Technological Impacts

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Dimensional Reasoning
1. What are the units of Youngs Modulus? 2.
Are these equations correct? 3. What is the
common problem in the two images below?
Sign outside New Cuyama, CA
1998 Mars Polar Orbiter
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1. What are the units of Youngs Modulus?
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2. Are these equations correct?
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3. What is the common problem in the two images
below?
Pounds-force Newtons-force
125mil error Instead of passing about 150 km
above the Martian atmosphere before entering
orbit, the spacecraft actually passed about 60 km
above the surfaceThis was far too close and the
spacecraft burnt up due to friction with the
atmosphere. BBC News
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Dimensional Reasoning
Lecture Outline 1. Units base and
derived 2. Units quantitative
considerations 3. Dimensions and Dimensional
Analysis fundamental rules and uses 4.
Dimensionless Quantities 5. Scaling, Modeling,
and Similarity
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Dimensional Reasoning
Measurements consist of 2 properties 1. a
quality or dimension 2. a quantity expressed in
terms of units Lets look at 2 first
THE INTERNATIONAL SI SYSTEM OF MEASUREMENT IS
COMPRISED OF 7 FUNDAMENTAL (OR BASE) QUANTITIES.
THE ENGLISH SYSTEM, USED IN THE UNITED STATES,
HAS SIMILARITIES AND THERE ARE CONVERSION FACTORS
WHEN NECESSARY.
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Dimensional Reasoning
2. a quantity expressed in terms of units
THE INTERNATIONAL SI SYSTEM OF MEASUREMENT IS
COMPRISED OF 7 FUNDAMENTAL (OR BASE) QUANTITIES.
BASE UNIT A unit in a system of measurement
that is defined, independent of other units, by
means of a physical standard. Also known as
fundamental unit. DERIVED UNIT - A unit that is
defined by simple combination of base units.
Units provide the scale to quantify measurements
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• SUMMARY OF THE 7 FUNDAMENTAL SI UNITS
• LENGTH
  - meter
• MASS
  - kilogram
• TIME
  - second
• ELECTRIC CURRENT -
  ampere
• THERMODYNAMIC TEMPERATURE - Kelvin
• AMOUNT OF MATTER - mole
• LUMINOUS INTENSITY - candela

Quality (Dimension) Quantity Unit
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Units provide the scale to quantify measurements
LENGTH
YARDSTICK
METER STICK
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Units provide the scale to quantify measurements
MASS
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Units provide the scale to quantify measurements
TIME
ATOMIC CLOCK
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Units provide the scale to quantify measurements
ELECTRIC CURRENT
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Units provide the scale to quantify measurements
THERMODYNAMIC TEMPERATURE
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Units provide the scale to quantify measurements
AMOUNT OF SUBSTANCE
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Units provide the scale to quantify measurements
LUMINOUS INTENSITY
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Units
1. A scale is a measure that we use to
characterize some object/property of
interest. Lets characterize this plot of
farmland
The Egyptians would have used the length of their
forearm (cubit) to measure the plot, and would
say the plot of farmland is x cubits wide by y
cubits long. The cubit is the scale for the
property length
y
x
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Units
7 historical units of measurement as defined by
Vitruvius Written 25 B.C.E. Graphically
depicted by Da Vincis Vitruvian Man
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Units

• Each measurement must carry some unit of
  measurement (unless it is a dimensionless
  quantity well get to this soon).
• Numbers without units are meaningless.
• I am 72 tall
• 72 what? Fingers, handbreadths, inches,
  centimeters??

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Units
3. Units can be algebraically manipulated also,
conversion between units is accommodated. Factor
-Label Method Convert 16 miles per hour to
kilometers per second
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Units
4. Arithmetic manipulations between terms can
take place only with identical units. 3in 2in
5in 3m 2m 5m 3m 2in ? (use factor-label
method)
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2nd great unification of physics for
electromagnetism work (1st was Newton)
Dimensions are intrinsic to the variables
themselves
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Derived Base
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Dimensional Analysis
Fundamental Rules 1. Dimensions can be
algebraically manipulated.
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Dimensional Analysis
Fundamental Rules 2. All terms in an equation
must reduce to identical primitive (base)
dimensions.
Homogeneous Equation
Dimensional Homogeneity
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Dimensional Analysis
Opening Exercise 2
Non-homogeneous Equation
Dimensional Non-homogeneity
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Dimensional Analysis
Uses 1. Check consistency of equations
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Dimensional Analysis
Uses 2. Deduce expressions for physical
phenomena. Example What is the period of
oscillation for a pendulum? We predict that the
period T will be a function of m, L, and g
(time to complete full cycle)
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Dimensional Analysis
1. 2. 3. 4. 5. 6.
power-law expression
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Dimensional Analysis
6. 7. 8. 9.
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Dimensional Analysis
Uses 2. Deduce expressions for physical
phenomena. What weve done is deduced an
expression for period T. 1) What does it
mean that there is no m in the final
function? 2) How can we find the constant
C?
The period of oscillation is not dependent upon
mass m does this make sense? Yes, regardless of
mass, all objects on Earth experience the same
gravitational acceleration
Further analysis of problem or experimentally
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Dimensional Analysis
Uses 2. Deduce expressions for physical
phenomena. Chalkboard Example A mercury
manometer is used to measure the pressure in a
vessel as shown in the figure below. Write an
expression that solves for the difference in
pressure between the fluid and the
atmosphere.
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Dimensionless Quantities

• Few physical problems can be solved analytically.
  We often need to perform experiments to fully
  describe natural phenomena.
• Dimensional Reasoning then gives way to
• Dimensionless Quantities.
• Dimensional quantities can be made
  dimensionless by normalizing them with
  respect to another dimensional quantity of the
  same dimensionality.
• Ex. strain, percent, relative error, Reynolds ,
  Froude , etc.

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Dimensionless Quantities
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Dimensionless Quantities

• Dimensionless quantities can be defined as a
  quantity with the dimensions of 1 no M, L, T.
• Can be regarded as a ratio, percent
• Useful Properties
• Dimensionless variables/equations are independent
  of units
• Relative importance of terms can be easily
  estimated
• Scale is automatically built into the
  dimensionless expression
• Reduces many problems to a single, normalized
  problem

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Dimensionless Quantities
Example 1 Consider the steady flow of a fluid
through a pipe. An important characteristic of
this system, particularly to an engineer
designing a pipeline, is the pressure drop per
unit length that develops along the pipe as a
result of friction. Although this appears to be
a simple problem, it cannot generally be solved
analytically. Why? After an educated
prediction of factors affecting the system, the
pressure drop will be a function of 4 properties
pipe diameter, fluid density and viscosity, and
fluid velocity. In other words, designing an
experiment to hold any of these constant while
altering the others will take much time and money.
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Dimensionless Quantities
Example 1 Lets first attempt a dimensional
analysis of the problem and see where that gets
us Here is where our problem with analysis
lies. We have too many powers and will not have
enough equations. Remember, well only have 3
equations, at most, given by our 3 base
dimensions MLT. So what do we do?
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Dimensionless Quantities
Example 1 Buckingham Theorem If an
equation involving k variables is dimensionally
homogeneous, it can be reduced to a relationship
among k r independent dimensionless products,
where r is the minimum of reference dimensions
required to describe the variables. In our
problem, r 3 (MLT), and k 5 (
) Therefore, k r 2, so 2
dimensionless products will define our
problem. (For ALL PROBLEMS, if k r 1, then
dimensional analysis works)
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Dimensionless Quantities
Example 1 Step 1 Step 2 Step 3
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Dimensionless Quantities
Example 1 Step 4 Step 5 Step 6
Dependent variable always first Pick
other terms based on MLT simplicity
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Dimensionless Quantities
Example 1 Step 6 Step 7 Step 8
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Dimensionless Quantities
Example 1 Step 4 Step 5 Step 6
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Dimensionless Quantities
Example 1 Step 6 Step 7 Step 8
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Dimensionless Quantities
Example 1 Finally
Only experimentation will provide the form of the
function Phi
Possible because pi terms are dimensionless
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Dimensionless Quantities
Example 1 Whats the point?!?! We can now
compare those two pi terms in a meaningful way.
Where, originally, we had five variables to
assess, we now have two. Dimensionless quantities
often play an important, recurring role in
Engineering
The Reynolds
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Dimensionless Quantities
Example 2 Chalkboard Example A thin
rectangular plate having a width w and a height h
is located so that it is normal to a moving
stream of fluid. After consideration, we assume
the drag that the fluid exerts on the plate is a
function of w and h, the fluid viscosity and
density, and the velocity of the fluid
approaching the plate. Determine a suitable set
of pi terms to study this problem experimentally.
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Similarity, Modeling, and Scaling

• 3 types of similarity
• Geometric similarity linear dimensions are
  proportional, angles are the same
• Kinematic similarity includes proportional time
  scales (i.e., velocity is similar)
• Dynamic similarity includes force scale
  similarity (i.e., inertial, viscous, buoyancy,
  surface tension, etc.)

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Similarity, Modeling, and Scaling
Movies sometimes they look real, other
times something is not quite right any of the
three above similarities Distorted Model when
any of the three required similarities is
violated, the model is distorted. What movies
showcase accurate or distorted models? Titanic,
The Matrix, King Kong, Power Rangers, Star Wars
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Similarity, Modeling, and Scaling
This failed and abandoned Hydraulic Model of the
Chesapeake Bay (largest indoor hydraulic model in
the world) covered many parameters but failed
to model tides. Sometimes its necessary
to violate geometric similarity A 1/1000 scale
model of the Chesapeake Bay is 10x as deep as it
should be because the real Bay is so shallow that
the average depth would be 6mm too shallow to
exhibit stratified flow.
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Similarity, Modeling, and Scaling

• Dimensionless numbers (e.g., ratios and pi
  terms) make modeling simple.
• Dimensionless is independent of units /
  scale.
• Keep dimensionless s equals, your model is an
  accurate representation

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Similarity, Modeling, and Scaling
Chalkboard Example Whats the biggest elephant
on the planet?