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\title{Math 1a.  Lecture 12 \\
Applications to Physics and Engineering (Work)}

\author{T. Judson}

\date{Spring 2006}


\begin{document}

\maketitle


\section{Goals}

\begin{itemize}

\item
To understand and be able to apply the definite integral when computing work done.

\end{itemize} 



\section{Work}

\begin{itemize}

\item
If an object of mass $m$ moves along a straight line with position function $s(t)$, then the force on the object  (in the same direction) is given by Newton's Second Law of Motion
$$
F = m \frac{d^2s}{dt^2}.
$$
Force is measured in Newton's (N = kg$\cdot$m/s$^2$).  A force of one newton on a mass of one kilogram produces an acceleration of 1 m/s$^2$.

\item
For constant acceleration, force is constant and work is given by
$$
\mbox{Work} = \mbox{Force} \times \mbox{Distance}.
$$
We measure work in newton-meters or {\em joules}.  If force is measured in pounds and distance in feet, then we measure work in foot-pounds,
$$
1 \mbox{ ft-lb} \approx 1.36 \mbox{ J}.
$$

\item
The work done lifting a 30 lb toddler to hip height (3 ft) is
\begin{eqnarray*}
30 \mbox{ lb} \times 3 \mbox{ ft} & = & 90 \mbox{ ft-lb}\\
133.44 \mbox{ N} \times 0.914 \mbox{ m} & = & 121.964 \mbox{ J}
\end{eqnarray*}

\item
Suppose a continuous force $f(x)$ acts on an object over the interval $a \leq x \leq b$.  The work done on a subinterval is
$$
W_i \approx f(x_i^*) \Delta x_i,
$$
and the total work done can be approximated by
$$
W \approx \sum_{i = 1}^n  f(x_i^*) \Delta x_i.
$$
We now define work done by moving an object from $x=a$ to $b$ by
$$
W = \lim_{n\to \infty} \sum_{i = 1}^n  f(x_i^*) \Delta x_i = \int_a^b f(x) \, dx.
$$

\end{itemize}

\section{Examples}

\begin{enumerate}

\item
{\em Hooke's Law\/} tells us that the force created by a spring when it is stretched or compressed is proportional to the distance the spring is stretched or compressed,
$$
F(x) = kx.
$$
If a spring stretched 1 foot beyond its natural length requires 10 lbs of force, how much work is done stretching the spring?

\item
Is it more work to raise a 70 lb bucket from a 50 ft well or a 50 lb bucket from a 70 ft well if the rope weighs 0.25 lb per foot?

\item
A circular swimming pool has a diameter of 24 ft, the sides are 5 ft
high, and the depth of the water is 4 ft.  If the density of water is 62.5~lb/ft$^3$, how much work is required to pump all of the water out over the side?

\end{enumerate}

\section*{References}

\begin{itemize}

\item
\S 6.5 in 
James Stewart.
{\it Single Variable Calculus: Concepts \& Context}, third edition.
Brooks/Cole, Belmont CA, 2005.
ISBN 0--534--41022--7.


\end{itemize}



\section*{Notes}

\subsection*{March 8, 2006}



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