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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
r: 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
DRILL 2
1. Measurements consist of what 2 properties?
1. A quality or dimension
2. A quantity expressed in terms of units
3. (Magnitudes)
2. What are the two uses of Dimensional Analysis?
1. Check the consistency of equations
2. Deduce expressions for physical phenomena
3. Prove whether the following equations are consistent
(homogeneous) using 2 methods: Units and Dimensions.
W = F/t
P = W/t
Dimensional Reasoning
Lecture Outline:
1. Units – base and derived
2. Units – quantitative considerations
3. Dimensions and Dimensional Analysis
– fundamental rules and uses
4. Scaling, Modeling, and Similarity
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 –
we’ll get to this soon).
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)
QUIZ
Trig/Algebra
QUIZ
Complete the quiz on Engineering Paper:
1. LETTER 3 ways of solving systems of equations.
2. You work for a fencing company. A customer called this
morning, wanting to fence in his 1,320 square-foot garden. He
ordered 148 feet of fencing, but you forgot to ask him for the
width and length of the garden. What are the dimensions?
3. A backpacker notes that from a certain point on level ground,
the angle of elevation to a point at the top of a tree is 30o.
After walking 40 feet closer to the tree, the backpacker notes
that the angle of elevation is 60o. What is the height of the
tree?
4. At a joint conference of psychologists and sociologists, there
were 24 more psychologists than sociologists. If there were
90 participants, how many were from each profession?
QUIZ
Trig/Algebra
QUIZ
Complete the quiz on Engineering Paper:
1. LETTER 3 ways of solving systems of equations.
2. You work for a fencing company. A customer called this
morning, wanting to fence in his 260 square-foot garden. He
ordered 66 feet of fencing, but you forgot to ask him for the
width and length of the garden. What are the dimensions?
3. A backpacker notes that from a certain point on level ground,
the angle of elevation to a point at the top of a tree is 30o.
After walking 50 feet closer to the tree, the backpacker notes
that the angle of elevation is 60o. What is the height of the
tree?
4. At a joint conference of psychologists and sociologists, there
were 24 more psychologists than sociologists. If there were
90 participants, how many were from each profession?
“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? Yes, regardless of mass, all objects on
Earth experience the same gravitational acceleration
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.
QUIZ REVIEW
Topics Covered:
1.
2.
3.
4.
The properties of measurements
Difference between base and derived units
SI and English systems – Quality / Quantity matching
Problems – two uses of dimensional reasoning:
1. Check equation consistency
2. Deduce expressions for physical phenomena
QUIZ REVIEW
Practice Problems:
1. What are the units of Force? What are the dimensions
of force?
2. If Work = Force x Distance, what are the dimensions of
work?
3. If Power is Work / Time, what are the dimensions of
power?
QUIZ REVIEW
Practice Problems:
4. Which of the equations below is consistent?
Which is correct?
W = (1/2)mgh
W = Work
m = mass of object
h = height object is lifted
P = 2W / t
P = Power
W = Work
t = time length
QUIZ REVIEW
Practice Problems:
5. We have a wave traveling across a large body of water
such as the ocean. The wave has a well-defined
wavelength. The wavelength is reasonably long (20 cm
or more), but the wavelength is short compared to the
depth of the water. We want to know the speed of
propagation, vp of the wave. Intuition says that the
only relevant physical parameters are the wavelength λ,
the fluid density ρ, and the gravitational field strength
g. Deduce an expression relating the speed of
propagation to the relevant physical parameters.
Modeling: Similarity and Scale
Why we model:
1. Test design performance cheaply
2. Evaluate, promote, and sell the look of new construction
3. Closely imitate reality cheaply and easily (e.g.,
demonstrations, movies, etc.)
Modeling: Similarity and Scale
3 types of similarity:
1. Geometric similarity – linear dimensions are
proportional, angles are the same
Modeling: Similarity and Scale
1.
2.
Geometric similarity
Kinematic similarity – time scale is proportional
(i.e., geometry and velocity is similar)
Is this otto-engine animated model
kinematically similar?
Yes.
Although much slower than a real engine,
proportionality is accurate. IE, valves
open at correct piston position each cycle.
(Consider the explosion of Alderaan by the Death Star in Star Wars)
Modeling: Similarity and Scale
1. Geometric similarity – angles same, proportional lengths
2. Kinematic similarity – proportional time scale
3. Dynamic similarity – includes force scale similarity (i.e.,
inertial, viscous, buoyancy, surface tension, etc.)
Compare:
The Matrix
and
Mighty Morphin Power Rangers
New and Old King Kong
More Viscous Less Viscous
Modeling: Similarity and Scale
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
Modeling: Similarity and Scale
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 stratified flow.
Modeling: Similarity and Scale
Homework
Provide at least 1 movie example of each of the following:
1. Non-distorted geometric similarity
2. Distorted geometric similarity
3. Non-distorted kinematic similarity
4. Distorted kinematic similarity
5. Non-distorted dynamic similarity
6. Distorted dynamic similarity