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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 Content Goal 12 SWTJC STEM – ENGR 1201 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. Content Goal 12 SWTJC STEM – ENGR 1201 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). Content Goal 12 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. Content Goal 12 SWTJC STEM – ENGR 1201 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) Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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. Content Goal 12 SWTJC STEM – ENGR 1201 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. Content Goal 12 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 Content Goal 12 SWTJC STEM – ENGR 1201 Concept Electric circuit Principle Ohm's Law Example Concept– Electric Circuits Relation Property V=R.I voltage, V Content Goal 12 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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. Content Goal 12 SWTJC STEM – ENGR 1201 Measurement – Components A measurement consists of two components: • Numeric value • Measurement unit 65.8 meters Numeric value Measurement Unit Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 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) Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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! Content Goal 12 SWTJC STEM – ENGR 1201 Measurement – Significant Digits 5 6 5.74 3 significant digits! Exact Estimate Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 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 Content Goal 12 SWTJC STEM – ENGR 1201 Concept Principle Electric current Ohm's Law flowing in a circuit Unit – Electric Current Property Unit resistance R ohms (SI) ohms (USCS) Content Goal 12 SWTJC STEM – ENGR 1201 Concept Forces in static equilibrium Unit – Force Summation Principle Property Newton's First force F Law F2 Unit N (SI) lb (USCS) F3 F1 Content Goal 12 SWTJC STEM – ENGR 1201 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). Content Goal 12