Dimensional Reasoning

1. Dimensional Reasoning 1. What are the units of Young’s 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

2. 1. What are the units of Young’s Modulus?

3. 2. Are these equations correct?

4. 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 surface…This was far too close and the spacecraft burnt up due to friction with the atmosphere.” – BBC News

5. 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

6. 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.

7. 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

8. 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

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

10. Units provide the scale to quantify measurements MASS

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

12. Units provide the scale to quantify measurements ELECTRIC CURRENT

13. Units provide the scale to quantify measurements THERMODYNAMIC TEMPERATURE

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

15. Units provide the scale to quantify measurements LUMINOUS INTENSITY

16. 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

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

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

22. Derived Base

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

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

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

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

27. 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)

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

29. Dimensional Analysis 6. 7. 8. 9.

30. 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

31. 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.

32. 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.

33. Dimensionless Quantities

34. 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

35. 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.

36. 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?

37. 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)

38. Dimensionless Quantities Example 1: Step 1: Step 2: Step 3:

39. Dimensionless Quantities Example 1: Step 4: Step 5: Step 6: Dependent variable always first; Pick other terms based on MLT simplicity

40. Dimensionless Quantities Example 1: Step 6: Step 7: Step 8:

41. Dimensionless Quantities Example 1: Step 4: Step 5: Step 6:

42. Dimensionless Quantities Example 1: Step 6: Step 7: Step 8:

43. Dimensionless Quantities Example 1: Finally: Only experimentation will provide the form of the function Phi Possible because pi terms are dimensionless

44. 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 #

45. 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.

46. 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.)

47. 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

48. 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 stratified flow.

49. 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

50. Similarity, Modeling, and Scaling Chalkboard Example: What’s the biggest elephant on the planet?