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Second laboratory - Fall 2002
Due: Friday, 4 October 2002 at 11:30pm
This laboratory is concerned with the
behaviour of a weight on a spring
suspended from a support which is itself moving
up and down periodically. The relevant parts of
the textbook for this lab are sections 6.1 and 6.2.
Warning
For technical reasons, while working on
this lab, you
must not reload or leave this page within
your 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.
Another warning
This lab is more computationally
demanding on a computer than other ones.
For this reason, it is possible that you will
have trouble even viewing it from anywhere other
than the Mathematics Department system.
On the other hand, newer computers should cope.
The basic tool
In the following figure, press the
button to make the process run, and release it to make it stop.
Try running it for a few seconds, until the initial part
of the graph starts to disappear at the left.
(If it is extremely sluggish, just wait a while and try again.)
As it runs, two graphs are displayed:
- The graph of the location of the support is in gray.
- The graph of the location of the weight is displayed in blue.
The coordinate grid has horizontal unit equal to one second,
and vertical unit one centimetre. So the strong red horizontal lines
are separated by five centimetre intervals.
Exercise 1
In the basic tool,
you can choose the frequency at which the support moves
by moving the Frequency slider.
When the motion dies down, you can pump it
up again by changing the frequency and running it for a while.
- At start-up
the frequency is set at the first tick.
What frequency is this (in cycles per second)?
- The amplitude of the motion of the support is an integer.
What is it?
- As the process runs, the motion of the weight settles down
to a steady state oscillation. In the
initial configuration, what is the amplitude of this oscillation?
Exercise 2
No matter what frequency you set in the basic tool,
sooner or later the oscillation will settle down to
a steady state oscillation. The amplitude and phase
shift of this steady state, compared to the oscillation
of the support, will depend on frequency.
- Run the process for each one of the 16 frequencies
marked by a tick in the slider. Plot in the first figure below
the amplitude for each of those frequencies, and in the second figure
the phase shift. Note: When you make your final submission,
you should not have any extra points plotted
on either graph that you are not asked to put there!
Vertical units here are centimetres, as before.
- The phase shift is somewhat more difficult to estimate.
Exercise 3
The frequency for which the amplitude of the steady state oscillation is
a maximum is called the resonant frequency, when it exists.
- What is the resonant frequency of the system above?
- Add a point to the plot above
of amplitude versus frequency, corresponding
to the resonant frequency
- Add a point to the phase-shift plot, too, corresponding
to resonant frequency. (Making 17 in all on both plots.)
Exercise 4
By pumping up the weight
and then clamping the support down by
setting the support frequency equal to 0, you
can examine unforced oscillations fo the weight. The
displacement will then be the product of
a decreasing exponential function e^{-t/T} and
a simple periodic function C cos (wt - theta) .
The constant T is the relaxation time of the system
and the constant w is its quasi-frequency.
- What is the quasi-frequency of the system
(in cycles per second)?
- What is its relaxation time?
Sharpness of resonance
The resonance is said to be sharp
if the graph of amplitude versus frequency has a
sharp peak. The sharpness of resonance
depends only on the size of the friction
coefficient c relative to the mass
m and spring constant k .
More precisely, one can define a dimensionless number
which measures this sharpness---if is small
then the resonance is sharp, and if is large
there will be no resonance at all.
In the following figure you can adjust
to see the graph of amplitude versus frequency.
Exercise 5
By comparing the graph above to the plot you made earlier
of amplitude versus frequency, answer this question:
- What is for our system?
Submission
There is no reason in principle that
you will not be able to submit your answers from
home, but we cannot guarantee success.
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