%This is a template for LaTeXing homework for 
%the Winter 2008 classical mechanics course by John Baez

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\newcommand{\V}{V}            %potential energy
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	\begin{document}

       \begin{center}
      {\large Classical Mechanics Homework\\}  
      {\small January 16, 2008 \\}  
      {\small John Baez}  
      {\small homework by C. Pro}  
      \end{center}

\section*{Newton's 2nd law and constant acceleration.}
 A particle near the Earth's surface feeling only the force of
gravity.  This force is approximately independent of time and
position:$$F(t) = (0,0,-mg).$$  Here we are in $\R^3$
and $g$ is the downwards acceleration due to gravity - approximately
9.8 meters/second$^2$.\\\\ 
Solve Newton's second law $F =
ma$ for $q(t)\in\R^3$ for this $F(t)$ - find $q(t)$ in terms of the
initial position $q(0)$ and $\dot{q}(0)$.  Hint: the path it traces out is a parabola.\\\\

\sol Write $q(t)=(q_1(t),q_2(t),q_3(t))$ where $q_i\maps \R \to \R$. 
From Newton's 2nd law, we have	 $\ddot{q}_1(t)=\ddot{q}_2(t)=0$ and $\ddot{q}_3(t)=-g.$ 
It follows that $q_i(t)=q_i(0)+\dot{q}_i(0)t$ for $i=1,2$ and $q_3(t)=q_3(0)+\dot{q}_3(0)t-gt^2/2$. 
Therefore, $$q(t)=(q_1(0)+\dot{q}_1(0)t, q_2(0)+\dot{q}_2(0)t, q_3(0)+\dot{q}_3(0)t-\frac{g}{2}t^2).$$

\section*{Newton's 2nd law and linear acceleration.}
Consider a mass $m$ attached to an ideal spring. 
Define the position $q$ to be the distance of the mass relative to the equlibrium position 
(i.e., when the system is at rest). By Hooke's law, the force in this situation is approximately given by $$F(t)=-kq(t),$$ 
where $k\geq0$ is the spring constant.\\\\ 
Solve Newton's second law for $q(t)\in\R$ in terms of $m$, $k$, $q(0)$,
$\dot{q}(0)$ and find the period $P$ of the oscillation and the
frequency $\omega$, where $\omega = \frac{2\pi}{P}$ Hint: It
oscillates!\\\\

\sol Newton's second law says $$\ddot{q}(t)+\frac{k}{m}q(t)=0$$ which has a general solution of $$q(t)=A\cos\left(\sqrt{\frac{k}{m}}t\right)+B\sin\left(\sqrt{\frac{k}{m}}t\right).$$ With our initial conditions, we have $$q(t)=q(0)\cos\left(\sqrt{\frac{k}{m}}t\right)+\dot{q}(0)\sqrt\frac{m}{k}\sin\left(\sqrt{\frac{k}{m}}t\right).$$ Note the period and frequency are $P=2\pi\sqrt{m/k}$ and $\omega =\sqrt{k/m}$, respectively. 

%Copy relevant parts of the homework problem I assigned from the
%LaTeX file available at http://math.ucr.edu/home/baez/classical/texfiles

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