Transcript Physical Models

SWTJC STEM – ENGR 1201
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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SWTJC STEM – ENGR 1201
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.
Cross-section area
18 m
= D2/4 = (1.2m)2/4
= 1.13 m2 (indirect)
1.2 m (direct)
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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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Idea
Highest Abstraction
Levels of Abstraction

Thing
Lowest Abstraction
Concept  Principle  Relation  Property
18 m
The “idea” of a tree.
A “thing”, the tree’s height.
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SWTJC STEM – ENGR 1201 Example Concept – Rectilinear Motion
Concept
Principle
Uniform rectilinear Laws of Linear
motion
Motion
t0
Relation
Property
d = s .t
distance, d
t = t1 - t0
t1
s
d
t0
t1
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Concept
Electric circuit
Principle
Ohm's Law
Example Concept– Electric Circuits
Relation
Property
V=R.I
voltage, V
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SWTJC STEM – ENGR 1201 Example Concept – Force Summation
Concept
Forces in static
equilibrium
Principle
Relation
Newton's First F1 + F2 + F3 = 0
Law
F2 = 10 lb
Property
force, F
F3 = 12 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
Numeric
value
Measurement
Unit
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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".
A 2 l (liter) container is 2 times as
large as a 1 liter standard.
1 liter 2 liter
standard
1 meter standard (meter stick)
A 20 cm ruler is 0.20 times
as long as a 1 m standard.
0.20 meter
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Measurement – Numeric Value
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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SWTJC STEM – ENGR 1201 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
0.593
Leading zero
Period (decimal point)
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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
Times a power of ten
One digit to left
of decimal
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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 left
of decimal
Exponent a multiple
of 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 – Only
significant digits!
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Measurement – Significant Digits
5
6
5.74
3 significant digits!
Exact Estimate
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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!
12,300 = 1.230 · 104
A placeholder
digit only
Mantissa shows
4 significant digits!
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”.
People like to deal
with numbers from
Capacity:
1 to 999!
0.01 l
10 ml
Most appropriate unit
10 cc
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 Ratio distance to velocity v m/s (SI)
0
motion
time is a constant.
ft/s (USCS)
t = t1 - t0
v0
d
t0
t1
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Concept
Principle
Electric current
Ohm's Law
flowing in a circuit
Unit – Electric Current
Property
Unit
resistance R ohms (SI)
ohms (USCS)
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Concept
Forces in static
equilibrium
Unit – Force Summation
Principle
Property
Newton's First force F
Law
F2
Unit
N (SI)
lb (USCS)
F3
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).
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