% ============================================================================ % AUTO-GENERATED STUDENT LATEX TEMPLATE % This file is generated from the canonical assignment source. % All selected QABank files have been inlined and all solution environments % have been removed. It is therefore self-contained and safe to distribute. % Students may type their work directly into this copy. % Do not edit this generated file when maintaining the canonical assignment. % ============================================================================ \documentclass[11pt]{article} % >>> BEGIN EXPANDED INPUT: AssignmentCommon.tex % AssignmentCommon.tex % Shared formatting and reusable question-bank machinery for PHYS V1100. % Intended to be input from PSet*.tex and Exam*.tex master files. \usepackage[margin=1in]{geometry} \usepackage{amsmath,amssymb,mathtools,bm} \usepackage{enumitem} \usepackage{etoolbox} \usepackage{xparse} \usepackage{comment} \usepackage{xcolor} \usepackage{keyval} \usepackage{fancyhdr} \usepackage{tikz} \usetikzlibrary{arrows.meta,positioning,decorations.pathmorphing} % ----------------------------------------------------------------------------- % Global switches controlled by the master assignment/exam file. % ----------------------------------------------------------------------------- \newtoggle{ANS} \newtoggle{HINTS} % ----------------------------------------------------------------------------- % Guided multipart questions. % QA is deliberately a list environment: each part can scaffold the next part. % Two levels are defined for occasional subparts. % ----------------------------------------------------------------------------- \newlist{QA}{enumerate}{2} \setlist[QA,1]{ label=(\alph*), ref=\alph*, leftmargin=*, itemsep=0.9em, topsep=0.5em, parsep=0pt } \setlist[QA,2]{ label=(\roman*), ref=\roman*, leftmargin=*, itemsep=0.45em, topsep=0.35em, parsep=0pt } % Semantic wrapper for the common stem of a question. This is intentionally a % normal environment rather than a body-capturing macro, so bank questions remain % compatible with figures, listings, and other complex LaTeX content. \newenvironment{question}{}{} % Configure visibility after the master file sets ANS and HINTS. The comment % package skips hidden bodies completely and remains robust for long solutions, % nested lists, figures, and verbatim-style material. \newcommand{\ConfigureQAVisibility}{% \iftoggle{ANS} {% \specialcomment{answer} {\par\smallskip\begingroup\small\color{red}\noindent\textbf{Solution.}\quad} {\par\endgroup\smallskip}% } {\excludecomment{answer}}% \iftoggle{HINTS} {% \specialcomment{hint} {\par\smallskip\begingroup\small\color{black!65}\noindent\textit{Hint.}\quad} {\par\endgroup\smallskip}% } {\excludecomment{hint}}% } % ----------------------------------------------------------------------------- % Question selection from the bank. % Assignment-specific metadata belongs here, not inside the bank file. % % ----------------------------------------------------------------------------- \newcounter{problem} \makeatletter \define@key{QAUse}{label}{\def\QAUseLabel{#1}} \define@key{QAUse}{points}{\def\QAUsePoints{#1}} \newcommand{\QuestionHeader}[1]{% \par\bigskip \noindent {\large\bfseries Problem #1}% \ifdefempty{\QAUseLabel}{}{\hfill\fbox{\strut\sffamily\small \QAUseLabel}}% \ifdefempty{\QAUsePoints}{}{\hspace{0.75em}{\small[\QAUsePoints\ points]}}% \par\smallskip \hrule \medskip } % Begin a question without specifying where its body comes from. This public % wrapper is also used by the student-template generator after it inlines a bank % question into a single self-contained source file. \NewDocumentCommand{\BeginQuestion}{O{}}{% \def\QAUseLabel{}% \def\QAUsePoints{}% \setkeys{QAUse}{#1}% \refstepcounter{problem}% \QuestionHeader{\theproblem}% } \makeatother % Convenience notation used repeatedly in the mechanics assignments. \newcommand{\vect}[1]{\bm{#1}} \newcommand{\ones}{\vect{1}} % <<< END EXPANDED INPUT: AssignmentCommon.tex % ----------------------------------------------------------------------------- % Output mode. Change these two lines to generate alternate versions. % Student version: ANS=false, HINTS=false % Hinted version: ANS=false, HINTS=true % Solution version: ANS=true, HINTS=true % ----------------------------------------------------------------------------- \settoggle{ANS}{false} \settoggle{HINTS}{false} \ConfigureQAVisibility % Easy-to-edit assignment metadata. \newcommand{\coursename}{PHYS V1100: Analytical Dynamics} \newcommand{\semester}{Fall 2026} \newcommand{\instructor}{Professor Mark Shattuck} \newcommand{\assignmenttitle}{Problem Set 1} \pagestyle{fancy} \fancyhf{} \lhead{\coursename} \rhead{\assignmenttitle} \cfoot{\thepage} \setlength{\headheight}{14pt} \begin{document} \begin{center} {\Large\bfseries \assignmenttitle}\\[0.35em] {\large \coursename}\\[0.2em] \semester \hfill \instructor \end{center} \vspace{0.5em} \hrule \vspace{0.75em} \noindent Show enough reasoning that another person can follow your argument. Computation may be used where requested, but the purpose is to expose the mechanical structure, not just to obtain a numerical answer. % The master file chooses the questions and assigns semester-specific roles. % Reordering these lines redistributes the same bank questions without editing them. % ===================================================================== % FLATTENED QUESTION: QABank/Q01_01.tex % ===================================================================== \BeginQuestion % ID: Q01_01 % Lecture: 1 % Topic: state, deterministic evolution, projection % Type: conceptual % Difficulty: introductory % Computational: no % Last revised: 2026-09-03 \begin{question} A useful description of a dynamical system must contain enough information for the dynamics to close. In Lecture 1 we expressed this idea schematically as \[ S_{n+1}=F(S_n). \] For each situation below, distinguish a proposed state from a reduced or incomplete description. \begin{QA} \item A one-dimensional particle obeys \[ m\ddot{x}=-kx. \] Is \(x(t)\) alone a complete instantaneous state? Is \((x(t),\dot{x}(t))\) a complete state? Explain. \item Two masses are coupled by springs. Suppose the complete mechanical state is \[ S=(x_1,\dot{x}_1,x_2,\dot{x}_2), \] but an observer records only \[ s=(x_1,\dot{x}_1)=P(S). \] Why need there not exist an autonomous evolution law \(s_{n+1}=f(s_n)\)? \item Give one way in which the reduced description in part (b) could be made into a better state description without returning to the full microscopic state. \end{QA} \end{question} % ===================================================================== % FLATTENED QUESTION: QABank/Q02_01.tex % ===================================================================== \BeginQuestion % ID: Q02_01 % Lecture: 2 % Topic: fixed-fixed chain, incidence matrix, stiffness assembly % Type: derivation / interpretation % Difficulty: medium % Computational: no % Last revised: 2026-09-03 \begin{question} Consider four free masses arranged in a one-dimensional chain between two fixed walls. There are therefore six nodes in all: nodes 0 and 5 are fixed, while nodes 1--4 have displacements \[ \vect{u}=(u_1,u_2,u_3,u_4)^T. \] Five springs connect consecutive nodes and have spring constants \(c_1,\ldots,c_5>0\). Orient every spring from the lower-numbered node toward the higher-numbered node. \begin{QA} \item Construct the matrix \(A\) that maps the four free nodal displacements to the five oriented spring extensions. \item Define \[ C=\operatorname{diag}(c_1,c_2,c_3,c_4,c_5). \] Write the elastic potential energy in matrix form and explain the role of \(A\) and \(C\). \item Compute \(K=A^TCA\). \item Interpret the diagonal and off-diagonal entries of \(K\) mechanically. Why is there no rigid-translation zero mode for this system? \item If \(\vect{f}\) is the externally applied nodal load, write the static equilibrium equation and identify the internal restoring force. \end{QA} \end{question} % ===================================================================== % FLATTENED QUESTION: QABank/Q02_02.tex % ===================================================================== \BeginQuestion % ID: Q02_02 % Lecture: 2 % Topic: static constant-load chain, exact quadratic discrete solution % Type: analytic + computational % Difficulty: medium % Computational: MATLAB or Python/NumPy % Last revised: 2026-09-03 \begin{question} Consider \(M\) free masses between two fixed endpoints. There are \(M+1\) identical unit springs, so \(C=I\), and each free mass is acted on by the same unit external load, \[ \vect{f}=\vect{1}. \] Let the endpoint displacements be \(u_0=u_{M+1}=0\). \begin{QA} \item Show that the interior equations in \[ A^TA\vect{u}=\vect{1} \] can be written \[ 2u_i-u_{i-1}-u_{i+1}=1, \qquad i=1,\ldots,M. \] \item Verify directly that \[ u_i=\frac{i(M+1-i)}{2}, \qquad i=0,1,\ldots,M+1, \] satisfies both the boundary conditions and every discrete equilibrium equation. \item Reproduce the static solve in either MATLAB or Python/NumPy for \(M=30\). Plot the computed nodal displacements together with the quadratic expression from part (b). Your code should form \(A\), \(C\), \(\vect{f}\), and \(A^TCA\) explicitly. \item Copy your commands into a script. Change exactly one of the following and briefly describe what changes physically and in the solution: \begin{QA} \item the number of free masses \(M\); \item the diagonal entries of \(C\); \item the applied load \(\vect{f}\). \end{QA} \end{QA} \end{question} % ===================================================================== % FLATTENED QUESTION: QABank/Q02_03.tex % ===================================================================== \BeginQuestion[label=Oral Presentation] % ID: Q02_03 % Lecture: 2 % Topic: discrete fundamental theorem of calculus, boundary % Type: derivation / oral presentation candidate % Difficulty: medium % Computational: no % Last revised: 2026-09-03 \begin{question} Let \(A\in\mathbb{R}^{N\times(N+1)}\) be the full incidence matrix of an open chain with nodes \(0,1,\ldots,N\), with every edge oriented from left to right. Thus row \(i\) of \(A\) computes \(u_{i+1}-u_i\). Let \(\vect{1}_E\) be the vector of ones in edge space, and let \(\vect{e}^{(j)}\) be the nodal basis vector with components \((\vect{e}^{(j)})_i=\delta_{ij}\). \begin{QA} \item For a chain with four edges, write \(A\) explicitly and calculate \(A^T\vect{1}_E\). \item Generalize the result to \(N\) edges and show that \[ A^T\vect{1}_E=-\vect{e}^{(0)}+\vect{e}^{(N)}. \] What does the support of this vector tell you? \item Use only matrix identities to show \[ \vect{1}_E^TA\vect{u}=u_N-u_0. \] Then write the same identity as an explicit telescoping sum. \item Explain in words why this is a discrete fundamental theorem of calculus. Compare it with \[ \int_a^b u_x\,dx=u(b)-u(a). \] \item Now replace the open chain by a ring with consistently oriented edges. What do you expect for \(A_{\rm ring}^T\vect{1}_E\), and why? \end{QA} \end{question} % ===================================================================== % FLATTENED QUESTION: QABank/Q02_04.tex % ===================================================================== \BeginQuestion % ID: Q02_04 % Lecture: 2 % Topic: discrete summation by parts / adjoint incidence matrix % Type: derivation / interpretation % Difficulty: medium % Computational: no % Last revised: 2026-09-03 \begin{question} Use the same open-chain incidence matrix \(A\) as in the discrete fundamental theorem problem. Let \(\vect{v}\in\mathbb{R}^N\) be an arbitrary quantity living on the edges, with components \(v_0,\ldots,v_{N-1}\). \begin{QA} \item Explain why the scalar identity \[ \vect{v}^TA\vect{u} =\vect{u}^TA^T\vect{v} =(A^T\vect{v})^T\vect{u} \] is true without making any reference to calculus. \item Calculate the components of \(A^T\vect{v}\). Separate the two boundary components from the interior components. \item Expand \(\vect{v}^TA\vect{u}=\vect{u}^TA^T\vect{v}\) in components and show that \[ \sum_{i=0}^{N-1}v_i(u_{i+1}-u_i) = -\sum_{i=1}^{N-1}u_i(v_i-v_{i-1}) +u_Nv_{N-1}-u_0v_0. \] \item Compare the discrete identity with \[ \int_a^b v u_x\,dx =[vu]_a^b-\int_a^b v_xu\,dx. \] Which part of \(A^T\vect{v}\) acts like an interior derivative, and which part carries the boundary information? \item Recover the discrete fundamental theorem of calculus as a special case by choosing an appropriate \(\vect{v}\). \end{QA} \end{question} % ===================================================================== % FLATTENED QUESTION: QABank/Q02_05.tex % ===================================================================== \BeginQuestion[label=Challenge] % ID: Q02_05 % Lecture: 2 % Topic: paths, cycles, boundary of a discrete chain % Type: challenge % Difficulty: advanced / conceptual % Computational: no % Last revised: 2026-09-03 \begin{question} Let \(A\) be the incidence matrix of an oriented graph. An edge-space vector \(\vect{c}\) is used to describe an oriented path: its component is \(+1\) when the path uses an edge in the chosen edge orientation, \(-1\) when it uses the edge in the opposite direction, and \(0\) when it does not use that edge. \begin{QA} \item Consider a simple path beginning at node \(a\) and ending at node \(b\). Argue that all interior node contributions cancel in \(A^T\vect{c}\), and show that \[ A^T\vect{c}=-\vect{e}^{(a)}+\vect{e}^{(b)}. \] \item Use part (a) to show \[ \vect{c}^TA\vect{u}=u_b-u_a. \] Interpret the result. \item What changes if \(\vect{c}\) describes a closed cycle? \item Relate the result schematically to the idea \[ \partial^2=0, \] ``the boundary of a boundary is zero.'' No formal algebraic-topology machinery is required. \end{QA} \end{question} \end{document}