Everything about Dimensional Analysis totally explained
Dimensional analysis is a conceptual tool often applied in
physics,
chemistry, and
engineering to understand physical situations involving a mix of different kinds of physical quantities. It is routinely used by physical scientists and engineers to check the plausibility of
derived equations and computations. It is also used to form reasonable hypotheses about complex physical situations that can be tested by experiment or by more developed theories of the phenomena.
The
Buckingham π theorem is of central importance to dimensional analysis. This theorem describes how every physically meaningful equation involving
n variables can be equivalently rewritten as an equation of
n −
m dimensionless parameters, where
m is the number of fundamental dimensions used. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.
Introduction
The dimensions of a
physical quantity are associated with combinations of
mass,
length,
time,
electric charge, and
temperature, represented by symbols
M,
L,
T,
Q, and
Θ (respectively) each raised to rational powers. As examples, the dimension of the physical quantity
speed is "distance/time" (
L/
T or
LT−1), and the dimension of the physical quantity
force is "mass×
acceleration" or "mass×(distance/time)/time" (
ML/
T 2 or
MLT−2). In principle, other dimensions of physical quantity could be defined as "fundamental" (such as
momentum or
energy or
electric current) in lieu of some of those shown above. Some physicists have not recognized temperature,
Θ, as a fundamental dimension of physical quantity since it essentially expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). Still others don't recognize electric charge,
Q, as a separate fundamental dimension of physical quantity, since it has been expressed in terms of mass, length, and time in unit systems such as the electrostatic
cgs system. There are also physicists who have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity
The
unit of a physical quantity and its dimension are related, but not precisely identical concepts. The units of a physical quantity are defined by convention and related to some standard; for example length may have units of meters, feet, inches, miles or micrometres; but any length always has a dimension of
L, independent of what units are arbitrarily chosen to measure it. Two different units of the same physical quantity have
conversion factors between them. For example: 1
in = 2.54
cm; then (2.54 cm/in) is called a conversion factor (between two representations expressed in different units of a common quantity) and is itself dimensionless and equal to one. There are no conversion factors between dimensional symbols.
Dimensional symbols, such as
L, form a
group: there's an identity,
L0=1; there's an inverse to
L, which is 1/
L or
L−1, and
L raised to any rational power
p is a member of the group, having an inverse of
L−p or 1/
Lp. The operation of the group is multiplication, with the usual rules for handling exponents (
Ln ×
Lm =
Ln+m).
In
mechanics, the dimension of any physical quantity can be expressed in terms of base dimensions
M,
L and
T. This isn't the only possible choice, but it's the one most commonly used. For example, one might choose force, length and mass as the base dimensions (as some have done), with associated dimensions
F,
L,
M. The choice of the base set of dimensions is thus partly a convention, resulting in increased utility and familiarity. It is, however, important to note that the choice of the set of dimensions isn't
just a convention; for example, using length, velocity and time as base dimensions won't work well, because there's no way to obtain mass — or anything derived from it, such as force — without introducing another base dimension, and velocity, being derived from length and time, is redundant.
Depending on the field of physics, it may be advantageous to choose one or another extended set of dimensional symbols. In electromagnetism, for example, it may be useful to use dimensions of
M,
L,
T, and
Q, where
Q represents quantity of
electric charge. In
thermodynamics, the base set of dimensions is often extended to include a dimension for temperature,
Θ. In chemistry the number of
moles of substance (loosely, but not precisely, related to the number of molecules or atoms) is often involved and a dimension for this is used as well. The choice of the dimensions or even the number of dimensions to be used in different fields of physics is to some extent arbitrary, but consistency in use and ease of communications are important.
In the most primitive form, dimensional analysis may be used to check the plausibility of physical equations: the two sides of any equation must be
commensurable or have the same dimensions, for example, the equation must be dimensionally homogeneous. As a corollary of this requirement, it follows that in a physically meaningful expression, only quantities of the same dimension can be added or subtracted. For example, the mass of a rat and the mass of a flea may be added, but the mass of a flea and the length of a rat can't be meaningfully added. Physical quantities having different dimensions can't be compared to one another either. For example, "3 m > 1 g" isn't a meaningful expression.
Only like-dimensioned quantities may be added, subtracted, compared, or equated. When unlike dimensioned quantities appear opposite of the "+" or "−" or "=" sign, that physical equation isn't plausible, which might prompt one to correct errors before proceeding to use it. When like-dimensioned quantities or unlike dimensioned quantities are multiplied or divided, their dimensional symbols are likewise multiplied or divided. When dimensioned quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities.
Scalar arguments to
exponential,
trigonometric,
logarithmic, and other
transcendental functions must be
dimensionless quantities. This requirement is clear when one observes the
Taylor expansions for these functions (a sum of various powers of the function argument). For example, the logarithm of 3 kg is undefined even though the logarithm of 3 is nearly 0.477. An attempt to compute ln 3 kg would produce
» where
is the dimension of the lattice.
It has been argued by some physicists, for example Michael Duff, that the laws of physics are inherently dimensionless. The fact that we've assigned incompatible dimensions to Length, Time and Mass is, according to this point of view, just a matter of convention, borne out of the fact that before the advent of modern physics, there was no way to relate mass, length, and time to each other. The three independent dimensionful constants:
c,
hbar, and
G, in the fundamental equations of physics must then be seen as mere conversion factors to convert Mass, Time and Length into each other.
Just like in case of critical properties of lattice models, one can recover the results of dimensional analysis in the appropriate scaling limit. E.g. dimensional analysis in mechanics can be derived by reinserting the constants
, c, and G (but we can now consider them to be dimensionless) and demanding that a nonsingular relation between quantities exists in the limit
,
and
. In problems involving a gravitational field the latter limit should be taken such that the field stays finite.
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