Problem Solving: Methods, Formats, and Conventions Introduction to Mechanical Engineering The University of Texas-Pan American College of Science and Engineering

Objectives  Describe the methods used for solving engineering problems: Analytic and Creative.  Explain how to determine significant figures in calculations.  Explain the proper use of scientific and engineering notation.  Explain the difference between accuracy and precision.  Explain the different types of errors that may be encountered in engineering measurements. Assignment: See HW#2 in my website:

Introduction  The most outstanding trait of engineers is their ability to solve problems.  Successful problem solving involves following a sequence of steps that lead to a logical solution.  It also involves careful documentation of the problem and its solution.  Reliable results are achieved by paying careful attention to physical quantities, units of measurement, numeric measurements and understanding accuracy and precision.

Introduction  To become a successful engineer one must:  Develop good work habits.  Adopt a positive attitude towards school and engineering.  Take pride in his work.

Analytic Problem Solving Technique  Read the entire problem carefully.  Restate the problem, diagram and describe.  Determine additional data needed.  Determine the quantities that must be calculated.  State assumptions.  Apply theory and equations.  Complete calculations.  Check or verify the numerical answer and units and discuss your findings.

Standard Format for Engineering Calculations  Use significant figures (if required).  Use scientific or engineering notation for large numbers and small decimal fractions.  Solution of a problem involving multiplication, division and/or unit conversion is easier to follow when is written as multiplication of fractions.

Significant Figures (intro)  Numbers are the basis of data collected in the field or a laboratory, numerical calculations, statistical analysis, etc…  Engineers and scientists must master dealing with numbers in terms of significant figures, rounding, errors, accuracy and precision.

Significant Digits  The rule to follow in determining significant figures (digits) in a number is that all digits are significant except zeros necessary to locate the decimal point.

Example: Determining Number of Significant Figures Number Significant Figures

Determine the Number of Significant Figures NumberSignificant Figures

Rounding of Numbers  The generally accepted rule for rounding is to round up if the digit to the right of the last significant digit is 5 or greater. Otherwise, round down.

Example of Rounding of Numbers Original NumberRounded NumberFigures Desired

Try Rounding the Following Numbers Original NumberRounded NumberFigures Desired

Arithmetic Operations and Significant Figures  The final result of an engineering calculation should be rounded to account for the significant figures of the original numbers.  The rounding procedure is different for addition/subtraction and multiplication/division.

Arithmetic Operations and Significant Figures  The result of a multiplication/division calculation, including taking numbers to a power or taking the root of a number, should be rounded to the number of significant figures exhibited by the original number with the least number of significant figures.  Rounding should be done after the calculation is complete.

Multiplication/Division Examples 36.5 x = = / 31.2 = = = = (1/5) = = 6.73

Arithmetic Operations and Significant Figures  When whole numbers are added and subtracted, all numbers are considered significant.  When decimal numbers are added or subtracted, the answer should be rounded to the number of digits to the right of the decimal point exhibited by the original number with the least digits to the right of the decimal point.

Addition/Subtraction Examples = 5, = 5, = = 17.2

Arithmetic Operations and Significant Figures  When addition/subtraction operations are embedded in multiplication/division calculations, the addition/subtraction should be undertaken first and rounded. Then the multiplication/division is carried out and final rounding is completed.

Embedded Operations I ( ) / 16.2 = / 16.2 = / 16.2 = = 9.95

Embedded Operations II {( ) / 2.935} + (16.12 – 2.521)(1.0032) = = / x = 16.0 / x = = = 19.09

Scientific and Engineering Notation  In scientific notation a number is written as n.nnn X 10 N where N is a positive or negative integer.  In engineering notation a number is written as n.nnn X10 E where E is a positive or negative multiple of three.  Numbers larger than 999 and less than 0.01 should be written using scientific notation.

Accuracy and Precision  Accuracy is a measure of how close a measurement is to the true value.  Precision refers to the reproducibility of a measurement.

Accuracy and Precision

Accidental and Systemic Error  Accidental errors are random (will have positive and negative values).  Systemic errors are due to a characteristic of the measuring system and always have the same sign. Systemic errors are usually due to an error or inaccuracy in calibration of the measuring devise or use of the device under conditions different form calibration conditions.

The Creative Method  Many engineering problems are open-ended problems.  Just as with analytic problem solving, developing a systematic approach that uses creativity will pay dividends in better solutions.  A creative problem-solving process focuses on answering these five questions:  What is wrong?  What do we know?  What is the real problem?  What is the best solution?  How do we implement the solution?

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