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PART IA: DIMENSIONAL ANALYSIS
HANDOUT 1
AIMS OF THE COURSE
To introduce and illustrate the use of Dimensional Analysis.
To develop an understanding of the principle of dimensional consistency and how it can
be used:
- to check the units of an equation,
- to convert from one system of measuring units to another.
To develop the techniques required to form dimensionless quantities and relationships.
To explain how Dimensional Analysis can be used:
- to simplify problems by reducing the number of parameters,
- to correlate experimental data,
- to assist in the design of models for testing.
LABORATORY EXPERIMENTS
Use of Dimensional Analysis in model testing to obtain general expressions for a number of
problems:
Dimensional Analysis 1: The deflection of an elastic beam under load.
Dimensional Analysis 2A: The temperature variation in two blocks initially at
different temperatures.
Dimensional Analysis 2B: The flow over a “V” notch weir.
RECOMMENDED READING
Book Title: Dimensional Analysis and Scale Factors
Author: R.C. Pankhurst
Publisher: Chapman and Hall
Shelf Mark: EV 8
Guide: Guide to Units, Unit Symbols and Abbreviations
Author: G.T. Parks
Source: CUED Data Book Folder
PART IA: DIMENSIONAL ANALYSIS
1. INTRODUCTION
Dimensional Analysis uses our knowledge of the systems of measuring units and the
dimensions of physical quantities in the solution of engineering problems. If we have a
theory, dimensional analysis complements it, but we can also get close to the answer
without one.
Dimensional Analysis allows us to:
Convert from one system of units to another.
Check the units of an equation.
Simplify problems by reducing the number of parameters.
Plan experiments so as to reduce the effort required to investigate the situation under
consideration.
Design and use models for experimental tests.
Correlate experimental data.
Graphically present the results of experimentation or analysis more concisely.
2. UNITS
Basic units are defined as standards for the purpose of comparison or measurement. In
many systems units of
mass , length , time , electric current , temperature and luminous intensity
are chosen. The S.I. (Système International) system of units (also known as the metric
system) takes as basic units:
Quantity Name Symbol Units
mass kilogram kg kg
length metre m m
time second s s
electric current ampere A A
absolute temperature kelvin K K
luminous intensity candela cd cd
Derived units are formed from the basic units either by definition or through some physical
law.
4. CONVERSION OF UNITS
Different systems of units (e.g. S.I., Imperial) have different definitions. By comparing the
definitions in different systems of units, we can convert between the systems. For the S.I.
and Imperial systems, the data books give:
1 lb = 0.45359237 kg (approximately)
1 ft = 0.3048 m (exactly!!)
So the feet-to-metres length conversion factor is 1 ft
- 3048 m = 1.
Example: A typical car is 14 ft long. Find its length in m.
14 ft = 14 ft × 1 ft
- 3048 m = 4.267 m
This is a typical conversion between basic units, involving a single conversion factor.
Conversions between derived units are slightly more complicated, generally requiring the
use of both conversion factors between basic units and scale factors.
Example: A typical car speed is 35 mph. Find this speed in m s
35 mph = 35 hr
mile × 3600 s
1 hr × 1 mile
5280 ft × 1 ft
- 3048 m = 15.65 m s
To minimise the chance of errors in conversion, keep unit symbols in your arithmetical
working (as in the examples above). If the conversion is set up correctly, unwanted units
should cancel. For example:
35 mph = 35 hr
mile × 3600 s
1 hr × 1 mile
5280 ft × 1 ft
- 3048 m = 15.65 m s
Now have a go at Questions 1 and 2 on the Examples Paper.
5. AN EXAMPLE OF WHY THE FOREGOING IS IMPORTANT
For more information about this story see:
http://news.bbc.co.uk/1/hi/sci/tech/514763.stm
http://www.cnn.com/TECH/space/9909/30/mars.metric.02/
To read about another famous “conversion factor” incident see:
http://www.wadenelson.com/gimli.html
Friday, October 1, 1999 LA Times
Mars Probe Lost Due to Simple Math Error
By ROBERT LEE HOTZ, Times Science Writer
Why it is so crucial to always double check your data entries and data manipulations.
NASA lost its $125-million Mars Climate Orbiter because spacecraft engineers failed
to convert from English to metric measurements when exchanging vital data before the
craft was launched, space agency officials said Thursday.
A navigation team at the Jet Propulsion Laboratory used the metric system of newtons
and meters in its calculations, while Lockheed Martin Astronautics in Denver, which
designed and built the spacecraft, provided crucial data in the English system of
inches, feet and pounds.
As a result, JPL engineers mistook readings measured in English units of pound-
seconds for a metric measure of impulse called newton-seconds. In a sense, the
spacecraft was lost in translation.
"That is so dumb," said John Logsdon, director of George Washington University's
space policy institute…
7. THE PRINCIPLE OF DIMENSIONAL CONSISTENCY
A complete statement of a physical law is independent of the system of measurement.
This implies that any consistent system of units may be used to substitute in a proper
(dimensionally consistent) algebraic equation.
As a demonstration, consider the simple pendulum:
Quantity Symbol Typical Units Dimensions
period t s T
length l m L
gravitational acceleration g m s
In one system of units, we find, by experiment, that the period t is given by:
Equation t = 2 π g
l
Dimensions T 2 LT
L
!
= T
Both the LHS and RHS have dimensions T, so the equation is dimensionally consistent.
(Being a pure number, the term “2π” is dimensionless, of course.)
Now, consider using another set of measuring units, say, “heart-beats” (symbol hb) and
“arm-lengths” (symbol al):
Quantity Symbol Typical Units Dimensions
period t! hb T
length l! al L
gravitational acceleration g! al hb
We would have time (1 hb =! s) and length (1 al = !m) conversion factors:
time: 1 1 hb
s
length: 1 1 al
m
Now: t !hb = t! hb × 1 hb
! s = t "! s = t s
So, we have t = " t !and, similarly, l = " l !, while:
g! al hb
al × 1 al
! m ×
2
s
1 hb !
2 !
g # m s
Hence g = g!
" 2 #$.
The equation in the original measuring units is:
g
l t = 2!
On changing the measuring units we obtain:
g
l t !
2
i.e. g
l t !
!^ = 2 " THE SAME!!
So the principle of dimensional consistency is demonstrated.
In reality, any valid equation must be dimensionally consistent.
There are many equations that, on first inspection, appear not to be dimensionally
consistent. Consider, for example, the equation for the gravitational force between two
bodies, of masses m and M , a distance r apart:
Equation F = 2 r
GMm
Dimensions
2 MLT
! 2 L
(M)(M)
2 2 M L
!
These expressions appear not to be dimensionally consistent. So what is wrong?
The answer is that the constant G is not dimensionless, as has been assumed here. In fact, it
has dimensions L
3 M
Equation F = 2 r
GMm
Dimensions
2 MLT
! 2
3 1 2
L
(L M T )(M)(M )
!!
2 MLT
!
The practical significance of this is that, although if a given system of units is used G will
have a constant value, if a different system of units is used the governing equation will be of
the same form, BUT G will take a different value, G !, say. The value of G! can be found
from the value of G using the rules for conversion of units shown earlier.
Conversely, if a constant appearing in an equation is dimensionless, then its value will be
the same whatever system of units is used.
Now have a go at Questions 3 and 4 on the Examples Paper.
Example: For the simple pendulum we have, in different systems of units:
g
l t = 2! and g
l t !
These can be rearranged to give the dimensionless form:
l
g t and =! "
l
g t
Thus, the dimensionless group ( t gl ) has the same value in both measuring systems.
9. DIMENSIONLESS EQUATIONS
The observation that the value of a dimensionless group does not depend on the system of
measuring units suggests that this type of quantity could be “fundamental” in physical
problems. The principle of dimensional consistency (PDC) tells us that:
In a complete statement of a physical law,
it is possible to rearrange the terms so that all groups or quantities are dimensionless.
Example: Consider a car moving at an initial velocity u which is then accelerated at a
constant rate a. We require the velocity v after a time t.
Equation v = u + at
Typical units m s
- 1 m s - 1 m s - 2 s = m s - 1
Dimensions LT
- 1 LT - 1 LT - 2 T = LT - 1
The equation is, of course, dimensionally consistent and we can rearrange it to give:
Equation u
v = 1 + u
at
Inspection shows that both vu and atu are dimensionless. Thus, this dimensionless
equation contains all the information in the original equation, but in just two dimensionless
groups ( vu and atu ) and one dimensionless quantity (1).
