\documentclass{article}
\usepackage{exam3}
\usepackage{amssymb}
\usepackage{graphicx}
\usepackage{amsmath}
\usepackage{epstopdf}
\usepackage{hyperref}
\usepackage{enumerate}
\usepackage{color}
\paper{PHYS 2210}               % <- do not include FC, FT etc

\title{Homework 7}
\time{Start of class on Thurs. Mar 1}
\semester{Spring}
\year{2012}                     % <- default is the current year
%\campus{City}
%\note{ Be sure to show your work and explain what you are doing. (Correct answers, for which we cannot follow the work, are worth no credit). }
\begin{document}

\begin{questions}

\question Electrical force is given by $\vec F = q \vec E$, where $\vec E$ is the electric field. Electric force is conservative in ``electrostatic" situations (but is {\it not} conservative in all situations!) 


\begin{enumerate}
\item Consider a laboratory setup in which a charge q sits in a field $\vec E = (z^2+1) {\hat k}$.  Prove that the resulting force is conservative, and deduce a formula for the potential energy as a function of position of this charge

\item You release the charge (it has mass m) at the origin, starting from rest. Describe its motion qualitatively. How fast is it going when it reaches a distance h from the starting point? If the object was in the shape of a bead ({\it same} mass m and charge q) threaded on a curved, frictionless, rigid wire which started at the origin $(0,0,0)$  and ended at some point $(x_0,y_0,h)$  would it also have that same speed you just calculated, or not? (Why?) 
%% Comment for next time. It's ok, but perhaps making the field depend on x or y too would prevent this from being overly simple? 

\end{enumerate} 

\question Electrical potential energy is given by $U = q V$, where $V$ is the voltage. Consider a charge q moving in a region of space where the voltage is approximated by the formula $V(x,y,z) =  c(2x  z +x^2) $ (Assume the constant c has appropriate units so U is in Joules when x,y, z are in meters) 
\begin{enumerate}
\item
Find the force on the charge q at the origin, and the point (1,1,1)
\item Does this situation produce a conservative electrical force? Clearly justify your answer.
\item For an arbitrary scalar function f(x, y, z), evaluate the components of
$\vec{\nabla} \times \vec{\nabla} f$ in Cartesian coordinates and show that the result is 0. Does the formal result relate to (b)? (If not, why not? If so, briefly comment) 

\item Explicitly compute the work done by the force in part a on a charge q as you follow a path from the origin (0,0,0)  to (1,1,1) along a direct, straight line path. Then, check your answer very simply using the potential function given. Explain the idea of your check. 
%% Comment for next time. I would like to make the starting point something other than 0,0,0 so they have to think about "subtracting the starting point". Maybe make the starting point (-1, -1, -1), so the straight path is still nontrivial (you must parametrize z(x) or x(z) to do the integrals - but those are easy enough!! 


\end{enumerate}


%%NOTE: This question is very easy for most students, it's good but very much a quirky!!

\question The diagram below shows a region of space. The dashed curves indicate  positions of equal potential and are labeled with the value of the potential at that curve. Three vectors originating from point Y are also shown in blue. The vector B points directly from point Y to Z. If you want a particle to be launched from Y and reach  Z, which vector represents best the initial velocity? Explain your reasoning.  Then, make a (very crude, no calculations required!) sketch/guess of the {\it path} you expect the particle to follow if launched from point Y with initial {\bf v} given by vector A. ( Explain your reasoning, briefly, so we know what you were thinking about)  
 %\begin{figure}[htc]
%\centering
\begin{center}
{\includegraphics[width=2in]{Hom7fig1.pdf}}%\caption{}
\end{center}
%\end{figure}

\question Each diagram in Fig 2 (next page) depicts a force field in a region of space. 
\begin{enumerate}
\item For which force fields  can you identify a closed path over which  $\oint \vec{F}\cdot d\vec{l}\neq 0$? For each such case, clearly indicate an appropriate path on the diagram.
\item For each case indicate if $\nabla \times \vec{F}=0$ everywhere in the box? Also, for each case, could the force  depicted in the diagram  be conservative? Briefly, explain.  
\item For each case, is it possible to draw a self-consistent set of
equipotential contours for that situation? If so: Draw them! (Each drawing should clearly show the correct shape of the contour lines,
 the correct relative spacing of the contours, and label the regions that
correspond to highest and lowest potential energy) If not: Explain why drawing such contours is impossible.
\end{enumerate}

\begin{center}
{\includegraphics[width=2.5in]{Hom7fig2.pdf}}%\caption{}
\end{center}





\question  A driver traveling {\it downhill} slams on the
brakes and skids 40 m on before hitting a parked car. You have been hired as a physics expert to help the insurance investigators decide if the driver had been traveling faster than the 25 MPH speed limit at the start of this event.  The slope of the hill is 5 degrees.   Assuming braking friction has the usual form $\mu N$, what is the ``critical value" of $\mu$ for which you would conclude the driver was speeding? %Is this a lower or upper bound for $\mu$? 
Can you convince the investigators this driver was speeding, or do you need more information? (Do you need to know the mass of either vehicle? Road conditions?)   While there are
multiple ways to solve this problem, {\it please solve it using  work and energy}



%This next question is going back to Ch 1 stuff (it was worked out there) but adding in the "energy" part. It also leads into the next topic of the course, SHM, I think it's great timing and a good problem. We added the MMA portion, which is also nice, though we noticed a fair number of students punting it. Not sure why! 
\question  A simple pendulum consists of  a point mass $m$ fixed  to the end of a massless rod (length $l$), whose other end is pivoted from the ceiling to let it swing freely in the vertical plane. The pendulum's position can be identified simply by its angle  $\phi$ from the equilibrium position.
\begin{enumerate}
\item  Write the equation of motion  for $\phi$ using Newton's second law.  Assuming that the angle $\phi$ remains small throughout  the motion, solve for $\phi(t)$ and show that the motion is periodic. What is the period of oscillation?
\item Show that the pendulum's potential energy  (measured from the equilibrium level ) is
$ U(\phi)=A (1-\cos \phi)$. Find A in terms of $m$, $g$ and $l$. Then,  write down a formula for the {\it total energy} as a function of $\phi$ and $\dot{\phi}$.
\item Show that by differentiating the energy with respect to $t$ you can recover  the equation of motion you found in (a).  (What basic principle of physics are you using, here?) 


%
\item Picking simple values for parameters (e.g, $m=l=1, g=10$), use your favorite computational environment (e.g., NDSolve in Mathematica) to solve for $\phi(t)$ given a fairly small starting angle $\phi_0$. Provide output that clearly shows that the period of oscillation is very close to the theoretical prediction from part d. Repeat with $\phi_0=2$ rad (which is quite large), and show that the period now deviates from ideal.  (Does the period get {\it larger} or {\it smaller} as $\phi_0$ increases?) 
\item Using your numerical solution to part (d) and your formulas for energy from part (b), generate a single plot which graphs KE, U, and total E as a function of time, for one period, in the case where $\phi_0=2$ rad is NOT small. Briefly, comment! (Do we still conserve energy when we cannot use the small angle approximation any more?)  


%\begin{figure}[htc]
%\centering
%{\includegraphics[width=2in]{Homfig3.eps}}
%\end{figure}
\end{enumerate}





\end{questions}
\end{document}

