
\begin{align*}
\mbox{length} &= 1.61 \cdot 10^{-35}\, m,
\\
\mbox{mass} &= 1.18 \cdot 10^{-8}\,kg ,
\\
\mbox{time} &= 5.39 \cdot 10{-44} s,
\\
\mbox{charhe} &= 1.88 \cdot 10^{-18} C,
\\
\mbox{temperature} &= 1.42 \cdot 10{32} K,
\end{align*}
then the constants
\begin{align*}
&\mbox{gravitational constant},
\\
&\mbox{Planck constant},
\\
&\mbox{speed of light},
\\
&\mbox{Coulomb constant},
\\
&\mbox{Boltzman constant}
\end{align*}
all get the numerical value 1. This choice leads to enormous simplifications in the algebraic manipulations that are required for predicting events in, for instance, high energy physics.
In addition to base or primary quantities, we have derived quantities. These are known two kinds of latter. For example, if we have
\[
\mbox{base quantities} = \begin{cases} \ell &= 3\,m , \\ t &= 60\,s, \end{cases}
\]
then
\[
\mbox{derived quantities of first kind} =
\begin{cases} A &= \ell\,t = 3 \cdot 60 = 180 ,
\\
B &= \frac{1}{2}\,\ell \, t^2 = \frac{1}{2} \cdot 3 \cdot 60^2 = 5400 .
\end{cases}
\]
In order to keep track of how the numerical values of a derived quantity changes when we change units for the basic quantities, we introduce the dimension for a derived quantity.
First, all basic physical quantities are assigned letters chosen by convention. For example, we have
\[
\begin{array}{lcl}
\mbox{length} & \rightarrow & L \\
\mbox{time} & \rightarrow &T \\
\mbox{mass} & \rightarrow & M \\
\mbox{force} & \rightarrow & F .
\end{array}
\]
Next, a derived physical quantity is assigned a monomial of letters based on the mathematical formula defining the quantity. We use notation [
A] to denote the dimension of a physical quantity
A. In general, if
A is a physical quantity defined by a monomial
\[
A = c\, a_1^{k_1} a_2^{k_2} \cdots a_n^{k_n} ,
\]
then its dimension is given by
\[
\left[ A \right] = \left[ a_1 \right]^{k_1} \left[ a_2 \right]^{k_2} \cdots \left[ a_n \right]^{k_n} .
\]
For example
\[
V= \ell_1 \ell_2 \ell_3 \qquad \Longrightarrow \qquad \left[ V \right] = L^3 .
\]
The dimensions are used to keep track of how the numerical values of derived physical quantities are changed when we change units for the basic quantities. For instance, you want to switch
\[
\begin{array}{lcl}
\ell = 1\,\mbox{cm} & \qquad\Longrightarrow \qquad & \ell = 10\,\mbox{mm} ,
\\
m = 1\,\mbox{kg} & \qquad\Longrightarrow \qquad & m = 1000\,\mbox{g},
\\
t = 1\,\mbox{sec} & \qquad\Longrightarrow \qquad & t = \frac{1}{3600}\,\mbox{hours} ,
\end{array}
\]
then quantity with dimensions
\[
\left[ A \right] = M^{1/2} L^2 T^3
\]
becomes
\[
\left[ A \right] = 1^{1/2} 1^2 1^3 = 1 \mbox{kg}^{1/2} \mbox{m}^2 \mbox{s}^3 = \left( 1000 \right)^{1/2} \left( 10 \right)^2 \left( \frac{1}{3600}\right)^3 \approx 6.78 \cdot 10 ^{-8} \mbox{g}^{1/2} \mbox{mm}^2 \mbox{hours}^3 .
\]
The number of base quantities and the choice of their units depends on what kind of object and/or events are of interest, For this reason, there are many such systems of units in use.
Classical mechanics describes force as
\[
F = m\,a = m\,\dot{v} = m \,\frac{{\text d} v}{{\text d} t} ,
\]
which is a physical law discovered by I. Newton. Therefore, in SI system force is a derived quantity. However, in British Engineering System, force is a base quantity and Newton's second law reads
\[
F = c\,m\,a ,
\]
where
c is dimensional constant with dimensions
\[
\left[ c \right] = F\,M^{-1} L^{-1} T^2 \approx 0.031 .
\]
In engineering, applied mathematics, and physics, the
Buckingham π theorem is a key theorem in
dimensional analysis. It is a formalization of
Rayeigh's method of dimensional analysis.
Although named for
Edgar Buckingham (1867--1940), the π theorem was first proved by the French mathematician
Joseph Bertrand (1822--1900) in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modeling physical phenomena.
Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters π1, π2, … , πp constructed from the original variables, where k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.
The number p of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and k is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
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