Dimensional Reasoning

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

2

r

vmF atd

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1

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

2. Are these equations correct?

2

2

r

vmF atd

2

1

3. 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

Dimensional Reasoning

Lecture Outline:

1. Units – base and derived2. Units – quantitative considerations3. Dimensions and Dimensional Analysis

– fundamental rules and uses4. Dimensionless Quantities5. Scaling, Modeling, and Similarity

Dimensional Reasoning Measurements consist of 2 properties:

1. a quality or dimension2. 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


MASS


TIME

ATOMIC CLOCK


ELECTRIC CURRENT


THERMODYNAMIC TEMPERATURE


AMOUNT OF SUBSTANCE


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:

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 = 5in3m + 2m = 5m3m + 2in = ?

(use factor-label method)

Dimensions are intrinsic to the variables themselves

“2nd great unification of physics” for electromagnetism work (1st was Newton)

Der

ived

Bas

e

Characteristic DimensionSI

(MKS) English

Length L m foot

Mass M kg slug

Time T s s

Area L2 m2 ft2

Volume L3 L gal

Velocity LT-1 m/s ft/s

Acceleration LT-2 m/s ft/s

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

Dimensional Analysis

Fundamental Rules:1. Dimensions can be algebraically manipulated.


Fundamental Rules:2. All terms in an equation must reduce to identical

primitive (base) dimensions.

221 attvdd oo

22T

T

LT

T

LLL

Dimensional Homogeneity

Homogeneous Equation


Opening Exercise #2:

Dimensional Non-homogeneity

Non-homogeneous Equation


Uses:1. Check consistency of equations:

atd2

1

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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 1.

2.

3.

4.

5.

6.


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

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.

Dimensionless Quantities1. Few physical problems can be solved analytically. We

often need to perform experiments to fully describe natural phenomena.

2. Dimensional Reasoning then gives way to… Dimensionless Quantities.

3. 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.

Dimensionless Quantities

Dimensionless QuantitiesDimensionless quantities can be defined as a quantity with the dimensions of “1” – no M, L, T.

Can be regarded as a ratio, percent

Useful Properties1. Dimensionless variables/equations are independent of units2. Relative importance of terms can be easily estimated3. Scale is automatically built into the dimensionless expression4. Reduces many problems to a single, normalized problem

Dimensionless QuantitiesExample 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.


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?


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)


Step 1:

Step 2:

Step 3:


Step 4:

Step 5:

Step 6:

Dependent variable always first; Pick other terms based on MLT simplicity


Step 6:

Step 7:

Step 8:


Step 4:

Step 5:

Step 6:


Step 6:

Step 7:

Step 8:


Finally:

Possible because pi terms are dimensionless

Only experimentation will provide the form of

the function Phi


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 #

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.

Dimensionless Quantities

Similarity, Modeling, and Scaling3 types of similarity:

1. Geometric similarity – linear dimensions are proportional, angles are the same

2. Kinematic similarity – includes proportional time scales (i.e., velocity is similar)

3. Dynamic similarity – includes force scale similarity (i.e., inertial, viscous, buoyancy, surface tension, etc.)

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


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.

Similarity, Modeling, and Scaling1. Dimensionless numbers (e.g., ratios and pi terms) make

modeling simple.2. Dimensionless # is independent of units / scale.3. Keep dimensionless #s equals, your model is an accurate

representation


Chalkboard Example:

What’s the biggest elephant on the planet?

Documents

Dimensional Reasoning