(*^ ::[ 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, 12; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, italic, L1, 12, "Times"; ; fontset = leftfooter, 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; paletteColors = 128; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] KIND OF A DRAG :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] Kinematic descriptions of motion for a point on the dragging end of a stick which is hinged to a point on a rolling disk are developed and used to determine velocity and acceleration quantities. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsubsection; inactive; preserveAspect; endGroup] FileName: DRAGSTIK Full title: Kind of a Drag Developers: Ken Kerr, Glenbrook South HIgh School, 4000 W. Lake Ave., Glenview IL 60025 USA. Brian J. Winkel, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Contact: Brian J. Winkel, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8412. Email: winkel@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] Consider a wheel of radius r = 10 cm and a stick of length l = 40 cm. (In general prescribe l >= 2r for this problem.) The stick is attached to the edge of the wheel with a pin which permits it to freely move while the wheel roles along a flat surface (say the x-axis). In fact, as the wheel turns along the flat surface one end of the stick merely drags along the surface while the other end (attached to the pin) rides up and down. :[font = subsection; inactive; preserveAspect] Suppose we start the wheel with the stick attachment pin at the origin (0, 0) and center at (0, 10), positive x-axis to the right, and the stick laying on the x-axis to the left, with its left hand end-point at (-40, 0). :[font = subsection; inactive; preserveAspect] Further, suppose the wheel roles to the right, without friction, at a rate of t radians/sec, i.e. it takes 2 Pi sec to complete one revolution at which point we find the stick-pin-wheel configuration looking exactly the same (pin on the x-axis - only at (2 Pi 10,0) now, stick on the x-axis from pin to point (2 Pi 10 - 40, 0), and the wheel center at (2 Pi 10, 10). :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 1. Write a function to describe the position of the tip of the stick not attached to the wheel as the wheel is rolling on a plane, i.e. in terms of the parameter t, the angle of rotation of the wheel. We suggest a step by step approach in (a) - (e) below. :[font = subsubsection; inactive; preserveAspect] a. In terms of the angle of rotation, what is the horizontal displacement of the point where the stick is attached to the wheel ? :[font = subsubsection; inactive; preserveAspect] b. In terms of the angle of rotation, what is the vertical displacement of the point where the stick is attached to the wheel? :[font = subsubsection; inactive; preserveAspect] c. In terms of the angle of rotation, what is the horizontal displacement of the tip of the stick from the point where it is attached to the wheel? :[font = subsubsection; inactive; preserveAspect] d. In terms of the angle of rotation, what is the horizontal displacement of center of the rolling wheel? :[font = subsubsection; inactive; preserveAspect; endGroup] e. What is the total horizontal displacement of the dragging tip of the stick that is not attached to the wheel? :[font = subsection; inactive; preserveAspect] Here are some questions concerning the dragging tip of the stick. :[font = subsection; inactive; preserveAspect] 2. Plot a graph of the horizontal displacement of the dragging tip of the stick. We know the vertical position is constantly y = 0 as it drags along the flat surface of the x-axis. A reasonable interval over which to plot this function is [0, 4 Pi], i.e. two full rotations of the wheel. Does this graph make sense? Recall where it should start, where it should be at t = 0, at t = 2 Pi, at t = 4 Pi. Explain some of the features of this curve in terms of the motion of the dragging tip of the stick. Does the dragging tip of the stick ever go backwards? How would you characterize its horizontal motion? :[font = subsection; inactive; preserveAspect] 3. Plot a graph of the horizontal velocity on the dragging tip of the stick. Explain some of the features of this curve in terms of the motion of the dragging tip of the stick. :[font = subsection; inactive; preserveAspect] 4. Plot a graph of the horizontal acceleration on the dragging tip of the stick. Explain some of the features of this curve in terms of the motion of the dragging tip of the stick. :[font = subsection; inactive; preserveAspect] 5. For what time(s)/angle(s) is the the horizontal velocity on the dragging tip of the stick a maximum and a minimum? What are these maxima and minima values of the velocity? :[font = subsection; inactive; preserveAspect] 6. State other observations, conjectures, issues, or questions you have concerning the motion of the dragging tip of the stick. :[font = subsection; inactive; preserveAspect] 7-12 Now do the problem with length of the stick (l = 40) equal to the length of the diameter (d = 2 r = 40). There is something odd about some of the graphs of horizontal displacement, velocity, and acceleration. If the domain of the functions is [0, 4Pi] find and explain any discontinuities in these plots? What is going on in the motion at these points of discontinuity? :[font = subsection; inactive; preserveAspect; endGroup] 13 (For the dimensions used in 7-12 only.) You may see a place where the derivative of a discontinuous function actually appears to exist at the point of discontinuity. What theorem on derivatives and continuity seems to be violated? Explain why this violation is only apparently so and not in fact true. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Trigonometric functions, parametric equations, derivative, (first and second), velocity, acceleration :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect; endGroup] To start with make sure the length of the stick is greater than the diameter of the wheel as prescribed in 1 - 10. We consider a wheel of radius r = 10 cm and a stick of length l = 40. In general prescribe l >= 2r for this problem. Be sure the problem solvers understand the physical situation - a model of cardboad may be appropriate to get the feel for the wheel/stick configuration. The stick is attached to the edge of the wheel with a pin which permits it to freely move while the wheel roles along a flat surface (say the x-axis). In fact, as the wheel turns along the flat surface one end of the stick merely drags along the surface while the other end (attached to the pin) rides up and down. There are three components to the horizontal position of the tip of the stick. Imagine the wheel is fixed first and write a position function for the pin contact point between wheel and stick and then add the rolling part. :[font = subsection; inactive; preserveAspect] Prerequisites :[font = subsection; inactive; preserveAspect] Time allotment - time management :[font = subsection; inactive; preserveAspect] Expectations :[font = subsection; inactive; preserveAspect] Future payoffs :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect; endGroup] VARIATION: Make the circle move along one path either by prescribing the base along which it moves or the path of its center and/or make the surface on which the stick rests vary as well. E.g. make the path of the circle up a ramp and drag the stick up the ramp. This should be no different except there will be a rising term in all equations. One could move the circle along a sinusoidal path - move center along this with rotation and follow the stick as it lays along the ground. Here the length of the stick would have to be greater than the height of the center of the circle plus the amplitude of the sinusoidal track. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Dragging Stick problem (15.53, p. 724) from the text Vector Mechanics for Engineers: Dynamics, Fifth Edition, by F. P. Beer and E. R. Johnston, Jr., McGraw-Hill: New York. 1988. :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] Write up of first situation - Problems 1-6: :[font = subsection; inactive; preserveAspect; startGroup] We first define the variables: radius = constant radius of circle length = constant length of dragged stick theta = the angle (we presume the disk rotates at 1 radian/sec) xwheel(theta) = the horizontal distance of pin from the vertical radius through the center of the rolling circle. ywheel(theta) = the vertical distance of pin from the x-axis xstick(theta) = the horizontal distance from the pin to the dragging tip of the stick :[font = input; preserveAspect; startGroup] radius = 10; length = 40; :[font = message; inactive; preserveAspect; endGroup; endGroup] General::spell1: Possible spelling error: new symbol name "length" is similar to existing symbol "Length". :[font = subsection; inactive; preserveAspect; startGroup] We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0). :[font = input; preserveAspect] xwheel[theta_]= radius*Sin[theta]; :[font = input; preserveAspect; startGroup] ywheel[theta_]= radius*(1 - Cos[theta]); :[font = message; inactive; preserveAspect; endGroup] General::spell1: Possible spelling error: new symbol name "ywheel" is similar to existing symbol "xwheel". :[font = input; preserveAspect; endGroup] xstick[theta_]= Sqrt[length^2 - ywheel[theta]^2]; :[font = subsection; inactive; preserveAspect; startGroup] xroll(theta) = the distance the circle has rolled along the horizontal when it rotates through angle theta. :[font = input; preserveAspect; endGroup] xroll[theta_]= radius*theta; :[font = subsection; inactive; preserveAspect; startGroup] xtotal(theta) = the actual x-coordinate of the dragging tip of the stick. We obtain xtotal(theta) by subtracting xwheel(theta) (the horizontal distance of pin from the vertical radius through the center of the rolling circle) and xstick(theta) (the the horizontal distance from the pin to the dragging tip of the stick) from xroll(theta) (the the distance the circle has rolled along the horizontal when it rotates through angle theta). :[font = input; preserveAspect; endGroup] xtotal[theta_] = -xwheel[theta] - xstick[theta] + xroll[theta]; :[font = subsection; inactive; preserveAspect; startGroup] Here is a plot of the horizontal displacement of the dragging tip of the stick. :[font = input; preserveAspect; endGroup] Plot[xtotal[theta],{theta,0,4Pi}]; :[font = subsection; inactive; preserveAspect; startGroup] We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick. :[font = input; preserveAspect; startGroup] vxtotal[theta_]=xtotal'[theta]; Plot[vxtotal[theta],{theta,0,4Pi}]; :[font = message; inactive; preserveAspect; endGroup; endGroup] General::spell1: Possible spelling error: new symbol name "vxtotal" is similar to existing symbol "xtotal". :[font = subsection; inactive; preserveAspect; startGroup] It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0. We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at theta = 2 Pi. :[font = input; preserveAspect; startGroup] axtotal[theta_] = vxtotal'[theta]; Plot[axtotal[theta],{theta,0,4Pi}]; :[font = message; inactive; preserveAspect; endGroup] General::spell: Possible spelling error: new symbol name "axtotal" is similar to existing symbols {vxtotal, xtotal}. :[font = input; preserveAspect; startGroup] solmin = FindRoot[axtotal[theta] == 0,{theta,6.4}] :[font = output; output; inactive; preserveAspect; endGroup] {theta -> 6.28318530719527} ;[o] {theta -> 6.28319} :[font = input; preserveAspect; startGroup] 2 Pi//N :[font = output; output; inactive; preserveAspect; endGroup] 6.283185307179586 ;[o] 6.28319 :[font = input; preserveAspect; startGroup] vxtotal[theta]/.solmin[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0. ;[o] 0. :[font = subsection; inactive; preserveAspect; startGroup] Now as to the maximum velocity we find that this is 21.3541 cm/sec and this occurs at theta = 2.70334 when the wheel has not yet finished a half a revolution, i.e,. the maximum velocity DOES NOT occur when the pin is at the top of the wheel, while the minimum velocity DOES occur when the pin is at the bottom of the wheel. :[font = input; preserveAspect; startGroup] solmax = FindRoot[axtotal[theta] == 0,{theta,2}] :[font = output; output; inactive; preserveAspect; endGroup] {theta -> 2.703336559648136} ;[o] {theta -> 2.70334} :[font = input; preserveAspect; startGroup] vxtotal[theta]/.solmax[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 21.35411563599737 ;[o] 21.3541 :[font = subsection; inactive; preserveAspect; startGroup] We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact. :[font = input; preserveAspect; startGroup] Plot[{xtotal[theta], vxtotal[theta],axtotal[theta]}, {theta,0,4 Pi}, PlotStyle-> {Thickness[.005], Thickness[.01],Thickness[.015]}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect] The thickest curve is acceleration, the middle thickness curve is velocity, and the thinnest curve is position. We see the usual confirmations of one function's derivative being zero at its relative maxima and minima. We see some other interesting things here. :[font = subsection; inactive; preserveAspect; startGroup] From the graph of the acceleration we note that acceleration peaks at 1.46974 sec to a value of 12.8564 cm/sec^2, again we have an asymmetry with respect to time, i.e. things do not happen at equal intervals in the period of the rolling wheel [0, 2 Pi] . :[font = input; preserveAspect; startGroup] jxtotal[theta_] = axtotal'[theta]; :[font = message; inactive; preserveAspect; endGroup] General::spell: Possible spelling error: new symbol name "jxtotal" is similar to existing symbols {axtotal, vxtotal, xtotal}. :[font = input; preserveAspect; startGroup] solamax = FindRoot[jxtotal[theta] == 0,{theta,1.5}] :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "solamax" is similar to existing symbol "solmax". :[font = output; output; inactive; preserveAspect; endGroup] {theta -> 1.469736269815843} ;[o] {theta -> 1.46974} :[font = input; preserveAspect; startGroup] axtotal[theta]/.solamax[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 12.85638692775777 ;[o] 12.8564 :[font = subsection; inactive; preserveAspect] The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.46974]) and then decelerates. :[font = subsection; inactive; preserveAspect; startGroup] We animate the motion of the stick and the disk. :[font = input; preserveAspect; endGroup] Do[Show[Graphics[{PointSize[.02],Point[{xtotal[theta],0}], Point[{radius theta - xwheel[theta],ywheel[theta]}], Circle[{radius theta,radius},radius], Line[{{-40,0},{8 Pi radius + radius,0}}], Line[{{xtotal[theta],0}, {radius theta - xwheel[theta],ywheel[theta]}}]}], PlotRange->{{-length - 10,8 Pi radius+ radius}, {-5,2 radius + 5}}, AspectRatio->Automatic], {theta,0, 8 Pi, Pi/8}] :[font = subsection; inactive; preserveAspect] Write up of second situation - Problems 11-20: :[font = subsection; inactive; preserveAspect; startGroup] We consider the following situation where the length of the stick is equal to the diameter (l = 2 r). First we remove variable from the previous problem and redefine all the variables and functions. :[font = input; preserveAspect] Remove[radius,length,xwheel,ywheel,xstick,xroll,vxtotal,axtotal,jxtotal] :[font = input; preserveAspect; startGroup] radius =20; length = 40; :[font = message; inactive; preserveAspect; endGroup; endGroup] General::spell1: Possible spelling error: new symbol name "length" is similar to existing symbol "Length". :[font = subsection; inactive; preserveAspect; startGroup] We start the disk rolling at origin (0, 0) and the stick laying down l = 40 units to the left of the origin, i.e. the dragging tip of the stick is at (-40, 0). :[font = input; preserveAspect] xwheel[theta_]= radius*Sin[theta]; :[font = input; preserveAspect; startGroup] ywheel[theta_]= radius*(1 - Cos[theta]); :[font = message; inactive; preserveAspect; endGroup] General::spell1: Possible spelling error: new symbol name "ywheel" is similar to existing symbol "xwheel". :[font = input; preserveAspect] xstick[theta_]= Sqrt[length^2 - radius^2(1-Cos[theta])^2]; :[font = input; preserveAspect] xroll[theta_]= radius*theta; :[font = input; preserveAspect; endGroup] xtotal[theta_] = -xwheel[theta] - xstick[theta] + xroll[theta]; :[font = subsection; inactive; preserveAspect; startGroup] Here is a plot of the horizontal displacement of the dragging tip of the stick. :[font = input; preserveAspect; endGroup] Plot[xtotal[theta],{theta,0,4Pi}]; :[font = subsection; inactive; preserveAspect; startGroup] We compute the velocity and acceleration of the horizontal displacement of the dragging tip of the stick. :[font = input; preserveAspect; startGroup] vxtotal[theta_] = xtotal'[theta]; axtotal[theta_] = vxtotal'[theta]; :[font = message; inactive; preserveAspect] General::spell1: Possible spelling error: new symbol name "vxtotal" is similar to existing symbol "xtotal". :[font = message; inactive; preserveAspect; endGroup; endGroup] General::spell: Possible spelling error: new symbol name "axtotal" is similar to existing symbols {vxtotal, xtotal}. :[font = subsection; inactive; preserveAspect; startGroup] We compute and plot the velocity of the horizontal displacement of the dragging tip of the stick. :[font = input; preserveAspect; endGroup] vxtotal[theta_]=xtotal'[theta]; Plot[vxtotal[theta],{theta,0,4Pi}]; :[font = subsection; inactive; preserveAspect; startGroup] It appears that the dragging tip of the stick comes to a complete stop because the minimum value of the velocity looks to be 0. We confirm this below by determining where the derivative of velocity, acceleration, is 0. Indeed we see that the tip of the stick is at rest at theta = 2 Pi. :[font = input; preserveAspect] axtotal[theta_] = vxtotal'[theta]; Plot[axtotal[theta],{theta,0,4Pi}]; :[font = input; preserveAspect; startGroup] solmin = FindRoot[axtotal[theta] == 0,{theta,6.4}] :[font = output; output; inactive; preserveAspect; endGroup] {theta -> 6.283185309000783} ;[o] {theta -> 6.28319} :[font = input; preserveAspect; startGroup] 2 Pi//N :[font = output; output; inactive; preserveAspect; endGroup] 6.283185307179586 ;[o] 6.28319 :[font = input; preserveAspect; startGroup] vxtotal[theta]/.solmin[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 0. ;[o] 0. :[font = subsection; inactive; preserveAspect; startGroup] Now as to the maximum velocity we find that this is 31.547 cm/sec and this occurs at theta = Pi/2 when the wheel has finished a half a revolution, i.e,. the maximum velocity occurs when the pin is at the top of the wheel and the minimum velocity occurs when the pin is at the bottom of the wheel. We note that the maximum velocity is greater in this case and occurs just when the pin is just at the top path, as the stick is being "pulled upright" most rapidly. :[font = input; preserveAspect; startGroup] vxtotal[Pi/2]//N :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 31.54700538379251 ;[o] 31.547 :[font = subsection; inactive; preserveAspect; startGroup] We plot the displacement, velocity, and acceleration of the function xtotal(theta), the location of the dragging stick contact. :[font = input; preserveAspect; startGroup] Plot[{xtotal[theta], vxtotal[theta],axtotal[theta]}, {theta,0,4 Pi}, PlotStyle-> {Thickness[.005], Thickness[.01],Thickness[.015]}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsection; inactive; preserveAspect] The thickest curve is acceleration, the middle thickness curve is velocity, and the thinnest curve is position. We see sharp cusp on the position function at theta = Pi/2 (when the pin is at the top of the wheel and the stick is fully vertical). This gives rise to a discontinuity in the derivative, indeed, there is no derivative at these point. When we take the second derivatives there appears to be a second derivative at these points of discontinuity in the first derivative. But the second derivatives are both approaching 0 from the left and from the right and so we seem to have a derivative at these points, but the second derivative does not exist at these points. :[font = subsection; inactive; preserveAspect; startGroup] From the graph of the acceleration we note that acceleration peaks at 1.66536 sec to a value of 35.5782 cm/sec^2, (greater than in the shorter stick version above) again we have an asymmetry with respect to time, i.e. things do not happen at equal intervals in the period of the rolling wheel [0, 2 Pi] . :[font = input; preserveAspect; startGroup] jxtotal[theta_] = axtotal'[theta]; :[font = message; inactive; preserveAspect; endGroup] General::spell: Possible spelling error: new symbol name "jxtotal" is similar to existing symbols {axtotal, vxtotal, xtotal}. :[font = input; preserveAspect; startGroup] solamax = FindRoot[jxtotal[theta] == 0,{theta,1.5}] :[font = output; output; inactive; preserveAspect; endGroup] {theta -> 1.665362795850795} ;[o] {theta -> 1.66536} :[font = input; preserveAspect; startGroup] axtotal[theta]/.solamax[[1]] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] 35.57819325326904 ;[o] 35.5782 :[font = subsection; inactive; preserveAspect] The motion would appear to be stop and go, with an initial fast go (due to increasing acceleration in the interval [0, 1.66536]) and then decelerates. :[font = subsection; inactive; preserveAspect; startGroup] We animate the motion of the stick and the disk. :[font = input; preserveAspect; endGroup; endGroup; animationSpeed = 5] Do[Show[Graphics[{PointSize[.02],Point[{xtotal[theta],0}], Point[{radius theta - xwheel[theta],ywheel[theta]}], Circle[{radius theta,radius},radius], Line[{{-40,0},{8 Pi radius + radius,0}}], Line[{{xtotal[theta],0}, {radius theta - xwheel[theta],ywheel[theta]}}]}], PlotRange->{{-length - 10,8 Pi radius+ radius}, {-5,2 radius + 5}}, AspectRatio->Automatic], {theta,0, 8 Pi, Pi/8}] :[font = section; inactive; preserveAspect; endGroup] ISSUES IN SOLUTION ^*)