# 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

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## 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