(************** Content-type: application/mathematica ************** Mathematica-Compatible Notebook This notebook can be used with any Mathematica-compatible application, such as Mathematica, MathReader or Publicon. The data for the notebook starts with the line containing stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). 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For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. *******************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 27515, 866]*) (*NotebookOutlinePosition[ 28285, 892]*) (* CellTagsIndexPosition[ 28241, 888]*) (*WindowFrame->Normal*) Notebook[{ Cell["Lab 8: Masses and Centroids and and Moments, Oh My!", "Title", Background->RGBColor[0, 0, 1]], Cell[TextData[{ "Math 233\tFall 2001\t", StyleBox["Chapter 12, Sections 2 and 5", FontFamily->"Arial", FontSize->16, FontWeight->"Bold"], " \n", StyleBox["Modification of a lab written by Marie Vanisko", FontSize->12, FontSlant->"Italic"] }], "Subtitle", TextAlignment->Left, TextJustification->0], Cell[CellGroupData[{ Cell[TextData[StyleBox["Instructions!", FontColor->RGBColor[1, 0, 0]]], "Section"], Cell[TextData[{ "Work in ", StyleBox["groups of 2 or 3", FontWeight->"Bold"], " today, going through the entire notebook. If you have questions, flag me \ down. Write up your answers (complete sentences, please) for a ", StyleBox["group", FontWeight->"Bold"], " hand in on ", StyleBox["Monday", FontWeight->"Bold"], " of next week (", StyleBox["19 October)", FontWeight->"Bold"], ". \n\nHomework exercises that you should work on to build your skills \ are: ", StyleBox["Section 12.2 (p. 997)", FontWeight->"Bold"], ": 9, 19, 23, 32, 33. ", StyleBox["Section 12.5 (p. 1021)", FontWeight->"Bold"], ": 3, 5, 13, 15abc. ", StyleBox["Section 12.6 (p. 1036)", FontWeight->"Bold"], ": 67, 75, 79 (ignore radius of gyration), 83", ". These problems won't be handed in, but you ", StyleBox["should", FontWeight->"Bold"], " try them all (and hopefully ", StyleBox["complete", FontWeight->"Bold"], " them all too if you wish to do well on the Exam we have on Tuesday, Nov \ 20). I'll post solutions soon." }], "Subsubtitle"] }, Closed]], Cell[CellGroupData[{ Cell["Introduction", "Section"], Cell[TextData[{ "OBJECTIVE: Today, you will learn:\n\t1. The basics of calculating Masses, \ Centroids, and Moments using ", StyleBox["Mathematica", FontSlant->"Italic"], ".\n\t2. A few applications of these ideas to real world problems \t" }], "Text"], Cell[CellGroupData[{ Cell["Technology Guidelines", "Subsection", CellDingbat->"\[LightBulb]"], Cell[TextData[{ StyleBox["NOTE: If you have just finished a module, restart ", CellFrame->True, Background->None], StyleBox["Mathematica", CellFrame->True, FontSlant->"Italic", Background->None], StyleBox[" before executing a new module.\nTO OPEN CELLS, put your cursor \ on the right cell bracket and double click.", CellFrame->True, Background->None], "\nTO STOP AN EXECUTION\n\tSelect the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu and click on ", StyleBox["Abort Evaluation.\n", FontSlant->"Italic"], "ORDER OF EXECUTION\n\tExecute cells in the order given. Do not skip any \ Input cells within a given notebook.\nSAVING NOTEBOOKS\n\tYou can save \ anytime to any directory you choose, and it is wise to save often.\n\t\ However, before you do your final save, delete all your output by selecting \ the \n\t ", StyleBox["Delete All Output", FontSlant->"Italic"], " selection under the ", StyleBox["Kernel", FontSlant->"Italic"], " pull-down menu.\nEXPERIENCING MAJOR PROBLEMS\n\tSave if appropriate, and \ then shut down ", StyleBox["Mathematica", FontSlant->"Italic"], " and start it up again." }], "Text"] }, Closed]] }, Closed]], Cell[CellGroupData[{ Cell["Part I: Masses and Moments", "Section"], Cell["Mass", "Subsubsection"], Cell[TextData[{ "We recall from basic physics that Mass = Density x Volume. In the simple \ world, we always have a ", StyleBox["constant ", FontSlant->"Italic"], "density. What if we have a density that changes depending upon our \ position in an object (e.g. the density of the earth varies as we move \ through the different materials composing it). If our density function is \ \[Delta] (typical units would be gm/", Cell[BoxData[ \(TraditionalForm\`cm\^3\)]], "), then we can evaluate a ", StyleBox["triple integral ", FontSlant->"Italic"], "that defines the volume we wish to calculate, using the density function \ as the integrand. The result will be the ", StyleBox["mass ", FontSlant->"Italic"], "of the object.\n\t\t\t\t\t\t\t\[Integral]\[Integral]", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_R\ \[Delta]\ dV\)]] }], "Text"], Cell[TextData[{ "For example, we could solve the following problem:\n\t\tWrite an integral \ representing the mass of a sphere of radius 3cm if the density, in ", "gm/", Cell[BoxData[ \(TraditionalForm\`cm\^3\)]], ", of the sphere at any point is twice the distance of that point from the \ center of the sphere." }], "Text"], Cell[BoxData[ \(\[Integral]\_0\%\(2 \ \[Pi]\)\(\[Integral]\_0\%\[Pi]\(\[Integral]\_0\%3\((2 \[Rho]*\(\[Rho]\^2\) Sin[\[Phi]])\) \[DifferentialD]\[Rho] \[DifferentialD]\[Phi] \ \[DifferentialD]\[Theta]\)\)\)], "Input"], Cell["\<\ So the density would be 162\[Pi] gm. \ \>", "Text"], Cell["Center of Mass", "Subsubsection"], Cell[TextData[{ "The motion of a solid object can be analyzed by thinking of the mass as \ concentrated at a single point, which we call the ", StyleBox["center of mass", FontSlant->"Italic"], ". If the object has density \[Delta](x,y,z) at the point (x,y,z) and \ occupies the region ", StyleBox["R", FontSlant->"Italic"], ", then the coordinates (", Cell[BoxData[ \(TraditionalForm\`x\&_, \ y\&_, \ z\&_\)]], ") of the center of mass are given by:\n\t\t\t", Cell[BoxData[ \(TraditionalForm\`x\&_ = \(1\/mass\) \(\[Integral]\_R\ x\[Delta]\ dV\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(y\&_ = \(1\/mass\) \(\[Integral]\_R\ y\[Delta]\ dV\)\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`z\&_ = \(1\/mass\) \(\[Integral]\_R\ z\[Delta]\ dV\)\)]], ", where the mass is given by ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_R\ \[Delta]dV\)]] }], "Text"], Cell["\<\ Find the center of mass of the solid bounded below by the square z = 0, 0 \ \[LessEqual] x \[LessEqual] 1, 0 \[LessEqual] y \[LessEqual] 1 and above by \ the surface z = x + y + 1. Assume that the density is given by \ \[Delta](x,y,z) = 1. \ \>", "Text"], Cell[BoxData[ \(mass\ = \ \[Integral]\_0\%1\(\[Integral]\_0\%1\(\[Integral]\_0\%\(x + \ y + 1\)1 \[DifferentialD]z \[DifferentialD]y \[DifferentialD]x\)\)\)], "Input"], Cell[BoxData[ \(xbar\ = \ \(1\/mass\) \(\[Integral]\_0\%1\(\[Integral]\_0\%1\(\ \[Integral]\_0\%\(x + y + 1\)\((x*1)\) \[DifferentialD]z \[DifferentialD]y \ \[DifferentialD]x\)\)\)\)], "Input"], Cell[TextData[{ "Go ahead and enter the formulas for ", Cell[BoxData[ \(TraditionalForm\`y\&_\)]], " and ", Cell[BoxData[ \(TraditionalForm\`z\&_\)]], "." }], "Text"], Cell[BoxData[ \(\ \)], "Input"], Cell[TextData[StyleBox["Question 1:\tWhat is the center of mass of this \ object?", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]], Cell["Moments", "Subsubsection"], Cell[TextData[{ "The ", StyleBox["moment of inertia", FontSlant->"Italic"], " of a solid body about an axis in 3-space gives the ", StyleBox["angular acceleration", FontWeight->"Bold"], " about this axis for a given torque (force twisting the body). The \ moments of inertia about the coordinate axes of a body of constant density \ and mass ", StyleBox["m", FontSlant->"Italic"], " occupying a region ", StyleBox["R", FontSlant->"Italic"], " of a volume ", StyleBox["V", FontSlant->"Italic"], " are defined to be:\n\t\t\t", Cell[BoxData[ \(TraditionalForm\`I\_x = \[Integral]\_R\ \((y\^2 + z\^2)\) \[Delta]\ dV\)]], ", ", Cell[BoxData[ \(TraditionalForm\`\(\(\ \ \)\(I\_y = \[Integral]\_R\ \((x\^2 + z\^2)\) \[Delta]\ dV\)\)\)]], ", ", Cell[BoxData[ \(TraditionalForm\`I\_z = \[Integral]\_R\((x\^2 + y\^2)\) \[Delta]\ dV\)]] }], "Text"], Cell[TextData[{ "Let's find the moments of intertia of the object described in the ", StyleBox["center of mass", FontSlant->"Italic"], " section. Here is the problem again: \n\tThe solid bounded below by the \ square z = 0, 0 \[LessEqual] x \[LessEqual] 1, 0 \[LessEqual] y \[LessEqual] \ 1 and above by the surface z = x + y + 1.\n\tAssume that the density is given \ by \[Delta](x,y,z) = 1. " }], "Text"], Cell[BoxData[ \(isubx\ = \ \[Integral]\_0\%1\(\[Integral]\_0\%1\(\[Integral]\_0\%\(x + \ y + 1\)\((y\^2 + z\^2)\) \[DifferentialD]z \[DifferentialD]y \ \[DifferentialD]x\)\)\)], "Input"], Cell[TextData[{ "Go ahead and find the moments about the ", StyleBox["y- ", FontSlant->"Italic"], "and ", StyleBox["z-", FontSlant->"Italic"], " axes." }], "Text"], Cell[BoxData[ \(\ \)], "Input"], Cell[TextData[StyleBox["Question 2:\tWhat are the moments of intertia of this \ object?", FontWeight->"Bold"]], "Text"], Cell[TextData[{ StyleBox["(Type your answer here)", FontSlant->"Italic"], "\n" }], "Commentary", Background->RGBColor[0, 1, 1]] }, Closed]], Cell[CellGroupData[{ Cell["Part II: Masses and Moments Applied to Probability Functions", "Section"], Cell["\<\ In multivariable probability, probability density functions are defined over \ specified regions (domains) in the same way that density functions are \ defined over regions of a solid. These probability functions are designed so \ that the total \"mass\" is always 1. Probabilities are then defined to be the \ integrals of the density functions over a particular subregion of the domain. \ \ \>", "Text"], Cell[TextData[{ StyleBox["Background:", FontWeight->"Bold"], " A function ", StyleBox["p(x,y)", FontSlant->"Italic"], " is called a ", StyleBox["joint density function", FontWeight->"Bold"], " for ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " if:\n[the fraction of the population with ", StyleBox["x", FontSlant->"Italic"], " between ", StyleBox["a", FontSlant->"Italic"], " and ", StyleBox["b", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " between ", StyleBox["c", FontSlant->"Italic"], " and ", StyleBox["d", FontSlant->"Italic"], "] is the same as [the volume under the graph of ", StyleBox["p", FontSlant->"Italic"], " above the rectangle ", StyleBox["a \[LessEqual] x \[LessEqual] b ", FontSlant->"Italic"], " and ", StyleBox["c \[LessEqual] y \[LessEqual] d", FontSlant->"Italic"], "] which is defined mathematically as ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_a\%b\ \(\[Integral]\_c\%d\ \(p(x, y)\)\ dy\ dx\)\)]], "\n", "Where we must have: ", Cell[BoxData[ \(TraditionalForm\`\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \(\ \[Integral]\_\(-\[Infinity]\)\%\[Infinity]\ \(p(x, y)\)\ dy\ dx\)\ = \ 1\)]], " and ", StyleBox["p(x,y) \[GreaterEqual] ", FontSlant->"Italic"], "0 for all ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], "." }], "Text", CellFrame->True, Background->GrayLevel[0.849989]], Cell[TextData[{ "Consider the following example:\nSuppose that a national fast-food outlet \ is interested in the joint behavior of the random variables ", StyleBox["x", FontSlant->"Italic"], ", defined as the total time between a customer's arrival at the store and \ departing from the service window, and ", StyleBox["y", FontSlant->"Italic"], ", the time that a customer waits in line before reaching the service \ window. Since ", StyleBox["x", FontSlant->"Italic"], " contains the time a customer waits in line, ", StyleBox["x", FontSlant->"Italic"], " must be greater than ", StyleBox["y", FontSlant->"Italic"], ". The relative frequency distribution of observed values of ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " can be modeled by the probability density function: \n", Cell[BoxData[ FormBox[ RowBox[{\(f(x, y)\), " ", "=", " ", FormBox[\( .04 e\^\(\(- .1\) x - \(\(.3\) \(y\)\(\ \)\)\)\), "TraditionalForm"]}], TraditionalForm]]], " for ", Cell[BoxData[ \(TraditionalForm\`0\ \[LessEqual] \ y\ \[LessEqual] \ x\ < \ \[Infinity]\)]], " and 0 elsewhere, where ", StyleBox["x", FontSlant->"Italic"], " and ", StyleBox["y", FontSlant->"Italic"], " are measured in minutes." }], "Text"], Cell[BoxData[ \(f[x_, y_] := .04 E\^\(\(- .1\) x - \(\(.3\) \(y\)\(\ \)\)\)\)], "Input"], Cell["\<\ We examine the domain of this function. First we must load a package to \ assist us in graphing.\ \>", "Text"], Cell[BoxData[ \(<< \ Graphics`FilledPlot`\)], "Input"], Cell[BoxData[ \(\(FilledPlot[x, {x, 0, 50}, AxesLabel -> {"\