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\title{\vspace*{-1.5in}\textbf{\S6.6 Part I: Work}\\\normalsize (Created by Faan Tone Liu)}
\date{}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-007: Calculus II}
\rhead{February 13, 2018}
\chead{\S6.6 Part I: Work}
\vspace{-.8in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item \framebox{$W = \text{Force}\times\text{Distance} = F\cdot d$}
\item Units:
\begin{tabular}[t]{l|c|c|c}
&$F = \text{Force}$ & $d = \text{Distance}$ & $W = \text{Work}$\\\hline
Metric &\phantom{Newtons} & \phantom{meters} & \phantom{$\text{Newtons} \cdot \text{meters} = \text{Joules}$}\\
&\phantom{$\text{N} = \frac{\text{kg}\cdot \text{m}}{\text{sec}^2}$}  & \phantom{m} & \phantom{$\text{Nm} = \text{J}$}\\\hline
U.S. Units&\phantom{lbs} & \phantom{ft} & \phantom{$\text{ft}\cdot\text{lbs}$}\\&&&
\end{tabular}
\item Now, what if $F$ is not constant?\vskip1cm

\item Dealing with springs - \textbf{Hooke's Law:}
$$F = \phantom{kx}\quad,$$
where $x$ is the distance stretched or compressed past the natural (equilibrium) length, and $k$ is the spring constant.
\item Dealing with the force of gravity (metric system):
$$ F = \phantom{mg}\quad,$$
where $m$ is the mass of the object and $g = 9.8 \frac{\text{m}}{\text{sec}^2}$.
\item Dealing with the force of graity (U.S. system):
$$F = \phantom{weight in lbs}\qquad\quad.$$
\end{itemize}


\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item A box is slid $3$ meters across a carpet against a force of kinetic friction of $45$N. How much work is done?\vskip2cm

\item I am pushing my sister across a 10 foot room. She pushes back with increasing ferocity, with a force of $20 + \frac{x^2}{2}$ pounds, where $x$ is how far I have pushed her. How much work do I do?\newpage

\item A $30$-centimeter long spring with a spring constant of $k = 120 \frac{\text{N}}{\text{m}}$ is compressed to $20$cm. Calculate the work done. \vskip2in

\item A force of $10$ lbs is required to hold a spring stretched to $6$ inches past its natural length. Calculate the work required to stretch it $8$ inches past its natural length.\vskip2in

\item How much energy is required to hoist a $3$-kilogram pumpkin $15$ meters to the roof of the math building?\vskip1.5in

\item How much energy is required to carry a $44$-lb stack of books up to the third floor of the math building? (30 ft.)\newpage

\item A $6$-kg chain is $3$ meters long. How much work is done lifting it from the ground until its lower end is $2$ meters off of the ground?\vskip3.65in

\item How much work is done emptying a $2\times 2\times 3$-ft rectangular tank? The water must be pumped to a point in the upper corner of the tank.\newpage

\item A tub has the shape of the solid of revolution formed by rotating around the $y$-axis the portion of the curve $y=2x^4$ that lies between $x = 0$ and $x = 1$. (Draw a picture.) How much work is done to empty the tank? All of the water must be pumped out of the top of the tank.
 
\end{enumerate}
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