Dimensional Reasoning DRILL 1. Is either of these equations correct? 2. What is the common problem in the two examples below? http://en.wikipedia.org/wiki/New_Cuyama,_California http://news.bbc.co.uk/2/hi/science/nature/514763.stm Sign outside New Cuyama, CA 1998 Mars Polar Orbiter

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

UNITS! 2. What is the common problem in the two images below? Pounds-force Newtons-force UNITS! http://en.wikipedia.org/wiki/New_Cuyama,_California http://news.bbc.co.uk/2/hi/science/nature/514763.stm $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

Units 7 historical units of measurement as defined by Vitruvius Written ~25 B.C.E. Graphically depicted by Da Vinci’s Vitruvian Man http://en.wikipedia.org/wiki/File:Vitruvian_Man_Measurements.png

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

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 http://en.wikipedia.org/wiki/New_Cuyama,_California http://news.bbc.co.uk/2/hi/science/nature/514763.stm

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: LENGTH - meter MASS - kilogram TIME - second ELECTRIC CURRENT - ampere THERMODYNAMIC TEMPERATURE - Kelvin AMOUNT OF MATTER - mole 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 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:

(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)

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?

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?

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

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

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

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

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

Dimensional Analysis Uses: 1. Check consistency of equations:

(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) http://upload.wikimedia.org/wikipedia/en/4/4f/SimplePendulum01.JPG

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? 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 http://upload.wikimedia.org/wikipedia/en/4/4f/SimplePendulum01.JPG 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. Sonntag and Borgnakke – Introduction to Engineering Thermodynamics

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

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?

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

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.

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.