swtjc stem – engr 1201 content goal 12 introducing mathematical models mathematical models are...

SWTJC STEM – ENGR 1201

Content Goal 12

Introducing Mathematical Models

Mathematical models are used extensively in engineering design. They provide a way to configure, simulate and test physical systems before actually building prototypes. Models come in several types:

•Traditional model

•Graphical model

•Object model

•Real-time model

•3D animated model


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Structure of a Model

Model consists of the following parts:

Concept - An idea that qualitatively describes the thing to be modeled.

Principle - A physical law that governs how the thing behaves.

Relation - A mathematical formula arising out of the physical law that quantitative describes the thing.

Property - Physical characteristics of the thing that can be measured and used in the formula.


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Example of a Model

Consider a car moving in a straight line at a constant speed.

Concept - Uniform rectilinear motion

Principle - The ratio of distance moved to time elapsed is a constant.

Relation - A formula given by d / t = s, a constant where d = distance, t = elapsed time, and s = speed.

Property - d is distance measured in miles (mi), t is elapsed time measured in hours (hr), and s is speed measured in miles per hour (mi/hr).


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Definition of Concept

A concept is the highest level of abstraction of an object. It is "a mental impression or image, a general notion or idea". Concepts are usually subjective; they are qualitative, rather than objective. An example concept is that of a "tree". We can easily picture it in our mind, but the specifics are left to each person's imagination.


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Definition of Physical Property

A physical property, on the other hand, is "a measurable characteristic or quantity of a thing or system. It is either measurable directly or through equations relating other measurable characteristics." Although similar to a concept, a property is objective and quantitative. The physical properties of a tree are such things as its height, girth, genetic makeup, and expiration rate.

18 m

1.2 m (direct)

Cross-section area

= D2/4 = (1.2m)2/4

= 1.13 m2 (indirect)


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Definition of Principle

A principle is "a fundamental truth or law of nature by which something operates". A principle is often derived from the application of one or more laws to a specific physical situation in which certain assumptions have been made.

Examples of principles:

•Laws of Linear Motion•Newton's Laws of Motion•Ohm's Law


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Definition of Relation

A relation is "a mathematical extension of one or more principles".

d = s . t (distance equals speed times time) is a relation of the physical properties of a moving object that follows from the Laws of Linear Motion.

F = m . a (force equals mass times acceleration) is a relation of the physical properties of a mass system that follows from Newton's Laws of Motion.

V = R . I (voltage equals resistance times current) is a relation of the physical properties of an electrical system that follows from Ohm's Law.


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Levels of Abstraction

Highest Abstraction Lowest Abstraction

Concept Principle Relation Property

Idea Thing

18 m

The “idea” of a tree. A “thing”, the tree’s height.


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Example Concept – Rectilinear Motion


Uniform rectilinear motion

Laws of Linear Motion

d = s . t distance, d

s

d

t = t1 - t0

t1t0

t0

t1


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Example Concept– Electric Circuits


Electric circuit Ohm's Law V = R . I voltage, V


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Example Concept – Force Summation


Forces in static equilibrium

Newton's First Law

F1 + F2 + F3 = 0 force, F

F3 = 12 lbF2 = 10 lb

F1 = 18 lb


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Define Measurement

Measurement is defined as "the process of quantifying a physical property by comparing it to a specified numerical standard".

"In physical science the first essential step in the direction of learning any subject is to find principles of numerical reckoning and practicable methods for measuring some quality connected with it. I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely in your thoughts advanced to the state of Science, whatever the matter may be." - Lord Kelvin


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Measurement – Made & Reported

A measurement is first made and then reported.

The process of "making" a measurement may be as simple as using a plastic ruler to measure a length of a pencil, or as complex as measuring the speed of light through a crystal in a scientific laboratory.

Historically, measurements were accomplished with mechanical instruments with results read on a continuous, analog scale. Today, many measurements are made using electronic sensors and reported digitally.


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Measurement – Components

A measurement consists of two components:

• Numeric value • Measurement unit

65.8 meters

Numericvalue

MeasurementUnit


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Measurement – Numeric Value

The numeric value of a measurement is "a quantity found by comparing the physical property to be measured to a standard".

1 literstandard

2 liter

A 20 cm ruler is 0.20 times as long as a 1 m standard.

1 meter standard (meter stick)

0.20 meter

A 2 l (liter) container is 2 times as large as a 1 liter standard.

The numeric value of a measurement can be expressed in one of three ways:

1. U. S. standard decimal notation; e.g.. 34,143.65 m

2. Scientific notation; e.g.., 3.414365.104 m

3. Engineering notation; e.g.., 34.14365 .103 m


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Measurement – Numeric Value


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Numeric Value – Report U.S. Standard

1. U. S. standard decimal notation where the comma ( , ) is used to indicate each third order of magnitude and the period ( . ) is used to indicate the decimal position. An example would be 32,143.65. It should be noted that a decimal fraction must always be written with a zero before the period as in 0.593, never as .593.

32,143.65

Comma

Period (decimal point)

0.593

Leading zero


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Numeric Value – Report Scientific

2. Scientific notation consisting of the product of a decimal number between 1 and 9.999... (called the mantissa) and a power of 10.

32,143.65 = 3.214365 · 104

One digit to leftof decimal

Times a power of ten


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Numeric Value – Report Engineering

3. Engineering notation is similar to scientific notation with the provision that the power of 10 is expressed as a multiple of 3 with the decimal number chosen appropriately between 1 and 999.999.... As we will see later, engineering notation accommodates the practice of expressing metric units in third order of magnitude steps; i.e., micro (10-6), milli (10-3), kilo (103), mega (106), etc

0.5931 l = 593.1 · 10-3 l = 593.1 ml

1-3 digit(s) to leftof decimal

Exponent a multipleof three


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Numeric Value – Significant Digits

Report only meaningful digits, or more properly significant digits. This is accepted to mean that we report all accurately known digits and the first digit that may contain an error.

Using scientific notation to express significant digits is preferred since, by definition, the mantissa (the numerical portion) may only contain significant digits.

32,143.65 m = 3.214365 · 104 m

Mantissa – Onlysignificant digits!


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Measurement – Significant Digits

5 6

5.74

Exact Estimate3 significant digits!


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Numeric Value – Ambiguous Digits

The number 12,300 could have three, four, or five significant digits depending on whether the zeros are placeholders. Reporting it in scientific notation as 1.230 · 103 clears up the matter quite nicely; only the first zero is significant!

Mantissa shows4 significant digits!

12,300 = 1.230 · 104

A placeholderdigit only

THE METER STICK


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Unit – Intra-unit Conversion

The measurement unit is "dictated by the physical property being measured and will vary depending on the nature of the physical property and the size of the measured quantity". For instance, the capacity of a test tube with a measured value of 0.01 liters might be more appropriately reported as 10 milliliters or 10 cubic centimeters. This is referred to an intra-unit conversion; i.e., “conversion within a measurement system”.

Capacity:0.01 l10 ml10 cc Most appropriate unit

People like to dealwith numbers from

1 to 999!

intra-unit conversion


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Unit– Inter-unit Conversion

Sometimes the property is measured in one system, then reported in another. A car's speedometer shows 65 mi/hr when stopped by a police officer in Mexico. The driver reports that his speed was 105 km/hr. He has performed an inter-unit conversion; i.e., “conversion between measurement systems”.

65 mi/hr = 105 km/hr

inter-unit conversion


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Unit– Rectilinear Motion

Concept Principle Property Unit

Uniform rectilinear motion

Ratio distance to time is a constant.

velocity v0 m/s (SI)

ft/s (USCS)

v0

d

t = t1 - t0

t1t0


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Unit – Electric Current


Electric current flowing in a circuit

Ohm's Law resistance R ohms (SI)

ohms (USCS)


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Unit – Force Summation


Forces in static equilibrium

Newton's First Law

force F N (SI)

lb (USCS)

F3F2

F1


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Measurement – Research Question

What are the standards used in each example measurements above? Is the standard a physical object or a laboratory method?

A good place to look is at the National Institute of Standards and Technology (NIST).

swtjc stem – engr 1201 content goal 12 introducing mathematical models mathematical models are...

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