Download mathematics of dimensional analysis and problem solving in physics

MATHEMATICS OF DIMENSIONAL ANALYSIS AND PROBLEM SOLVING IN
PHYSICS
Decio Pescetti, Dipartimento di Fisica, Università di Genova, Italy
1. Introduction
As is well known, the qualitative methods, based on the application of the principles of dimensional
homogeneity, continuity and symmetry, offer the opportunity for a truly fertile analysis of the
physical systems prior to their complete mathematical or experimental study [1-3]. In problem
solving, the qualitative methods enable us to deduce useful information about the dependence of a
physical quantity (the unknown) on other relevant quantities (the data) [4-8].
The complete description of a real physical system would require a great number of parameters. We
must select and take into account the important quantities and ignore those which have relatively
small effects. A given physical quantity is expressed by a number followed by the corresponding
unit of measurement. The radius of the Hearth is 6.4  106 m , but it is also 2.110 10 parsec ; from this
point of view it has no meaning to speak of big or small numbers. The order of magnitude of the
relative effect, on the value of a specified unknown X, of a neglected quantity can be expressed by
proper dimensionless products (pure numbers) of the system’s characteristics. For instance,
consider a damped simple harmonic oscillator. A body of mass m is acted on by an elastic restoring
force F=-kx, where x is the deplacement from the equilibrium position and k is the force constant.
What is the effect, on the period , of a viscous damping force, Fviscous  rdx / dt , where r is a
constant. As already remarked, to speak of small or big value of r has no meaning; a given
damping constant of value, for instance, 10 3 kg / s , has also the value 6.024 10 23 amu/s . As is well
known, such an effect is expressed by the dimensionless product P  r /(mk )1 / 2 , whose physical
meaning is: the ratio between the maximum value of the damping force and the maximum value of
the elastic restoring force.
We show that the origin of the information, on the problem’s solution, obtained by the qualitative
methods, is made more transparent by making a clear distinction between the mathematical
dimensional analysis results and the phenomenological assumptions and laws of nature relating
such results to the physical world. Obviously, wrong phenomenological assumptions plus correct
mathematical dimensional analysis will lead to erroneous results, as discussed in the comment to
example 4 of section 4.
Let us remark that the examples of dimensional analysis given in introductory physics textbooks
are likely to be misleading. The reader may be left with the impression that dimensional analysis is
a routine procedure. The practice of dimensional analysis requires a great deal of insight and
experience. For instance, such insight enters in a crucial way into the initial selection of variables to
be included in the analysis. Often considerable penetration is required to recognize when a
particular dimensional constant, such as the acceleration due to gravity, may be required. Failure to
include a relevant variable or dimensional constant will lead to an incorrect result. Unfortunately,
there is nothing in the nature of the mathematics of dimensional analysis to tell the practitioner that
a crucial variable has been omitted.
In section 2 we discuss the mathematical bases of dimensional analysis. In section 3 we present a
set of exercises in linear algebra, which are proposed as an help to understand the rational of the
paper: the necessity of a clear distinction between purely mathematical results and physical
assumptions. In fact, the answers of such exercises are involved in the qualitative solution of the
physical problems discussed in section 4. Section 5 is devoted to concluding remarks.
2. Dimensional analysis and continuity principle
The dimensions of physical quantities are represented by vectors in an abstract finite-dimensional
linear vector space, referred to as the dimension space. In Mechanics the dimension space has a
basis of three elements: length L, mass M, time T. The extension to electromagnetism requires a
basis of four elements (L, M, T, electrical current I). Finally, the extension to thermal phenomena
demands a five element basis: (L, M, T, I, absolute temperature ). For instance, in the linear vector
space of mechanics, the quantities length, mass, time, density, velocity, acceleration, force,
viscosity and elastic force constant are represented by the vectors (1,0,0), (0,1,0), (0,0,1), (-3,1,0),
(1,0,-1), (1,0,-2), (1,1,-2), (1,0,-1), and (0,1,-2) respectively.
The principle of dimensional homogeneity (PDH) states that in any legitimate physical equation the
dimensions of all terms which are added or subtracted must be the same.
The problem description indicates that a physical quantity X (the unknown) is a function of other
quantities A1 , A2 ,..., An (the data):
(2.1)
X  f ( A1 , A2 ,..., An ) .
The PDH allows us to study a function of fewer arguments:
(2.2)
Px   ( P1 ,P2 ,...,Pm ) ,
where Px is a dimensionless product between X (elevated at the first power) and some of the data,
and P1 ,P2 ,...,Pm are a complete set of independent dimensionless products between the data
themselves. One has (Buckingham’s theorem): m=n-r, where r is the rank of the matrix formed by
the dimensional exponents of the data.
Substantially, the Buckingham theorem is the following theorem of linear algebra: if the rank of the
matrix (aij ),i  1,2,...,n and j  1,2,...,q , associated with the n vectors ( xi1 , xi 2 ,..., xiq ) is r<q, there are
exactly r vectors which are linearly independent while each of the remaining m=n-r vectors can be
expressed as a linear combination of these r vectors. In other words, it can be formed a complete set
of m independent linear combinations between the n vectors ( xi1 , xi 2 ,..., xiq ) , such as
h1 ( x1 )  h2 ( x2 )  ...  hn ( xn )  0
For every set
h1 , h2 ,..., hn , there corresponds an unique
dimensionless product between the data A1 , A2 ,..., An : P  A1h1 A2h2    Anhn
A complete set of independent Px is formed by m+1 elements.
The principle of continuity states that small causes produce small effects. In teaching physics, this
principle is tacitly taken for granted. Usually, its statement is not explicitly given. Perhaps because
there are exceptions [1,9], especially for complex systems (butterfly effect). In fact, arbitrarily small
causes might produce finite effects, but this cannot be the rule. The idealizations of physics such as
frictionless planes, inextensible string, massless pulleys etc. find their justification and validity in
the continuity principle. Such idealizations imply that we may have experimental conditions in
which the friction of the plane, the extensivity of the string, the mass of the pulley etc., have
negligible effects, with respect to those of other causes and, of course, with respect to the finding of
a well defined unknown.
Let us consider, for the sake of simplicity, the particular case m=1 in eq. (2.2),
Px   (P) .
Let us suppose, without loss of generality, P<<1. We say that Px has an essential type dependence
on P if
0
lim  ( P) 
.
P 0

On the contrary, the dependence is non-essential if
lim  ( P) c ,
P 0
where c is a dimensionless positive constant.
The criteria for the evaluation of the orders of magnitude [7,8] are: a) The dimensionless constants
c are of the order of unity. b) A dimensionless product P very different from unity is irrelevant to
the solution of the problem. The criterion b) is satisfied under the following conditions: i)
continuity of function  ii) a non-essential dependence of  on P; iii) a correct choice of the
dimensionless product Px , that is in Px should appear the dominant data. By dominant data, with
respect to the prediction of the unknown X, we mean a subset of s ≤ (dimension of the linear space)
relevant parameters for which the phenomenology under study is preserved, even if all the other
parameters are absent (in the sense that they are not influential).
3. Linear algebra exercises
Exercise 1. Find the coordinates of the vector (0,0,1) relative to the basis (0,1,0), (0,1,-2), (1,0,1).
Answer: (1/2, -1/2, 0).
Exercise2. Find the coefficients hi of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0),
(0,1,-1) such as: h1 (0,1,0)  h2 (0,1,2)  h3 (1,0,0)  h4 (0,1,1)  0 .
Answer: h1  h,h2  h,h3  0,h4  2h .
Exercise 3. Express the vector (0,0,1) as a linear combination of any set of no more than three of the
following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,-1).
Answer; i) (0,0,1)=1/2(0,1,0)-1/2(0,1,-2); ii) (0,0,1)=(0,1,0)-(0,1,-1); iii) (0,0,1)=-(0,1,-2)+(0,1,-1).
Let us remark that answer iii) is not independent of answers i) and ii).
Exercise 4. Find the coefficients hi of a linear combination of the vectors (0,1,0), (0,1,-2), (1,0,0),
(0,1,0) such as: h1 (0,1,0)  h2 (0,1,2)  h3 (1,0,0)  h4 (0,1,0)  0 .
Answer: h1  h,h2  0, h3  0,h4  h .
Exercise 5. Express the vector (0,0,1) as a linear combination of any set of no more than three of the
following vectors: (0,1,0), (0,1,-2), (1,0,0), (0,1,0).
Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2).
Exercise 6. Find the coefficients hi of a linear combination of the vectors (1,0,0), (1,0,0), (0,1,0),
(1,0,-2) such as: h1 (0,1,0)  h2 (1,0,0)  h3 (0,1,0)  h4 (1,0,2)  0 .
Answer: h1  h,h2  h,h3  0,h4  0 .
Exercise 7. Express the vector (0,0,1) as a linear combination of any set of no more than three of the
following vectors: (1,0,0), (1,0,0), (0,1,0), (1,0,-2).
Answer: (0,0,1)=1/2(1,0,0)-1/2(1,0,-2).
Examples
Example1. (Harmonic oscillator) A body of mass m is acted on by an elastic force F  kx , where
x is the deplacement from the equilibrium position and k is the force constant.. The body is relaxed
from the rest at an initial position x0 . The body executes an oscillatory motion. Find the period  .
Solution: The relevant parameters are m, k, x0 ; so
  f (m, k , x0 ) .
By PDH and answer of algebra exercise 1, one finds,
  c m/k ,
where c is a dimensionless constant. The period  is independent of motion amplitude x0 . By
criterion a) for the evaluation of the orders of magnitude, the constant c is of the order of the unity.
As is well known, the exact value of c is 2.
Example2. (Damped harmonic oscillator) Find the effect, on period of a viscous damping
force F  rdx / dt 
Solution: The relevant parameters arem, k, x0, r. So,
  f (m, k , x0 , r ) .
By PDH and answer of algebra exercise 2, one has P  [r 2 /( mk )]h . A proper choice of the constant
h is h=1/2. In fact P  r m 1 / 2 k 1 / 2 is the ratio between the maximum value of the damping force
and the maximum value of the elastic restoring force. [cause: friction)  0  [ P  0] .
By PDH and answers i), ii) and iii) of algebra exercise 3, one has:
i )  m / k ,ii )  m / r ,iii )  r / k .
Representation i) is correct. In fact, the problem of finding the period makes sense also in absence
of the friction force. Representations ii) and iii) are senseless from the physical point of view. Let us
remark that a complet set of independent dimensionless products Px is formed by two elements. For
instance:  /( m / r ), /( r / k ) . Obviously, there are infinite other possibilities; for instance:
[ / m / k , /( m / r )]or[ /( r / k ), /( m 4 / 5 r 3 / 5 k 1 / 5 )] . The correct choice of Px is a matter of physics;
it does not follow from the mathematics of dimensional analysis.
In conclusion
m

 (P) .
k
The dependence of  ( P) on P is non-essential .
As is well known, the complete exact mathematical solution of the problem leads to:
2
 ( P) 
.
[1  ( p / 2) 2 ]1 / 2
Example3. (Harmonic oscillator with mspring ) A body of mass m is placed on an horizontal air track.
The body is attached to an horizontal spring of force constant k and massa ms. The body is released
from the rest at a position in which the spring is stretched by x0 . The body executes an oscillatory
motion. Find the period .
Solution: The relevant parameters are m, k, x0, ms. So,
  f (m, k , x0 , ms ) .
By PDH and answer of exercise 4, one finds that there is the following dimensionless product
between the data: (m / ms )h , where h is a constant. A proper choice of h is h=-1. Then: P=m/ms .
[(cause : spring mass)  0]  ( P  0) .
By PDH and answer of algebra exercise 5, one finds: i)  m / k and ii) i)  m s / k .
The dominant parameters are m, k, x0 . Reprensentation i) is correct. In conclusion, one has
  m / k ( P) ,
where  is a dimensionless undetermined function of P. There is a non-essential dependence of
 (P ) on P: lim  ( P)  positive constant .
P 0
A complete exact mathematical analysis yields to:  ( P)  2 (1  P / 3) 1 / 2 .
Example 4. (Flexible chain) A flexible chain of total length l and mass m rests on a frictionless
table with length x0 overhanging the edge. At time t=0 the system is released. Find time  after
which the entire chain leaves the table.
Solution: The relevant parameters are: x0 ,l ,m and gravity g . So,
  f ( x0 , l , m, g ) .
By PDH and answer of algebra exercise 6, one finds: P  ( x0 / l ) h . A proper choice of h is h=1.
Then P  x 0 / l .
By PDH and answer of algebra exercise 7, one finds:   x0 / g or   l / g .
In conclusion, the correct representation is
  l / g ( P) .
The sliding time does not depend on the mass m of the chain.
It appears quite natural, from the physical point of view, to predict the following special cases: a)
( P  0)  [ ( P)  ] , b) ( P  1)  [ ( P)  0] . Therefore there is an essential dependence of
 (P) on P.
The solution can also be written
  (lx 0 / g 2 )1 / 4 1 P) ,
where 1 ( P)  P 1 / 4 ( P) . The essential dependence on P is preserved. The parameters x0 , m and
g are a set of dominant parameters, but condition ii) for the validity of criterion b) for the
evaluation of the orders of magnitude is not satsfied.
The function  (P) is an “universal” function. The function  (P) cannot be obtained by the
qualitative methods. The function (P) can be found:
i) by the analytical solution of the equation of motion F=ma;
ii) by numerical solution of the equation of motion F=ma;
iii) experimentally.
One finds:  (0)  ; (0.1)  2.99; (0.5)  1.32; (0.9)  0.467; (1)  0 .
Remark: [(wrong phenomenological assumptions)+(correct mathematical dimensionl ansalysis)]
 [erroneous (senseless) information on the problem solution].
Wrong assumption: the motion of the chain is periodic; wrong question: find period By PDH
etc.   l / g ( P);P  x0 / l 
Erroneous conjecture: lim  ( P)  positive constant  c . In conclusion: senseless result: period
P 0
 c l/g .
4. Conclusions
The application of the qualitative methods in problem solving is put into execution according to the
scheme: [(1) Phenomenological behaviour of the system]  [(2) Identification of the unknown X
and of the relevent parameters A1, A2, ..., An]  [(3) linear algebra (Buckingham’s theorem)]  [(4)
Complete set of independent dimensionless products P1, ..., Pm; complete set of Px independent
dimensionless products]  [(5) Proper choice of P1, ..., Pm; correct choice of Px]  [(6) Physical
exploitation of the qualitative solution].
The mathematics of dimensional analysis is involved in steps (3) and (4) only. The preliminary step
(1) requires sound information about the system’s phenomenological behaviour. Step (2) demands
the knowledge of the physical relations that must be invoked to work out a complete quantitative
study of the system. In step (5) the mathematical dimensional analysis results are coupled with the
physics of the problem, and finally in step (6) an explicit interpretation of the qualitative solution is
given.
Let us remark that, when studying a physical system, the information obtained by the qualitative
analysis might be increased by breaking the problem into subproblems, even if additional quantities
must be introduced [5].
The notion of similarity is closely related to dimensional analysis and model building. One of the
reasons that dimensional analysis is a useful method in physics is that many variables are combined
into one or few dimensionless numbers (products). It is significant that these dimensionless
products serve often as a quantitative measure of similarity.
It is surprising that no introductory university algebra textbook mentions the dimension space of the
physical quantities, as an example of linear vector space. We hope that this paper will contribute in
filling the gap between mathematics and physics in the teaching/learning of this important topic.
The sistematic resort to qualitative methods along the lines presented in this paper should be the
source of a fertile reflection on the mathematical modeling of physical systems and on the modelreality relationship.
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