% SIMIODETemplate-TeacherORStudent.tex and simiode.cls for LaTeX2e
% Version 29 July 2015.
% 
% This class (simiode.cls) is a modification of that used by 
% the journal PRIMUS - Problems, Resources, and Issues in 
% Mathematics Undergraduate Studies. 
% 
% See http://www.tandfonline.com/loi/upri .
% 
% The class was developed by  Marc Frantz with suggestions of the 
% PRIMUS Editorial Board, especially John Thoo.
%
% This template is to be used with the  class file "simiode.cls". 
%  
% The latest version of this class and template files 
% can be found at
% 
%       <http://www.SIMIODE.org> in the Author section.
%
% A "Quick Start Guide" appears below.
%
% This file must be processed with version LaTeX2e or higher.
%
\documentclass{SIMIODE}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% With this version of the tex file and the associated SIMIODE.cls file you can produce BOTH a Student Version and a Teacher Version of your modeling scenario or technique narrative.
%
% Basically the Student Version contains the title STUDENT VERSION along with the NAME OF SCENARIO, with no abstract, no keywords, no tags, no author identifiers, and no COMMENTS section. The Student Version contains only the STATEMENT of the modeling scenario with appropriate figures, data, instructions, and perhaps a bibliography appropriate for student resourcing, but nothing in the bibliography which would "give away" the nature of the modeling scenario, etc.
%
%The Teacher Version contains the material in the Student Version, namely the title NAME OF SCENARIO, now headed with   TEACHER VERSION, but also full author  identification, abstract, key words, tag words, and, following the STATEMENT, a section of COMMENTS which can be discussion of presentation  ideas, "solutions," techniques, etc.
%
% Authors can prepare a full Teacher Version using the notions immediately below  to create a pdf for review (but in review please eliminate all information about the author as we use a double-blind referee system. Such information will be restored when accepted and published in SIMIODE at www.simiode.org.
%
%***********   Toggle the following line to turn the teacher version on (1) or off (0). ***********
%
%
\def\TeacherEdition{0} % Toggle this line's value.
\newcommand{\teacher}[1]{\ifnum\TeacherEdition=1 #1 \fi}
%
%
% The \teacher{ } command wraps any text that will appear only in the teacher version of
% the document. See this file for examples. This has been done for you in the document below.   
% Rename the pdf after you compile each final version.
% This option was created  by Eric Sullivan, Carroll College, Helena MT.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage{amsmath,amssymb}
\usepackage{color}
\usepackage{graphicx}
%\usepackage{hyperref}
\usepackage{url}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding double spacing %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Manuscripts should be submitted doubled spaced.  This is 
% accomplished via the \baselinestretch command below.
% However, you may obtain an idea of how your article will 
% look in final form, so that you can better plan your 
% presentation.  To see your manuscript formatted very 
% similarly to the on-line published format of  SIMIODE, 
% comment the following line out by inserting a percent sign % (%) at its beginning.

\renewcommand{\baselinestretch}{1.3}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% NOTE regarding macros %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Any macros defined for your paper should be contained in 
% the top matter. Likewise, any environment definitions such 
% as \newtheorem or \newenvironment should also go in the 
% top matter. 
% 
% Do not \input your macros as separate files within your 
% final version because more files make it harder for the 
% editors and users to keep track of and/or modify and 
% append your paper.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Top matter is above and 
% the beginning of the document below.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\newpage
% You only need to change the NAME OF SCENARIO in the next line and delete the footnote. See above for instructions on toggling from Teacher Version to Student Version.
\title{\ifnum\TeacherEdition=1 TEACHER VERSION  \else STUDENT VERSION \fi\\CHORD PATH TIME}

\markboth{Chord Path Time }{Chord Path Time }






\author{Brian Winkel, Director SIMIODE\\
Chardon OH 44024  USA\\
BrianWinkel@simiode.org
}



% \makeStitle
% replaces: \makeStitle
% Choose only one of the following:
\makeStitlePDFLaTex  % <--- use this for PDFLaTex (jpg and pdf image files etc.)
% \makeStitleLaTex    % <--- use this for LaTex (eps image files)


\begin{abstract}
Using Newton's Second Law of motion students can determine the time it takes for a mass to slide along a chord of a vertical circle from high point to any point along the circle.  They come to a nonintuitive conclusion through their analysis, but upon reflection the conclusion does make sense.
\end{abstract}



\section*{SCENARIO DESCRIPTION}

A mass particle starts from rest at the highest  point, $O = (0,1)$, of a vertical circle and slides along a chord  only effected by the acceleration due to  gravity to another terminal point $A$ on the circle. See Figure~\ref{fig:3-20-ChordPathTime}. 
What can you say about the time(s) it takes for the mass to hit the terminal points in terms of a changing angle $\theta$? Reflect on your answer and make comments on the reasonableness and interpretations we might offer for your solution. This modeling opportunity appears in \cite[p.~59]{O'Neil2003} and other textbooks. However, we believe the use of elementary vector mathematics to obtain a differential equation and then solve the equation and interpret the solution is  a worthy modeling task.

\begin{figure}[h]
\centering
\includegraphics[width=13cm]{3-20-ChordPathTime.eps}


\parbox{9.8 true cm}{\caption{\label{fig:3-20-ChordPathTime}  Useful figure in  determining the time it takes for a mass particle to slide along the chord from point ) at the top of a circle to an arbitrary point A along the circle.}}




\end{figure}

\teacher{
\section*{NOTES FOR TEACHERS}

This is a nice, short modeling scenario in which one application of Newton's Second Law of Motion and a bit of trigonometry gets the student to a reasonable second order differential equation which can then be solved. Once the solution is at hand for the position of the mass particle as a function of time $t$ that solution yields an $(x,y)$ coordinate in terms of time of travel, $t$. This placed into the defining equation of the circle, $x^2 + y^2 = 1$ gives rise to two meaningful solutions, one for $t = 0$ and one for (surprisingly) $ t = 2\sqrt{\frac{r}{g}}$ - alas, independent of $\theta$. After some discussion the result might not appear so surprising for the steeper (more vertical) the chord is the longer the path is, but the velocity is increasing as the mass particle moves along that longer track due to gravitational acceleration so it is conceivable that the time for all chords is comparable, even the same!!! 

We offer a complete analysis in a Mathematica notebook accompanying this modeling scenario, 30200T-ChordPathTime-TeachersVersion.nb and a pdf version for those who do not have access to Mathematica. Of course, this analysis could be done by hand or in some other computer algebra system.

In Figure~\ref{fig:3-20-ChordPathTime} consider the vertical circle with radius $r = 1$ meter and the release point at $O = (0, 1)$. 

Point $P$ is the current position of the mass along the chord of the circle.

$s(t)$ is the distance along that chord traversed to point $P$ at time $t$ seconds.
$\theta$ is the angle the chord makes with the vertical line through the point $O$.
$m\cdot g$ is the force due to gravity acting on the mass - always vertical and always downward. The value of $g$ in SI units is $g = 9.8 m/(s^2)$.

We can decompose the gravity force into a component normal to the chord and a component in the direction of the chord. Using trigonometry we  find the magnitude of the component in the direction of the chord to be $m\cdot g \cdot \cos(\theta)$.

So, considering only the motion along the chord and using Newton's Second Law of Motion which says that the sum of the forces acting on a body are equal to the product of the body's mass and acceleration; in this case along the direction of the chord of the circle, we have the following second order differential equation with initial conditions $s(0) = 0$, i.e., the distance traversed along the chord at the start $t=0$ is 0 and $s'(0) = 0$ the initial velocity is for we simply release the mass to travel along the chord.


\begin{equation}\label{eq:basicEq}
m  \cdot s''(t) = g  \cdot m  \cdot \cos(\theta)\, , 
\end{equation}


We can solve this differential equation, (\ref{eq:basicEq}), to obtain


\begin{equation}
s(t) = 4.9 \, t^2 \cos (\theta )\, ,
\end{equation}
and thus we have the distance traversed,$s(t)$,  along the chord at time $t$.


Thus we can locate the point $P = P(t, \theta)$ as a function of the time from release, $t$ in seconds,  and the angle, $\theta$, of the chord with the vertical line through $(0, r = 1)$.

This means 

\begin{equation}\label{eq:eqP}
P = P(t, \theta)= (0, r) +  4.9 \, t^2 \cos (\theta ) \cdot (\sin(\theta), -\cos(\theta))
\end{equation}
where $(\sin(\theta), \cos(\theta))$  is a unit vector in the direction of the chord and $s(t) = 4.9 \, t^2 \cos (\theta )$ is the distance traversed along the chord of the circle at time $t$.

We seek to determine the time, $t_i$, at which the mass impacts the circle on the path of the chord. So we seek to determine the value of $t$ in $P(t, \theta)$ when the point $P(t, \theta)$ lies on the circle, again. We substitute the $x$ and $y$ coordinates of $P(t, \theta)$ into the equation of the circle of radius $r = 1$ centered at the origin, i.e., $x^2 + y^2 = 1$ and find that

\begin{equation}\label{eq:intCirc}
24.01 t^4 \sin ^2(\theta ) \cos
   ^2(\theta )+\left(1-4.9 t^2 \cos
   ^2(\theta )\right)^2=1\, .
\end{equation}

Now when we attempt to solve the quartic equation in $t$ to determine the time at which the mass particle sliding along the chord actually hits the circle again we obtain

\begin{equation}\label{eq:Hit}
t_i = 0.638877\, .
\end{equation}

This means, $\theta$ , the angle for the chord has nothing to do with the time of impact! Indeed, if we solve the differential equation with general $r$ and $g$, determine  $P(t, \theta)$, and then determine time of impact, $t_i$, we find


\begin{equation}\label{eq:HitGen}
t_i = 2 \sqrt{\frac{r}{g}}\, .
\end{equation}

Thus, again, we confirm the time of impact of the mass particle on the circle as it slides along the chord is constant, independent of the angle, $\theta$, of the chord.

Now, if we examine the form of (\ref{eq:HitGen}) we note that if we increased $g$ this time will decrease.  This makes sense as it is the size of the force vector's component along the wire that then increases. Also, if we increase the radius of our circle, $r$, this time will also increase as the circle will be bigger.
}


\begin{thebibliography}{99}


\bibitem{O'Neil2003}  O'Neil, P.~V. 2003.  {\it Advanced Engineering Mathematics, Fifth Edition\/}. Pacific Grove CA: Brooks/Cole-Thomson Learning.  p.~59. 

\end{thebibliography}






\end{document}
