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# Third laboratory - Fall 2002

### Due: Friday, 18 October, 2002 at 11:30 pm

##
Warning

*For technical reasons, while working on
this lab, ***you
must not reload or leave this page** within
your web browser. If you do leave and want to come
back again, you should quit your browser and
restart it, either from the background menu or
a terminal window.
Also, **save
your answers frequently** by using the submittor
at the end of the page.
The part of the textbook relevant to this lab is
Chapter 6 on spring problems.

### Beats

The graph shows the solution of an undamped forced oscillator:
with initial conditions *y*(0) = *y'*(0)=0. The blue vertical lines are at one second intervals. The radian frequency
of the forcing can be adjusted with the slider.

### Exercise 1

- Find two different values of
at which the beats occur two seconds apart.
- Use this to determine the natural frequency
of the oscillator.

### Design of a Measuring Instrument

A measuring instrument typically
shows its value using a needle that swings across a scale. In some cases
the needle may oscillate about the correct reading, and it may be some
time until it settles down enough for an accurate reading to be made.
On a typical bathroom scale, for example, this may take several seconds.
In cases where speed and accuracy are important, such a delay may be
unacceptable, and instruments
should be designed to produce an accurate reading as quickly as possible.
We will model the measuring instrument as a mass on a spring.
The equilibrium position represents the quantity to be measured. The
mass *m*=1 and spring constant *k*=1 are given, but the
damping constant
*c* is under our control. The system will start at some
value *y*(0)=*y*_{0} > 0 with *y'*(0)=0,
and at some time we will take the
measurement *y*(*t*). We want to be sure that *y*(*t*)
is within .01 *y*_{0} of the equilibrium value 0. In order to
ensure this, we should wait until some time *t*_{1}, long
enough that
|*y*(*t*)| < = .01 *y*_{0} for all *t* >
*t*_{1}.
It is not good enough just to have |*y*(*t*)| < .01 *y*_{0}
at some time *t* if this is not always true later.
We want to choose *c* to minimize *t*_{1}.

In the applet
below, we see part of the graph of the solution of the initial value
problem *y''* + *c y'* + *y* = 0,*y*(0) = 1, *y'*(0) = 0,
as a function of *t*. The slider adjusts the value of c. The two green
lines are *y*= 0.01 and
*y* = -0.01. The blue vertical lines are at intervals of 1.
If you click on the
graph, the coordinates of the arrowhead where
you clicked are shown at the top left.

### Exercise 2

- Find the value of
*c* for which the first minimum is at *y*= -.01,
and the time *t*_{1} for this *c*. This is the optimal solution
to our problem.
- What is
*t*_{1} if *c* is 0.02 less than the
optimal *c*?
- What is
*t*_{1} if *c* is 0.02 greater than the
optimal *c*?
- What is
*t*_{1} for the case of critical damping?

Various things are not working as they should for the next two questions.
**So ignore them if you wish. They will NOT be counted.**

- Let
*t*_{2} be the time at which the first minimum is
reached. Express *t*_{2} in terms of the quasi-frequency.
(The text denotes the quasi-frequency
. In your answer,
write the quasi-frequency as *u* ).
- Use this to obtain a formula for the optimal
*c*.

## Submission

There is no reason in principle that
you will not be able to submit your answers from
home, but we cannot guarantee success.
If a submission from within the Mathematics Department system
is not successful, tell the TA.

##
Comments?

Send questions about problems with submitter to
the lab TA

Send mathematical questions about the labs to
the professor in charge of Math 256:
fournier@math.ubc.ca