% Simplified version of an UMTYMP template.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%
% If the text is too small, change 10pt to 12pt
\documentclass[11pt]{article}
\RequirePackage{amsmath,amsthm,amssymb,latexsym,amsthm,verbatim,xspace,amstext,xspace,setspace}

% Here's where you edit the class, section and title information

% Set a fixed margin along all sides
\usepackage[margin=1in, nohead, pdftex]{geometry}
\usepackage{enumerate,graphicx}
% Re-adjust the top margin if desired
\topmargin -0.2in
% These packages used for spacing in the header and throughout



% Uncomment one of the following to set the spacing to single, 1.5 or double spacing
\singlespacing
%\onehalfspacing
%\doublespacing

% This prevents paragraphs from begin indented.  Comment it out to have indentation.
\parindent 0ex
\setlength{\textheight}{9.5in}
% Finally we start!
\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This is the "header" at the top.  You shouldn't have to edit these lines!
% Instead, chance the Class, Section and Title above.
\begin{center}{ \Large{Worksheets}  \\\large{05/03/2018} }\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


% At this point, you might have some introductory text, or jump straight
% into an ``enumerate'' environment for the problem statements.
\bigskip
\begin{enumerate}
\item Suppose that
\[y(t)=3{{e}^{-2t}}\cos t-2{{e}^{-2t}}\sin t+\sqrt{2}\cos (5t-\delta )\]is the solution of a mass-spring system
\[mx''+bx'+kx={{F}_{0}}\cos {{\omega }_{f}}t.\]
\begin{enumerate}
\item What part of the solution is the transient solution? What part is the steady-state solution?\vspace{1in}
\item If the mass is 1 kg, what are the damping constant $b$ and the restoring constant $k$?\vspace{1.5in}
\item Is the system underdamped, critically damped, or overdamped?\vspace{0.5in}
\item What is the time-varying amplitude of the transient solution?\vspace{1in}
\item Find the angular frequency $\omega_f$ of the forcing function. Find the forcing amplitude $F_0$.\vspace{2in}
\end{enumerate}
\item Determine whether the following matrix is diagonalizable. If it is, diagonalize it (i.e. determine an invertible matrix $P$ and a diagonal matrix $D$ such that $A=PDP^{-1}$).\[A=\left[ \begin{matrix}
   4 & 2 & 0  \\
   -1 & 1 & 0  \\
   0 & 1 & 2  \\
\end{matrix} \right]\]
\newpage
\item \begin{enumerate}
\item Check if the following set of vectors are linearly independent. If not, express one vector as a linear combination of the others.
\[{{v}_{1}}=(1,0,1,1),~{{v}_{2}}=(-1,1,0,0),~{{v}_{3}}=(2,1,-1,1),~{{v}_{4}}=(-1,0,3,1)\]\vspace{3.5in}
\item Recall that the set $C(\mathbb{R})$ of continuous functions on the real line $\mathbb{R}$ is a vector space (in which each vector is a continuous function). Check if the following set of functions are linearly independent.\[{{f}_{1}}(t)=t,~~{{f}_{2}}(t)=\cos t\]
\end{enumerate}\newpage
\item Suppose two equal masses $m_1=m_2=1$ are attached to three springs, where two outside springs are attached to walls. The spring constants from the left to right are respectively $k_1=1$, $k_2=2$, $k_3=1$. Denote by $x(t)$ and $y(t)$ the positions of the respective masses $m_1$ and $m_2$ from their respective equilibrium positions.

The system is set in motion by holding the left mass in its equilibrium position while at the same time pulling the right mass to the right of its equilibrium a distance $y(0)=2$.
\begin{enumerate}
\item Write the differential equations for $x(t)$ and $y(t)$ together with the initial conditions.\vspace{3.5in}
\item Determine the Laplace transforms of $x(t)$ and $y(t)$.
\end{enumerate}
\end{enumerate}
\end{document}











