Dimensional Reasoning 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 DRILL

1. Is either of these equations correct? F: kg*m / s 2 m: kg r: m v: m / s a: m / s 2 kg*m / s 2 = kg*m 2 / s 2 m 2 = kg*m 2 s 2 m 2 kg*m / s 2 = kg / s 2

2. What is the common problem in the two images below? $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 Pounds-force Newtons-force UNITS!

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/tP = 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

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 provide the scale to quantify measurements

Units 1.A scale is a measure that we use to characterize some object/property of interest. Let’s characterize this plot of farmland: x y 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

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 30 o. After walking 40 feet closer to the tree, the backpacker notes that the angle of elevation is 60 o. 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 30 o. After walking 50 feet closer to the tree, the backpacker notes that the angle of elevation is 60 o. 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?

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

Derived Base CharacteristicDimension SI (MKS)English LengthLmfoot MassMkgslug TimeTss AreaL2L2 m2m2 ft 2 VolumeL3L3 Lgal VelocityLT -1 m/sft/s AccelerationLT -2 m/s 2 ft/s 2 ForceMLT -2 Nlb Energy/WorkML 2 T -2 Jft-lb PowerML 2 T -3 Wft-lb/s or hp PressureML -1 T -2 Papsi ViscosityML -1 T -1 Pa*slb*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. Dimensional Homogeneity Homogeneous Equation

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

Dimensional Analysis Uses: 1. Check consistency of equations:

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)

power-law expression Dimensional Analysis

Dimensional Analysis

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

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.

QUIZ REVIEW Topics Covered: 1.The properties of measurements 2.Difference between base and derived units 3.SI and English systems – Quality / Quantity matching 4.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)mghP = 2W / t W = WorkP = Power m = mass of objectW = Work h = height object is liftedt = 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, v p 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.)

3 types of similarity: 1.Geometric similarity – linear dimensions are proportional, angles are the same Modeling: Similarity and Scale

1.Geometric similarity 2.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 Modeling: Similarity and Scale More ViscousLess Viscous

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

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 Modeling: Similarity and Scale Homework