(*^ ::[ 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] Relative Motion of Rod and Yoke :[font = section; inactive; preserveAspect; startGroup] BRIEF ABSTRACT :[font = subsection; inactive; preserveAspect; endGroup] We seek to render parametric equations which describe the tip of a rod sitting astride a moving, circular-headed, piston, given the rules of motion of the piston. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: RODYOKE Full title: Relative Motion of Rod and Yoke Developer: Susan R. Clements, Terre Haute South Vigo High School, 3737 S. 7th Street, Terre Haute IN 47802 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 . ;[s] 3:0,0;55,1;67,2;803,-1; 3:1,12,9,Times,1,14,0,0,0;1,10,8,Times,1,12,0,0,0;1,12,9,Times,1,14,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] (See figure below.) A rod is attached to an overhead pivot at point A and hangs vertically. The angle at which the rod is hanging is being controlled by a disk of radius r centered at B attached to a yoke. The rod rests on the outer circumference of the disk at moving point C. As the yoke pushes up, the disk pushes the rod away from the vertical and as the yoke pushes down, the disk allows the rod to move closer to the vertical. The pivot of the rod, the center of the disk, and the yoke all lie on a vertical line AB. :[font = subsection; inactive; preserveAspect] To ease analysis we create functions and coordinatize the device as follows: :[font = subsection; inactive; preserveAspect] A = pivot point = (0, 20) dm (dm is a decimeter = 10 cm). :[font = subsection; inactive; preserveAspect] r = radius of disk = 8 dm. :[font = subsection; inactive; preserveAspect] cc(t) = center of disk = (0, M sin(w t)); M = 10 dm, w = 1 radian/sec. Notice that the disk is initially centered at the origin and begins its motion upward, thus pushing the rod out. :[font = subsection; inactive; preserveAspect] Draw the following diagram: On a line place point A, below it place point B. B is the center of the disk which moves vertically up and down. Draw a circle of radius r with center B. Draw a line segment from A tangent to circle B representing the rod. C is the point on the circle centered at B which is in contact with the rod. As the yoke is pushed up, disk centered at B pushes rod AC away from the vertical and as the yoke is pushed down, disk centered at B allows rod AC to move closer to the vertical. :[font = subsection; inactive; preserveAspect] We shall be interested in the angle between the vertical yoke and the rod (theta(t)) as well as the point on the disk, C, which the rod contacts as the yoke moves up and down, P(t) = (x(t), y(t)) . :[font = subsection; inactive; preserveAspect] 1. Derive an expression that relates the position of the following: center of disk (B) to pivot (A), radius of the disk (r), and the angle between the rod and the vertical (theta). :[font = subsection; inactive; preserveAspect] 2. Describe the relationship between the vertical velocity of the yoke (i.e. the rate of change point C moves) and the rate of change in the distance between the pivot (A) and the center of the disk (B). :[font = subsection; inactive; preserveAspect] 3. Derive an expression for the angular velocity of rod, i.e. the rate at which theta changes with respect to time t, in terms of the distance between the center of the disk and pivot (s) and the radius (r). :[font = subsection; inactive; preserveAspect] 4. Plot angle theta vs. time over the time interval [0, 4 Pi] sec. :[font = subsection; inactive; preserveAspect] 5. Plot angular velocity vs. time over the time interval [0, 4 Pi] sec. :[font = subsection; inactive; preserveAspect] 6. Plot angular velocity vs. distance from pivot to center of the disk over the time interval [0, 4 Pi] sec. This is called a phase portrait and you should describe how this curve is traced out as time increases, e.g., where on the curve would you find a point corresponding to t = 0, t = Pi/2, t = Pi, t = 3 Pi/2, t = 2 Pi, t = Pi/4? Recall that s(t) = s0 - M sin(w t) is the distance from pivot to the center of the disk. :[font = subsection; inactive; preserveAspect] 7. When is the angular velocity at a maximum size? :[font = subsection; inactive; preserveAspect] 8. Examine the angular acceleration and determine when it is at a maximum. :[font = subsection; inactive; preserveAspect; endGroup] 9. Determine the point (P(t)) on the disk where the rod touches the disk and plot its trajectory. :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Trigonometric functions, derivatives of trigonometric functions or derivatives of inverse trigonometric functions, related rates of change, angular velocity, angular acceleration, parametric equations. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect] The student would be aided in working this problem if a diagram were given. The basic problem is from the book Vector Mechanics for Engineers: Dynamics by Ferdinand P. Beer and E. Russell Johnston, Jr. (See references and sources below.) The student is challenged to examine the situation without concrete values given. ;[s] 3:0,0;112,1;153,2;323,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = subsubsection; inactive; preserveAspect] Inverse trigonometric functions are used and the student would need to be able to differentiate the ArcSin function. The problem could still be considered, however, by considering angular velocity as it relates to the size of the angle rather than the distance from the pivot to the yoke or the distance that the yoke has moved. :[font = subsubsection; inactive; preserveAspect; endGroup] The idea is to have the students relate their conception of the physical motion and see that it concurs with the mathematical description, e.g., does their model predict when the angle or angular velocity is a maximum? minimum? What do they make of the plots of angle, angular velocity, and angular acceleration plotted together? :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect; endGroup] Parametriuc description of motion, derivative, trigonometric functions. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] This problem can be formulated in 15 minutes in class IF a diagram is presented. :[font = subsection; inactive; preserveAspect] Expectations :[font = subsection; inactive; preserveAspect] Future payoffs :[font = subsection; inactive; preserveAspect] Extensions :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] References and Sources :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] The basic problem is found in the text Vector Mechanics for Engineers: Dynamics. Fifth Edition by Ferdinand P. Beer and E. Russell Johnston, Jr. New York: McGraw-Hill, 1988. Problem 15.110, p. 742. ;[s] 3:0,0;39,1;80,2;203,-1; 3:1,10,8,Times,1,12,0,0,0;1,10,8,Times,3,12,0,0,0;1,10,8,Times,1,12,0,0,0; :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 1. Derive an expression that relates the position of the following: center of disk (B) to pivot (A), radius of the disk (r), and the angle between the rod and the vertical (theta). :[font = subsubsection; inactive; preserveAspect] The disk is tangent to the rod as the yoke pushes upward. The relationship is therefore: Sin[theta] = radius(r)/s, (s = distance from A to center of yoke's circle B), which implies that theta = Arcsin(r/s), or s = r/Sin[theta], or r = s Sin(theta). In these equation, s , distance from A to center of yoke's circle B, is given by the equation: s = s0 - M Sin[w t] at any given time t, where M Sin[w t] is the up and down position of the yoke at time t and s0 is the original distance between the pivot (A = (0, 20)) and the center of the disk (B = (0, 0)). :[font = subsubsection; inactive; preserveAspect] To ease analysis we create functions and coordinatize the device as follows: :[font = subsubsection; inactive; preserveAspect] A = pivot point = (0, 20) dm. :[font = subsubsection; inactive; preserveAspect] r = radius of disk = 8 dm. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] cc(t) = center of disk = (0, M sin(w t)); M = 10 dm, w = 1 radian/sec. Notice that the disk is initially centered at the origin and begins its motion upward, thus pushing the rod out. :[font = input; preserveAspect] s0 = 20; M = 10; r = 8; w = 1; :[font = input; preserveAspect; endGroup; endGroup] theta[t_] = ArcSin[r/(s0-M Sin[w t])]; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 2. Describe the relationship between the vertical velocity of the yoke (i.e. the rate of change point C moves) and the rate of change in the distance between the pivot (A) and the center of the disk (B). :[font = subsubsection; inactive; preserveAspect] The velocity of the yoke (dB/dt) is the negative of the velocity of the pivot to disk center velocity (-ds/dt, where s is the distance between the pivot (A) and the center of the disk (B)). :[font = subsubsection; inactive; preserveAspect] 3. Derive an expression for the angular velocity of rod, i.e. the rate at which theta changes with respect to time t, in terms of the distance between the center of the disk and pivot (s) and the radius (r). :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Since we are relating the angular velocity to the yoke to pivot distance and the radius of the disk, we should use the version of the relationship which states that theta = Arcsin(r/s) or theta (t)= ArcSin(r/(s0-vot)) Differentiating both sides with respect to time we get: :[font = input; preserveAspect; endGroup; endGroup] vtheta[t_] = theta'[t]; :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 4. Plot angle vs. time. :[font = input; preserveAspect; endGroup] Plot[theta[t],{t,0,4 Pi}, AxesLabel ->{"t","theta"}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 5. Plot angular velocity vs. time. :[font = input; preserveAspect; endGroup] Plot[vtheta[t],{t,0,4 Pi}, AxesLabel ->{"t","theta'(t)"}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 6. Plot angular velocity vs. distance from pivot to center of the disk over the time interval [0, 4 Pi] sec. This is called a phase portrait and you should describe how this curve is traced out as time increases, e.g., where on the curve would you find a point corresponding to t = 0, t = Pi/2, t = Pi, t = 3 Pi/2, t = 2 Pi, t = Pi/4? Recall that s(t) = s0 - M sin(w t) is the distance from pivot to the center of the disk. :[font = input; preserveAspect] s[t_] = s0 - M Sin[w t]; :[font = input; preserveAspect; endGroup] ParametricPlot[{s[t],vtheta[t]},{t,0,4 Pi }, AxesLabel->{"s","theta'(t)"}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 7. When is the angular velocity at a maximum size? :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] From the plot of theta'[t] it would appear that near t = 1 and t = 3.3 there would be maxima and minima. :[font = input; preserveAspect; endGroup] Plot[vtheta[t],{t,0,4 Pi}, AxesLabel ->{"t","theta'(t)"}] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] To determine where vtheta[t] is at maximuum or minimum we need to examine when atheta[t] = theta''[t] = 0. :[font = input; preserveAspect] atheta[t_] = theta''[t]; :[font = input; Cclosed; preserveAspect; startGroup] sol1 = FindRoot[atheta[t]==0,{t,1}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 0.947280425693675} ;[o] {t -> 0.94728} :[font = input; Cclosed; preserveAspect; startGroup] sol2 = FindRoot[atheta[t]==0,{t,2}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 2.194312227896118} ;[o] {t -> 2.19431} :[font = input; preserveAspect; endGroup] t1 = t/.sol1[[1]]; t2 = t/.sol2[[1]]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] And we examine the v values of these two critical points. From this it would appear that the maximum size of the angular velocity is 0.447515 rad/sec. :[font = input; Cclosed; preserveAspect; startGroup] vtheta[t1] :[font = output; output; inactive; preserveAspect; endGroup] 0.447514980616627 ;[o] 0.447515 :[font = input; Cclosed; preserveAspect; startGroup] vtheta[t2] :[font = output; output; inactive; preserveAspect; endGroup; endGroup; endGroup] -0.4475149806166269 ;[o] -0.447515 :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 8. Examine the angular acceleration and determine when it is at a maximum. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We plot the angular acceleration. :[font = input; Cclosed; preserveAspect; startGroup] Plot[atheta[t],{t,0, 4 Pi}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] And in order to examine the maximum value of angular acceleration we take the derivative of angular acceleration and examine where it is 0. :[font = input; preserveAspect] jtheta[t_] = atheta'[t]; :[font = input; preserveAspect] Plot[jtheta[t],{t,0, 4 Pi}] :[font = input; Cclosed; preserveAspect; startGroup] sol1 = FindRoot[jtheta[t]==0, {t,.2}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 0.4404814390357749} ;[o] {t -> 0.440481} :[font = input; Cclosed; preserveAspect; startGroup] sol2 = FindRoot[jtheta[t]==0, {t,1.8}] :[font = output; output; inactive; preserveAspect; endGroup] {t -> 1.570796326794897} ;[o] {t -> 1.5708} :[font = input; preserveAspect; endGroup] t1 = t/.sol1[[1]];t2 = t/.sol2[[1]]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] Thus we see that the angular acceleration attains its maximum value at 0.440481 radians and this value is 0.298155 radians/sec^2 and its minimum value at 1.5708 = Pi/2 radians and this value is -1.33 radians/sec^2 :[font = input; Cclosed; preserveAspect; startGroup] atheta[t1] :[font = output; output; inactive; preserveAspect; endGroup] 0.2981549401707201 ;[o] 0.298155 :[font = input; Cclosed; preserveAspect; startGroup] atheta[t2] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] -1.333333333333333 ;[o] -1.33333 :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] EXTRA: We plot all three functions: angle (theta(t)) - thin, angular velocity (vtheta(t) = theta'(t)) - thicker, and angular acceleration (atheta(t) = theta''(t)) - thickest, on the same axes. :[font = input; preserveAspect; endGroup; endGroup] Plot[{theta[t],vtheta[t],atheta[t]},{t,0, 4 Pi}, PlotStyle->{Thickness[.005],Thickness[.01], Thickness[.015]}] :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 9. We determine the point (P(t)) on the disk where the rod touches the disk and plot its trajectory. :[font = subsubsection; inactive; preserveAspect] We need a few pieces of mathematical descriptions. :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] A = Pivot point . :[font = input; preserveAspect; endGroup] A = {0,20}; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] l = Length of rod for graphical purposes later. :[font = input; preserveAspect; endGroup] l = A[[2]] + M; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] cc[t] = Center of disk of radius r. :[font = input; preserveAspect; endGroup] cc[t_] = {0, M Sin[w t]}; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] s[t] = distance from pivot point A to center of disk cc[t]. :[font = input; preserveAspect; endGroup] s[t_] = s0 - M Sin[w t]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] theta[t] = angle between the rod and the vertical. :[font = input; preserveAspect; endGroup] theta[t_] = ArcSin[r/s[t]]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] c[t] = directed distance from pivot point A to center of disk cc[t]. :[font = input; preserveAspect; endGroup] c[t_] = cc[t] - A; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We find the point on the disk where the rod touches the disk (P(t)) by projecting the direction of c[t] onto the direction of the rod which is at angle theta[t] with the vertical. We call this latter (unit vector) direction vex[t]. :[font = input; preserveAspect] vec[t_] = {Sin[theta[t]], -Cos[theta[t]]}; :[font = input; preserveAspect; endGroup] P[t_] = A + c[t].vec[t] vec[t]; :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] We offer a ParametricPlot of the path of the point on the disk where the rod touches the disk, (P(t)). :[font = input; preserveAspect; endGroup] p = ParametricPlot[P[t],{t,0, 4 Pi}, PlotRange->{{-2 r, 2r},{-A[[2]] - M,A[[2]]+r}}, AspectRatio->Automatic] :[font = subsubsection; inactive; Cclosed; preserveAspect; startGroup] EXTRA: Finally we offer an animation of the disk, the rod, and the point of contact on the disk which the rod touches. :[font = input; preserveAspect; endGroup; endGroup; endGroup; animationSpeed = 7] Do[ Show[p, Graphics[ {PointSize[.05],Point[A], Point[P[t]],Point[cc[t]], Circle[{0,M Sin[t]},r], Line[{A,P[t]}], Line[{{0,M Sin[w t]}, P[t]}], Line[{P[t], P[t]+ (l - Sqrt[(P[t]-A).(P[t]-A)])* (P[t] - A)/Sqrt[(P[t]-A).(P[t]-A)]}], Line[{{0,-A[[2]] - M},{0,A[[2]]+r}}] } ], AspectRatio->Automatic, PlotRange->{{-2 r, 2r},{-A[[2]] - M,A[[2]]+r}} ], {t,0, 2 Pi,Pi/10} ] :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect; endGroup; endGroup] After a reasonable diagram has been drawn per the instructions the application of simple triangular trigonometric function sine and then the Arcsine function will determine the angle theta as a function of the parameter t (radians or seconds). From there on in it is repeated applications and scrutiny of derivatives. The location and mathematical description of the point P(t), the the point on the disk which the rod contacts can be accomplished in several ways; we chose to use projections. But this may be a challenge for students as they realize that what they want is an x-coordinate description and a y-coordinate description of the point P(t) in terms of the single parameter t. ^*)