\def\wsn{22}
\input worksheet.tex

\item{1.}  In physics, the equation $W=Fd$ gives the amount $W$ of
energy (or work) expended in moving a distance $d$ by applying a
constant force $F$.

\smallskip
\itemitem{a)}  A force of 10 N (Newtons) is applied to move an object
4 cm, then a force of 7 N is applied in moving it 6 more cm.  How much
energy $W$ was expended?

\smallskip
\itemitem{b)}  Now suppose the force over the first cm is 2 N, the
force over the second cm is 4 N, and so on.  How much energy $W$ is
expended over the entire 10 cm?

\medskip
\item{2.}  Hooke's law says that a spring requires a force $F=kx$ to
hold the end displaced by a distance $x$ from its equilibrium point.
Here $k$ is a constant, which depends on how ``stiff'' the spring is.

\smallskip
\itemitem{a)}  Explain why we cannot use the formula $W=fd$ to find
the amount of energy required to displace a spring 10 cm.

\smallskip
\itemitem{b)}  Describe a way to use the formula $W=Fd$ to find an
approximation to the amount of energy required to displace a spring 10
cm.  Find an approximation.

\medskip
\item{3.}  From physics recall that the force acting on an object is
equal to its mass times its acceleration, and that work is equal to
the force acting on an object multiplied by the distance the object
has traveled ($F=ma$ and $W=Fd$.)  In lifting a mass, the force to be
overcome is that of gravity which (on the surface of the Earth)
corresponds to an acceleration of 9.8 m/s$^2$.

\smallskip
\itemitem{a)}  Suppose a 1 kg bucket filled with 10 kg of water
(equals approximately 10 liters - isn't the metric system great?) is
lifted 10 m into the air.  How much work is done?


\smallskip
\itemitem{b)}  Now suppose a bucket of water is lifted 10 meters into
the air in such a way that (i) there are 10 kg of water in the bucket
at the start, (ii) the bucket is lifted at a constant rate, (iii) the
water leaks out of the bucket at a constant rate, (iv) the bucket
contains only 5 kg (how many liters?) of water when its is 10 m high.

\smallskip
\itemitem{}  Break the 10 m into eight segments of your choosing.  For
each segment find one value which is a good estimate for the force
required over that distance.  Use those values for $F$ to estimate the
work done over each interval.  Add up your eight estimates to obtain
an estimate for the total work required to lift the leaky bucket 10 m.

\medskip
\item{4.}  Aliens have kidnapped Marisa!  She will be set free only if
she can beat their ``hero'' at a game of Gorgo.  The game is simple.
A large metal bar 120 meters long (metric aliens!) is brought before
the contestants.  Whoever can guess the weight of the bar most
precisely, wins.
  
\smallskip
\item{}  When the day of the contest arrives, Marisa is shoved into a
huge arena before millions of screaming aliens.  While staring at the
long metallic Gorgo bar, she hears a low rumble begin, and a fearful
hush falls over the crowd.  As a door at the opposite end of the arena
lifts, Marisa gasps in horror: it's Elvis.

\smallskip
\item{}  Elvis looks at the bar for only a second, mutters something 
unintelligible, and the crowd breaks out in delirium throwing Twinkies
and little blue pills into the arena.  Now it is Marisa's turn.  She
goes to the bar, takes small samplings at 20 meter intervals, and
finds the density (in kilograms per centimeter).  Here is her data:
$$
\matrix{
\hbox { Meter mark } & \hbox{ Density } &\cr
        0            &     30.4         &\cr
       20            &     \cdots    &\hbox{ Data lost. (Stolen by aliens?)}\cr
       40            &     46.5         & \cr
       60            &     65.8         & \cr
       80            &     29.2         & \cr
      100            &     52.1         & \cr
      120            &     55.0         & \cr}
$$
Estimate the total weight of the Gorgo bar.

\medskip
\item{5.}    Suppose that a wind-up car is designed so that the velocity
$t$ seconds after it is started is given by $v(t)=\sin(\sqrt{\pi
t/10})$ (measured in m/sec.)  Your task is to estimate the position of
the car after 15 seconds.  Do this by estimating the velocity over
each one-second time interval by an appropriate constant velocity.
\eject



\bye