Third laboratory - Fall 2002

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

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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: $m y'' + k y = \cos(\omega t)$ with initial conditions y(0) = y'(0)=0. The blue vertical lines are at one second intervals. The radian frequency $\omega$ of the forcing can be adjusted with the slider.

Exercise 1

Find two different values of $\omega$ at which the beats occur two seconds apart.
Use this to determine the natural frequency $\omega_0$ 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)=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 enough that |y(t)| < = .01 y₀ for all t > t₁. 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.

Exercise 2

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 optimal c?
What is t₁ if c is 0.02 greater than the optimal c?
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 $\mu$ . In your answer, write the quasi-frequency as u ).
Use this to obtain a formula for the optimal c.

Submission

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Comments?

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