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Third laboratory - Fall 2002
Due: Friday, 18 October, 2002 at 11:30 pm
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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.
BeatsThe 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.
- 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 InstrumentA 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
c is under our control. The system will start at some
value y(0)=y0 > 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 y0 of the equilibrium value 0. In order to
ensure this, we should wait until some time t1, long
|y(t)| < = .01 y0 for all t >
It is not good enough just to have |y(t)| < .01 y0
at some time t if this is not always true later.
We want to choose c to minimize t1.
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.
- Find the value of c for which the first minimum is at y= -.01,
and the time t1 for this c. This is the optimal solution
to our problem.
- What is t1 if c is 0.02 less than the
- What is t1 if c is 0.02 greater than the
- What is t1 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 t2 be the time at which the first minimum is
reached. Express t2 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.
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.
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