(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "NeXT Mathematica Notebook Front End Version 2.2"; NeXTStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, noPageBreakBelow, noPageBreakInGroup, nohscroll, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 18, "Times"; ; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 14, "Times"; ; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 12, "Times"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, noPageBreakInGroup, nohscroll, preserveAspect, M7, italic, B65535, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 12, "Times"; ; fontset = leftheader, L0, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, L0, 12; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] ARCSLIDE :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We track an object moving along a circular path with a camera, ascertaining various quantities including angular acceleration of the camera's turning motion. Makes heavy use of parametric equation descriptors of position. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: ARCSLIDE Full title: Tracking an object on a circular arc and optimizing an angular velocity and angular acceleration. Last Revision Date: 30 May 1996. Developer: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Contact: Brian J. Winkel, Department of Mathematical Sciences, United States Military Academy, West Point NY 10996 USA. Phone: 914-938-3200. Email: ab3646@usma2.usma.edu. FAX: 914-938-2409. Aaron D. Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@Rose-Hulman.Edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] A small mass is moving along the circle x^2 + y^2 = 25 (distances in centimeters) at a rate of 2 cm/sec. The mass begins at the top of the circle, i.e. at point (0, 5) cm, and moves down the side of the circle in the first quadrant, i.e. x >0 and y> 0) until it stops at the point (5, 0) cm. A camera sits at point (10, 1) cm and we wish to aim the camera at the mass to capture the motion of the mass. :[font = subsection; inactive; preserveAspect; endGroup] (a) When will the mass arrive at the point (5, 0) cm? (b) Describe the motion of the mass as a position vector p(t) = (x(t), y(t)). (c) When will the camera first "see" the mass. (d) Let theta be the angle which the line of site of the camera when focused on the mass on the circle makes with the horizontal. Write theta as a function of time t. Plot theta(t). (e) Determine when the angular velocity theta'(t) and the angular acceleration theta''(t) are greatest and obtain these values. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Circular motion, parametric equations, angular velocity, angular acceleration, optimization. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect] ISSUES RELATED TO THE PROBLEM :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Parametric equation representation of circular motion of constant speed, angular velocity and angular acceleration, and optimization. :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This problem is very demanding of CAS effort, but the problem can be stated, formulated, and outlined in one class period and then offered as a one or two week homework project. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] Students will see the nature of the problem as a clasic "related rates" problem and understnad the subtleties introduced by the nature of the motions and the need for CAS efforts to facilitate solution. :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] Students can handle sophisitacted related rates or relative motion problems. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect] Further motions can be offered for camera tracking. :[font = subsubsection; inactive; preserveAspect] In addition, the camera can move. :[font = subsubsection; inactive; preserveAspect; endGroup] The distance from the mass to the camera - focal length - can be studied. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (a) The mass will arrive a the point (5, 0) cm in 3.93 sec. :[font = input; preserveAspect; startGroup] time = 5 Pi/2 / 2//N :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 3.926990816987241 ;[o] 3.92699 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (b) We can describe the motion of the mass as a position vector p(t) using circular functions. :[font = input; preserveAspect] p[t_] = {5 Sin[2/5 t], 5 Cos[2/5 t]}; :[font = subsubsection; inactive; preserveAspect; startGroup] We further compute the time at which the mass arrives at (5, 0) as a check on our parametric representation. :[font = input; preserveAspect; startGroup] stoptime = t/.FindRoot[p[t][[1]] == 5,{t,3}][[1]]//N :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 3.926545112976204 ;[o] 3.92655 :[font = subsubsection; inactive; preserveAspect; startGroup] We plot the motion of the mass. :[font = input; preserveAspect; startGroup] ParametricPlot[p[t],{t,0, time},AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (c) To determine when we first see the mass we need to determine the line from the cameraat (10,1) to the circle which is tangent to the circle. :[font = input; preserveAspect; startGroup] object = ParametricPlot[p[t],{t,0,time}, AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect] pt = Graphics[{PointSize[.015], Point[{10,1}]}]; :[font = input; preserveAspect; startGroup] Show[object,pt,AspectRatio->Automatic] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] We describe the curve non-parametrically. :[font = input; preserveAspect; endGroup] f[x_] = Sqrt[25 - x^2]; :[font = subsubsection; inactive; preserveAspect; startGroup] And we determine when the slope of the curve is equal to the slope of the tangent line from (10, 1) to the curve. :[font = input; preserveAspect; startGroup] sol = FindRoot[f'[x] == (f[x] - 1)/(x - 10),{x,2}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {x -> 2.043673371926666} ;[o] {x -> 2.04367} :[font = subsubsection; inactive; preserveAspect; startGroup] We now determine the x position and the starttime where and when respectively we first see the mass. :[font = input; preserveAspect; startGroup] xstart = x/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] 2.043673371926666 ;[o] 2.04367 :[font = input; preserveAspect; startGroup] starttime = t/.FindRoot[p[t][[1]] == xstart,{t,2}][[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 1.052668014233292 ;[o] 1.05267 :[font = subsubsection; inactive; preserveAspect; startGroup] We plot the situation. :[font = input; preserveAspect; startGroup] tanline = Graphics[Line[{{x,f[x]},{10,1}}]]/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = input; preserveAspect; endGroup; endGroup] Show[object,pt,tanline]; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (d) We determine and plot the angle which the line of site of the camera makes with the horizontal as a function of time t. :[font = input; preserveAspect; startGroup] angle[t_] = ArcTan[(p[t][[2]] - 1)/(10 - p[t][[1]] )] :[font = output; output; inactive; preserveAspect; endGroup] ArcTan[(-1 + 5*Cos[(2*t)/5])/(10 - 5*Sin[(2*t)/5])] ;[o] 2 t -1 + 5 Cos[---] 5 ArcTan[---------------] 2 t 10 - 5 Sin[---] 5 :[font = input; preserveAspect; endGroup] Plot[angle[t],{t,starttime,time}]; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] (e) We ascertain when the angular velocity and the angular acceleration of the camera is greatest. :[font = input; preserveAspect] Plot[angle'[t],{t,starttime,time}, AxesLabel->{"t","angle'(t)"}]; :[font = input; preserveAspect] acc[t_] = angle''[t]; :[font = input; preserveAspect; startGroup] solacc = FindRoot[acc[t]==0,{t,3.5}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 3.677819185759336} ;[o] {t -> 3.67782} :[font = input; preserveAspect; startGroup] angle'[t]/.solacc[[1]] :[font = output; output; inactive; preserveAspect; endGroup] -0.3960493584505498 ;[o] -0.396049 :[font = subsubsection; inactive; preserveAspect] We note that the maximum angular velocity (-0.40 rad/sec) at time 3.68 sec is NOT at the end of the motion but rather shortly before that time. :[font = subsubsection; inactive; preserveAspect; startGroup] We now determine the maximum angular acceleration. :[font = input; preserveAspect] Plot[angle''[t],{t,starttime,time}, AxesLabel->{"t","angle''(t)"}]; :[font = input; preserveAspect] jerk[t_] = angle'''[t]; :[font = input; preserveAspect; startGroup] soljerk = FindRoot[jerk[t]==0,{t,2.5}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 2.671344859711304} ;[o] {t -> 2.67134} :[font = input; preserveAspect; startGroup] angle''[t]/.soljerk[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] -0.2128649496108012 ;[o] -0.212865 :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Thus we determine the maximum acceleration is 0.213 cm/sec^2 and it is a deceleration experienced at time t = 2.67 sec. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect] This problem uses calculus and geometry to locate the point where the mass is first visible from the camera. :[font = subsection; inactive; preserveAspect] The solver is required to formulate a function to represent the angle between the line from the camera to the moving mass and the horizontal as a function of time. :[font = subsection; inactive; preserveAspect; endGroup; endGroup] This angular function is studied and its angular velocity and angular acceleration are optimized. ^*)