Dimensions Of Physical Quantities
The nature of a physical quantity is described
by its dimensions. All the physical quantities
represented by derived units can be expressed
in terms of some combination of seven
fundamental or base quantities. We shall call
these base quantities as the seven dimensions
of the physical world, which are denoted with square brackets [ ].
Thus, length has the
dimension [L], mass [M], time [T], electric current
[A], thermodynamic temperature [K], luminous
intensity [cd], and amount of substance [mol].
The dimensions of a physical quantity are the
powers (or exponents) to which the base
quantities are raised to represent that
quantity.
Note that using the square brackets
[ ] round a quantity means that we are dealing
with ‘the dimensions of’ the quantity.
In mechanics, all the physical quantities can
be written in terms of the dimensions [L], [M]
and [T].
For example, the volume occupied by
an object is expressed as the product of length,
breadth and height, or three lengths. Hence the
dimensions of volume are [L] × [L] × [L] = [L]$^{3}$ = [L$^{3}$ ].
As the volume is independent of mass and time,
it is said to possess zero dimension in mass [M°],
zero dimension in time [T°] and three
dimensions in length.
Similarly, force, as the product of mass and
acceleration, can be expressed as
Force = mass × acceleration
= mass × (length)/(time)$^2$
The dimensions of force are [M] [L]/[T]$^2$ =
[MLT$^{–2}$].
Thus, the force has one dimension in
mass, one dimension in length, and –2
dimensions in time. The dimensions in all other
base quantities are zero.
Note that in this type of representation, the
magnitudes are not considered. It is the quality
of the type of the physical quantity that enters.
Thus, a change in velocity, initial velocity,
average velocity, final velocity, and speed are
all equivalent in this context. Since all these
quantities can be expressed as length/time,
their dimensions are [L]/[T] or [LT$^{–1}$].
DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATIONS
The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity. For example, the dimensional formula of the volume is [M°L$^3$T°], and that of speed or velocity is [M°LT$^{-1}$]. Similarly, [M°LT$^{-2}$] is the dimensional formula of acceleration and [ML$^{-3}$T°] that of mass density.
An equation obtained by equating a physical
quantity with its dimensional formula is called
the dimensional equation of the physical quantity. Thus, the dimensional equations are
the equations, which represent the dimensions
of a physical quantity in terms of the base
quantities.
For example, the dimensional
equations of volume [V], speed [v], force [F] and
mass density [ρ] may be expressed as
[V] = [M°L$^{3}$T°]
[v] = [M°LT$^{-1}$]
[F] = [MLT$^{-2}$]
[ρ] = [ML$^{-3}$T°]
The dimensional equation can be obtained
from the equation representing the relations
between the physical quantities. The
dimensional formulae of a large number and
wide variety of physical quantities, derived from
the equations representing the relationships
among other physical quantities and expressed
in terms of base quantities are given in
Appendix 9 for your guidance and ready
reference.
DIMENSIONAL ANALYSIS AND ITS APPLICATIONS
The recognition of concepts of dimensions, which guide the description of physical behaviour is of basic importance as only those physical quantities can be added or subtracted which have the same dimensions. A thorough understanding of dimensional analysis helps us in deducing certain relations among different physical quantities and checking the derivation, accuracy and dimensional consistency or homogeneity of various mathematical expressions.
When magnitudes of two or more
physical quantities are multiplied, their units
should be treated in the same manner as
ordinary algebraic symbols. We can cancel
identical units in the numerator and
denominator. The same is true for dimensions
of a physical quantity. Similarly, physical
quantities represented by symbols on both sides
of a mathematical equation must have the same
dimensions.
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