1. Dimensional Reasoning
DRILL
1. Is either of these equations correct?
2. What is the common problem in the two examples below?
Sign outside New Cuyama, CA
1998 Mars Polar Orbiter

2. 1. Is either of these equations correct?
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

3. UNITS! 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

4. Units 7 historical units of measurement as defined by Vitruvius
Written ~25 B.C.E.
Graphically depicted by Da Vinci’s Vitruvian Man

5. 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

6. Dimensional Reasoning
DRILL 2
Measurements consist of what 2 properties?
A quality or dimension
A quantity expressed in terms of units
(Magnitudes)
What are the two uses of Dimensional Analysis?
Check the consistency of equations
Deduce expressions for physical phenomena
Prove whether the following equations are consistent (homogeneous) using 2 methods: Units and Dimensions.
W = F/t P = W/t

7. 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

8. 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.

9. 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

10. 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

11. Units provide the scale to quantify measurements
LENGTH
YARDSTICK
METER STICK

12. Units provide the scale to quantify measurements
MASS

13. Units provide the scale to quantify measurements
TIME
ATOMIC CLOCK

14. Units provide the scale to quantify measurements
ELECTRIC CURRENT

15. Units provide the scale to quantify measurements
THERMODYNAMIC TEMPERATURE

16. Units provide the scale to quantify measurements
AMOUNT OF SUBSTANCE

17. Units provide the scale to quantify measurements
LUMINOUS INTENSITY

18. Units
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??

19. 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:

20. (use factor-label method)
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)

21. QUIZ Trig/Algebra QUIZ
Complete the quiz on Engineering Paper:
LETTER 3 ways of solving systems of equations.
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?
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?
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?

22. QUIZ Trig/Algebra QUIZ
Complete the quiz on Engineering Paper:
LETTER 3 ways of solving systems of equations.
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?
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?
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?

23. Dimensions are intrinsic to the variables themselves
“2nd great unification of physics” for electromagnetism work (1st was Newton)
Dimensions are intrinsic to the variables themselves

24. Derived Base Characteristic Dimension SI (MKS) English Length L m foot
Mass M kg
slug
Time T s
Area L2 m2
ft2
Volume L3
gal
Velocity
LT-1
m/s
ft/s
Acceleration
LT-2
m/s2
ft/s2
Force
MLT-2 N lb
Energy/Work
ML2T-2 J
ft-lb
Power
ML2T-3 W
ft-lb/s or hp
Pressure
ML-1T-2 Pa
psi
Viscosity
ML-1T-1
Pa*s
lb*slug/ft
Derived Base

25. Dimensional Analysis Fundamental Rules:
1. Dimensions can be algebraically manipulated.

26. Dimensional Homogeneity
Dimensional Analysis
Fundamental Rules:
2. All terms in an equation must reduce to identical primitive (base) dimensions.
Homogeneous Equation
Dimensional Homogeneity

27. Dimensional Analysis Opening Exercise #2: Non-homogeneous Equation
Dimensional Non-homogeneity

28. Dimensional Analysis
Uses:
1. Check consistency of equations:

29. (time to complete full cycle)
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)

30. Dimensional Analysis 1. 2. 3. 4. 5. 6.
power-law expression

31. Dimensional Analysis 6. 7. 8. 9.

32. 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?
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

33. 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.
Sonntag and Borgnakke – Introduction to Engineering Thermodynamics

34. QUIZ REVIEW Topics Covered: The properties of measurements
Difference between base and derived units
SI and English systems – Quality / Quantity matching
Problems – two uses of dimensional reasoning:
Check equation consistency
Deduce expressions for physical phenomena

35. QUIZ REVIEW Practice Problems:
What are the units of Force? What are the dimensions of force?
If Work = Force x Distance, what are the dimensions of work?
If Power is Work / Time, what are the dimensions of power?

36. QUIZ REVIEW Practice Problems:
Which of the equations below is consistent?
Which is correct?
W = (1/2)mgh P = 2W / t
W = Work P = Power
m = mass of object W = Work
h = height object is lifted t = time length

37. QUIZ REVIEW Practice Problems:
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.

38. No calculators allowed.
QUIZ
No calculators allowed.
Take out a pencil.
HOMEWORK
By dimensional analysis we deduced the following expression for the period of oscillation of a pendulum:
Design an experiment that would provide the necessary information to solve for the constant, C.