(************** Content-type: application/mathematica ************** CreatedBy='Mathematica 5.2' 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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When the Reynolds number, ", Cell[BoxData[ \(TraditionalForm\`Re = \(\[Rho]\ v\ D\)\/\[Mu]\)]], ", is less than 0.5, the drag force is proportional to ", Cell[BoxData[ \(TraditionalForm\`v\)]], ". More specifically, according to Stoke's theory for low Reynolds number, \ the drag coefficient is ", Cell[BoxData[ \(TraditionalForm\`C\_D = 24\/Re\)]], ", which makes drag force equal to ", Cell[BoxData[ \(TraditionalForm\`f\_D = 3 D\ \[Pi]\ \[Mu]\ v\)]], ". Newton's second law states that mass times acceleration equals the sum \ of the forces. So, if a sphere is moving vertically through a fluid under \ the influence of a gravitational force and a drag force, and if displacement \ is measured upward, then the sphere's velocity must satisfy\n\n", Cell[BoxData[ \(TraditionalForm\`m\ dv\/dt = \(-mg\)\ - k\ v\)]], "\n\nwith ", Cell[BoxData[ \(TraditionalForm\`k = 3 D\ \[Pi]\ \[Mu]\)]], ". This is a differential equation for ", Cell[BoxData[ \(TraditionalForm\`v\)]], ". ", StyleBox["Mathematica", FontSlant->"Italic"], " can solve many differential equations using the ", StyleBox["DSolve", FontFamily->"Courier"], " command. The first argument to DSolve is the differential equation or \ list of differential equations and initial conditions, the second argument is \ the unknown function to be solved for, and the third argument is the \ independent variable. The solution is given as a replacement rule or a list \ of replacement rules." }], "Text"], Cell[BoxData[{ \(Clear[v, t, m, g, k]\), "\[IndentingNewLine]", \(vSolution = DSolve[m\ \(v'\)[t] \[Equal] \(-m\)\ g - k\ v[t], v[t], t]\)}], "Input"], Cell[TextData[{ "This is a first order differential equation, so the general solution \ includes one arbitrary constant, ", StyleBox["C[1]", FontFamily->"Courier"], ". " }], "Text"], Cell[TextData[{ "If we include an initial velocity, DSolve can find the value of ", StyleBox["C[1]", FontFamily->"Courier"], " to satisfy the initial condition. Notice that the first argument to ", StyleBox["DSolve", FontFamily->"Courier"], " is now a list of equations \[LongDash] the differential equation and the \ initial condition." }], "Text"], Cell[BoxData[{ \(Clear[v, t, m, g, k, v0]\), "\[IndentingNewLine]", \(vSolution = DSolve[{m\ \(v'\)[t] \[Equal] \(-m\)\ g - k\ v[t], v[0] \[Equal] v0}, v[t], t]\)}], "Input"], Cell["\<\ We can use this replacement rule to assign the solution to a \ function.\ \>", "Text"], Cell[BoxData[ \(v[t_] = v[t] /. vSolution[\([1]\)]\)], "Input"], Cell[TextData[{ StyleBox["Exercise 17.1:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " Graph the velocity of a falling sphere with mass ", Cell[BoxData[ \(TraditionalForm\`m\)]], " = 100 grams, ", Cell[BoxData[ \(TraditionalForm\`k\)]], " = 50 grams/sec, ", Cell[BoxData[ \(TraditionalForm\`g\)]], " = 980 cm/", Cell[BoxData[ \(TraditionalForm\`sec\^2\)]], ", and ", Cell[BoxData[ \(TraditionalForm\`v0\)]], " = 500 cm/sec for 0 < ", Cell[BoxData[ \(TraditionalForm\`t\)]], " < 20. ", "Use this graph to describe the motion of the falling sphere.", Cell[BoxData[ FormBox[Cell[""], TraditionalForm]]] }], "Text", Background->GrayLevel[0.750011]], Cell[TextData[{ "We can integrate the velocity to get the vertical displacement. If the \ initial position of the sphere is y[0] = y0, then the position at time ", Cell[BoxData[ \(TraditionalForm\`t\)]], " is" }], "Text"], Cell[BoxData[{ \(Clear[y, t]\), "\[IndentingNewLine]", \(y[t_] = \[Integral]\_0\%t v[\[Tau]] \[DifferentialD]\[Tau] + y0\)}], "Input"], Cell[TextData[{ StyleBox["Exercise 17.2:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " A room fogger sprays a flea-killing insecticide throughout a room at 20\ \[Degree]C with a 300 cm high ceiling. If the droplets have diameter 2.0 \ \[Times] ", Cell[BoxData[ \(TraditionalForm\`10\^\(-3\)\)]], " cm, how long should one wait before re-entering the room? Assume the \ density of the droplets is the same as the density of water. Densities and \ viscosities of selected fluids are listed in the table below." }], "Text", Background->GrayLevel[0.750011]], Cell[BoxData[GridBox[{ {"Fluid", \(Density, \[Rho]\[IndentingNewLine] \((gm/cm\^3)\)\), \(Viscosity, \[Mu]\[IndentingNewLine] \((gm/cm - sec)\)\)}, {\(Dry\ air\ at\ 5 \[Degree]C\ and\ 1\ atm\), \(1.27\ \[Times]\ 10\^\(-3\)\), \(1.74\ \[Times]\ 10\^\(-4\)\)}, {\(Dry\ air\ at\ 20 \[Degree]C\ and\ 1\ atm\), \(1.21\ \[Times]\ 10\^\(-3\)\), \(1.81\ \[Times]\ 10\^\(-4\)\)}, {\(Water\ at\ 50 \[Degree]C\), "1.00", \(1.51\ \[Times]\ 10\^\(-2\)\)}, {\(Water\ at\ 20 \[Degree]C\), "0.998", \(1.00\ \[Times]\ 10\^\(-2\)\)}, {"Glycerine", "1.26", "23.3"} }, GridFrame->True, RowLines->True, ColumnLines->True]], "Text"], Cell[TextData[{ StyleBox["Exercise 17.3:", FontWeight->"Bold", FontColor->RGBColor[0, 0.500008, 0.250004]], " In an electrical circuit with resistance ", Cell[BoxData[ \(TraditionalForm\`R\)]], ", in ohms, inductance ", Cell[BoxData[ \(TraditionalForm\`L\)]], ", in henries, capacitance ", Cell[BoxData[ \(TraditionalForm\`C\)]], ", in farads, and voltage source ", Cell[BoxData[ \(TraditionalForm\`v(t)\)]], ", in volts, Kirkoff's voltage law states that the charge on the capacitor \ ", Cell[BoxData[ \(TraditionalForm\`q\)]], ", in coulombs, satisfies the differential equation\n\n", Cell[BoxData[ \(TraditionalForm\`L \(\( d\^2\) q\)\/dt\^2 + R dq\/dt + \(1\/C\) q = v(t)\)]], ".\n\nFind the charge on the capacitor of a circuit with ", Cell[BoxData[ \(TraditionalForm\`R\)]], " = 70 \[CapitalOmega], ", Cell[BoxData[ \(TraditionalForm\`L\)]], " = 12 H, ", Cell[BoxData[ \(TraditionalForm\`C\)]], " = ", Cell[BoxData[ \(TraditionalForm\`1\/40\)]], " F, and voltage source ", Cell[BoxData[ \(TraditionalForm\`v(t) = 400 \( sin(10 t)\)\)]], ", if the capacitor has an initial charge of 5 C and the initial current, \ ", Cell[BoxData[ \(TraditionalForm\`dq\/dt\)]], ", is zero. Graph the solution. 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