\input amssym.def
\def\d{\displaystyle}
\input epsf
\def\d{\displaystyle}
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\font\largerbold = cmbx10 scaled \magstep2

\centerline {\largerbold Problems, Week 2}

\bigskip
%topics: work. average value. volume.

1.~Toxic waste (a homogeneous liquid whose density is 300 kg/${\rm m}^3$)
is stored in three buried containers. Each container, illustrated
below, is 10 meters tall. The top of each container is at ground
level. Each container has the {\it same} volume. The middle container
is a cylinder, and the other two are circular cones. Which container
needs the {\it least} amount of work to empty (that is, to pump the
toxic waste to ground level)?  Which container needs the {\it most}
work to empty? Justify your assertions by computing the
work necessary in each case\footnote*{\sevenrm You probably should
begin with the cylinder.}.

\bigskip

\centerline{\epsfbox{pic2-1.eps}}
\smallskip
\centerline{Toxic waste containers}
\smallskip

Note: the volume of a cylinder with base radius $r$ and height $h$ is
$\pi r^2 h$; the volume of a cone with base radius $r$ and height $h$ is
$\d{{1\over 3}\pi r^2 h}$.
 
\bigskip

2.~a) Suppose the area enclosed by $y=2x(1-x)$ and the $x-$axis is
moved so that its left corner is at $(3,2)$. Compute the volume
obtained when this translated area is revolved, first about the
$x-$axis and then about the $y-$axis.  

\medskip

\centerline{\epsfbox{pic2-2.eps}}
\smallskip

%the following part is very subtle for them, and much easier with the 
%theorem of pappus (later in the course).

b) Is it possible to move the area specified in a) so that its left
corner is at a point $(x_0,y_0)$ in the first quadrant {\it and} so
that the volumes resulting when the area is revolved about both of the
axes are equal? Explain your answer.

\bigskip


%what follows is problem 16, section 6.5, on p.410 of the text. it is
%(to me) quite subtle. 

3.~If a freely falling body starts from rest, then its displacement is
given by $\d{s= {1\over 2} gt^2}$. Let the velocity after a time $T$
be $v_T$. Show that if we compute the average of the velocities with
respect to $t$ we get $\d{v_{\rm ave}={1\over 2} v_T}$, but if we
compute the average of the velocities with respect to $s$ we get
$\d{v_{\rm ave} = {2\over 3} v_T}$. 

\bigskip

4.~A sort of raindrop is obtained by revolving the profile curve

$$y=\sqrt{x}(x-C)^2 \ {\rm for} \ 0\le x \le C$$

about the $x-$axis. Here $C$ is a positive constant.

\smallskip

a) Sketch the profile curve and the solid of revolution.

\medskip

b) For which value of $C$ will the raindrop have volume 1? What are
the rough dimensions (length and diameter) of this raindrop?

%easier to do the integral when we have int by parts, even though it
%is ``merely'' a polynomial.

\bigskip

5.~A wedge is cut from a cylindrical tree of radius 1. One cut is
parallel to the ground. The other cut is at an angle $\theta$ with the
ground, and the two cuts meet at the central axis of the tree. 
\smallskip

a) What is the area of the shaded triangular slice in terms of
$\theta$ and $x$? 

\medskip

b) What is the volume of the wedge?


\bigskip

\centerline{\epsfbox{pic2-3.eps}}
\bigskip

%two old Excel problems

6.~Electrons repel each other with a force which is inversely
proportional to the square of the distance between them; call the
proportionality constant $k$ in the units to be used. Suppose one
electron is fixed at $x=0$ on the $x-$axis.

\smallskip

a) Find the work done in moving a second electron along the $x-$axis
from the point $x=10$ to the point $x=1$.

\medskip

b) Find the work done in moving the second electron along the $x-$axis
from the point $x=M$ to the point $x=1$.

\medskip

c) What happens to your answer in b) (which should depend on $M$) as
$M \to +\infty$?

\bigskip

7.~Electrons repel each other with a force which is inversely
proportional to the square of the distance between them; call the
proportionality constant $k$ in the units to be used. Two electrons
are held fixed on the $x-$axis at the points $x=1$ and $x=-1$. Find
the work done in moving a third electron along the $x-$axis from the
point $x=10$ to the point $x=2$.



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