UNIVERSITY OF BRITISH COLUMBIA

Science 1 Math/Phys

Second Midterm

19 November 1998

Time: 50 minutes

Instructors: Mark MacLean, Jess H. Brewer & Domingo Louis-Martinez

1.
Sketch The Function   [30 marks]

Sketch the graph of     ${\displaystyle y \; = \; 5x^3 \; + \; 2x \; + \; {1 \over x} }$ .

Show all work related to your sketching process.

2.
Itchy & Scratchy   [30 marks]


\begin{figure} \begin{center} \epsfysize 2.25in \noindent \null \hfil \hbox{\epsfbox{../PS/itchy.ps}} \end{center}\end{figure}

Scratchy the cat is trapped at the bottom of a shaft as shown. Itchy the mouse gives a large thin-walled cylindrical shell of radius 0.5 m a push toward the ramp, letting go when it is at the bottom of the ramp. If the initial angular velocity   $\omega_\circ$  is 10 rad/s, who will get squashed? (Will the cylinder make it to the top and fall down on Scratchy, or roll back onto Itchy?) Assume that the cylinder rolls without slipping.

3.
``Quickie'' Questions   [40 marks]

(a)
Mathematics Quickies   [20 marks]

\begin{displaymath}{dx \over dt} \; = \; -3 \, x \qquad \qquad \hbox{\normalsize\rm and} \qquad \qquad x(0) = 2 \, . \end{displaymath}

i.
What is  x(t)?

ii.
Write down the formula for the $n^{\rm th}$ iteration of Euler's method for this case.

iii.
What kind of error do you expect if you use a timestep of  h = 0.1  for the interval   $0 \le t \le 1$?   Explain.

(b)
Physics Quickies   [20 marks]

i.
[8 marks]   Which of the following features are essential for SIMPLE HARMONIC MOTION? (Circle your answers.)
A linear restoring force. $\dot{x} = - k x$. Newton's First Law.
Viscous damping. A quadratic potential minimum. A spring.
$\ddot{x} = - \omega^2 x$. A driving force at the resonant frequency.


\begin{figure} \begin{center} \epsfysize 1.75in \noindent \null \hfil \hbox{\epsfbox{../PS/int2.ps}} \end{center}\end{figure}

The figure above shows the average intensity of ``some combination of waves'' plotted vertically as a function of [some variable] plotted horizontally.

ii.
[6 marks]   If the horizontal variable is a  spatial angle  $\theta$  (measured in milliradians [mr]), what phenomenon does this pattern describe and what can you tell about the sources of the waves that make it up?
iii.
[6 marks]   If the horizontal variable is  time  t  (measured in seconds [s]), what phenomenon does this pattern describe and what can you tell about the sources of the waves that make it up?