Dimensional Reasoning
DRILL
1. Is either of these equations correct?
1
v2
d  at
F m 2
2
d
2. What is the common problem in the two
examples below?
Sign outside New Cuyama, CA
1998 Mars Polar Orbiter
1. Is either of these equations correct?
2
v
F m 2
d
F: kg*m / s2
m: kg
d: m
v: m / s
a: m / s2
kg*m / s2 = kg*m2 / s2
m2
= kg*m2
s2 m2
kg*m / s2 = kg / s2
1
d  at
2
2. What is the common problem in the two images below?
Pounds-force  Newtons-force
UNITS!
$125mil error: “Instead of passing about 150 km above the Martian
atmosphere before entering orbit, the spacecraft actually passed about 60
km above the surface…This was far too close and the spacecraft burnt up
due to friction with the atmosphere.” – BBC News
Dimensional Reasoning
Lecture Outline:
1. Units – base and derived
2. Units – quantitative considerations
3. Dimensions and Dimensional Analysis
– fundamental rules and uses
Dimensional Reasoning
Measurements consist of 2 properties:
1. a quality or dimension
2. a quantity expressed in terms of “units”
Let’s 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.
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
SUMMARY OF THE 7 FUNDAMENTAL SI UNITS:
1. LENGTH
- meter
2. MASS
- kilogram
3. TIME
- second
4. ELECTRIC CURRENT
- ampere
5. THERMODYNAMIC TEMPERATURE - Kelvin
6. AMOUNT OF MATTER
- mole
7. LUMINOUS INTENSITY
- candela
Quality (Dimension)
Quantity – Unit
Units provide the scale to quantify measurements
LENGTH
YARDSTICK
METER STICK
Units provide the scale to quantify measurements
MASS
Units provide the scale to quantify measurements
TIME
ATOMIC CLOCK
Units provide the scale to quantify measurements
ELECTRIC CURRENT
Units provide the scale to quantify measurements
THERMODYNAMIC TEMPERATURE
Units provide the scale to quantify measurements
AMOUNT OF SUBSTANCE
Units provide the scale to quantify measurements
LUMINOUS INTENSITY
Units
1. A scale is a measure that we use to characterize
some object/property of interest.
Let’s 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
Units
7 historical units of
measurement as defined
by Vitruvius
Written ~25 B.C.E.
Graphically depicted by
Da Vinci’s Vitruvian Man
Units
2. Each measurement must carry some unit of
measurement (unless it is a dimensionless quantity).
Numbers without units are meaningless.
I am “72 tall”
72 what? Fingers, handbreadths, inches, centimeters??
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:
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)
“2nd great unification of physics”
for electromagnetism work
(1st was Newton)
Dimensions are intrinsic to the variables themselves
Base
Derived
Characteristic
Length
Mass
Time
Area
Volume
Velocity
Acceleration
Force
Energy/Work
Power
Pressure
Viscosity
Dimension
L
M
T
L2
L3
LT-1
LT-2
MLT-2
ML2T-2
ML2T-3
ML-1T-2
ML-1T-1
SI
(MKS)
m
kg
s
m2
L
m/s
m/s2
N
J
W
Pa
Pa*s
English
foot
slug
s
ft2
gal
ft/s
ft/s2
lb
ft-lb
ft-lb/s or hp
psi
lb*slug/ft
Dimensional Analysis
Fundamental Rules:
1. Dimensions can be algebraically manipulated.
Dimensional Analysis
Fundamental Rules:
2. All terms in an equation must reduce to identical
primitive (base) dimensions.
d  d o  vot  at
1
2
2
Homogeneous
Equation
L
L 2
L  L T  2 T
T
T
Dimensional
Homogeneity
Dimensional Analysis
Opening Exercise #2:
Non-homogeneous
Equation
Dimensional
Non-homogeneity
Dimensional Analysis
Uses:
1. Check consistency of equations:
1
d  at
2
d  d o  vot  12 at 2
Dimensional Analysis
Uses:
2. Deduce expressions for physical phenomena.
Example: What is the period of oscillation for a pendulum?
(time to complete full cycle)
We predict that the period T will be a function of m, L,
and g:
Dimensional Analysis
1.
2.
3.
4.
5.
6.
power-law expression
Dimensional Analysis
6.
7.
8.
9.
Dimensional Analysis
Uses: 2. Deduce expressions for physical phenomena.
What we’ve done is deduced an expression for period T.
1) What does it mean that there is no m in the final
function?
The period of oscillation is not dependent upon mass m –
does this make sense?
2) How can we find the constant C?
Further analysis of problem or experimentally
Uses:
Dimensional Analysis
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.
Dimensionless Quantities
1. Few physical problems can be solved analytically. We
often need to perform experiments to fully describe
natural phenomena.
2. Dimensional Reasoning then gives way to…
Dimensionless Quantities.
3. 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.
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
1.
2.
3.
4.
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
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.
Dimensionless Quantities
Example 1:
Let’s 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, we’ll only have 3 equations, at most,
given by our 3 base dimensions MLT.
So what do we do?
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)
Dimensionless Quantities
Example 1:
Step 1:
Step 2:
Step 3:
Dimensionless Quantities
Example 1:
Step 4:
Step 5:
Step 6:
Dependent variable always first;
Pick other terms based on MLT simplicity
Dimensionless Quantities
Example 1:
Step 6:
Step 7:
Step 8:
Dimensionless Quantities
Example 1:
Step 4:
Step 5:
Step 6:
Dimensionless Quantities
Example 1:
Step 6:
Step 7:
Step 8:
Dimensionless Quantities
Example 1:
Finally:
Only experimentation
will provide the form of
the function Phi
Possible because pi
terms are dimensionless
Dimensionless Quantities
Example 1:
What’s 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 #
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.
Similarity, Modeling, and Scaling
3 types of similarity:
1. Geometric similarity – linear dimensions are
proportional, angles are the same, roughness is the
same
2. Kinematic similarity – includes proportional time scales
(i.e., velocity and acceleration are similar)
3. Dynamic similarity – includes force scale similarity (i.e.,
inertial, viscous, buoyancy, surface tension, etc.)
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
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 it’s 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 correct flow.
Similarity, Modeling, and Scaling
1.
Dimensionless numbers (e.g., ratios and pi terms) make
modeling simple. We simply equate pi terms.
2. Dimensionless # is independent of units / scale.
3. Keep dimensionless #s equals, your model is an accurate
representation
Similarity, Modeling, and Scaling
Chalkboard Example:
What’s the biggest elephant on the planet?