\begin{homework}\
\label{homework:11}

\medskip


\exercise %1.
Suppose that the wheel in Example 11.2 has radius 1.

(a)~Graph the cycloid by plotting the points where the pebble is located
after the wheel rotates through all multiples of 30${}^\circ$ from 0 to
360${}^\circ$ (i.e., $\theta$ goes through all multiples of $\pi/6$ from 0
to $2\pi$).

(b)~Use the chain rule in the form 
\[
\frac{dy}{ dx}=\frac{dy/d\theta}{ dx/d\theta}
\] 
to find a formula for $dy/dx$ in terms of $\theta$.

(c)~Draw the tangent lines to the curve in part (a) at all of the plotted
points from $\theta=\pi/6$ to $\theta=11\pi/6$, and use part (b) to find
the slope of each tangent line.

\medskip


\exercise %2.
A spring is causing an object to oscillate up and down sinusoidally between
$y=3$ and $y=-3$.  At $t=0$ the object is at maximum height $y=3$, and it
makes one full motion (from 3 to $-3$ and back to 3) twice a second.
Meanwhile, the whole apparatus is moving at constant velocity 2 units/sec
down the $x$-axis, starting at 0 at time $t=0$.

(a)~Find $x(t)$.

(b)~Find $y(t)$.  Namely, use the information given to (i) graph $y$ as a
sinusoidal function of time in the $ty$-plane, and (ii) find $A$, $B$, $C$
and $D$ in the formula $y=A\sin\left(\frac{2\pi}{ B}(t-C)\right)+D$.

(c)~Graph the path of the object in the $xy$-plane.  Find the formula for
$y=f(x)$ in two ways: (i) Use the graph of the path to find $a$, $b$, $c$
and $d$ in $y=a\sin\left(\frac{2\pi}{ b}(x-c)\right)+d$ (where we're using
small letters for the constants, so as to avoid confusion with the
sinusoidal function in part (b)).  (ii) Eliminate $t$, and use part (b) to
write $y$ directly in terms of $x$.  Use both methods, and check that your
answers agree.

\medskip

\exercise %3.
The same as Problem 2, except suppose that the whole apparatus is moving
down the $x$-axis according to the formula $x(t)=t^2$.  Notice that the
vertical motion $y(t)$ as a function of time is unaffected by this change.
However, in the horizontal direction the object is {\bf accelerating},
i.e., its velocity, rather than being constant, is steadily increasing
according to the formula $v(t)=\dot{x}(t)=2t$; and the acceleration
$a(t)=\ddot{x}(t)=2$ is constant.  Here $x(t)=t^2$ can be thought of as
$\frac{1}{ 2}gt^2$, where the constant acceleration $g$ here is 2
units/sec${}^2$ (and of course, unlike gravity, which pulls in the negative
$y$-direction, here the constant acceleration is pulling in the positive
$x$-direction).  Also notice that your graph of the path $y=f(x)$ is no
longer sinusoidal: the distance between peaks becomes greater as the object
accelerates to the right.  The formula for $y=f(x)$ can be obtained by
substituting $t=\sqrt{x}$ in place of $t$ in the formula in part (b).

\medskip

\exercise %4.
A wheel of radius 2 ft is centered at the point $(0,100~\text{ft})$ and is
rotating clockwise at the constant rate of one revolution every 2 seconds.
At time $t=0$ a pebble on the perimeter of the wheel is located at the
point $(0,102)$.  At time $t=0$ the wheel is dropped.  Write down formulas
for the $x$- and $y$-coordinates of the pebble as functions of time, and
graph its trajectory for $t$ between 0 and $2.5$ sec (plot points at
intervals of 1/4 sec).

\medskip

\exercise %5.
A wheel rolls down the $x$-axis from left to right without slipping at
constant speed 6 m/sec.  The diameter of the wheel is 1.5 m.  As it passes
over the origin at time $t=0$ a pebble gets stuck in a spoke right under
the center at a distance 25 cm from the rim.  Find $x(t)$ and $y(t)$ for
the pebble.

\medskip

\exercise %6.
In an episode of the TV program ``Northern Exposure,'' the hero Chris and
a mathematician named Amy send his motorcycle careening
over a cliff.  We want to describe the path of the point $P$ at the top of
one of the wheels at the instant $t=0$ when the wheel starts over the
cliff.  Suppose that the motorcycle wheel has a radius 0.25 m, and until
time $t=0$ the motorcycle is traveling at 10 m/sec.  Neglect air
resistance, and take $g=9.8$ m/sec${}^2$.  Suppose that the wheel is
traveling in the $xy$-plane, the motorcycle is moving horizontally (along
the negative $x$-axis) until time $t=0$, and the origin is the point at the
edge of the cliff where the wheel is touching at the instant $t=0$.  Find
the coordinates $x(t)$, $y(t)$ of $P$ at time $t$ (while the motorcycle is
falling).

\end{homework}



