Instructions:
This "Resource Window" opens in front of the applet window. You can move this window
aside, resize it, or iconize it when you want it out of the way. To close it, use the
Close option from the File menu, not the Quit or Back commands.
The applet displays solutions of the differential equation
m d2y(t)/dt2 + c dy(t)/dt +k y(t) = 0.
This equation describes a damped harmonic oscillator with mass m, damping
constant c, and spring constant k.
You choose m, c, and k by using the sliders or by typing directly
in the right-hand control panels. You also choose the initial values y(0) and
y'(0). The applet updates its display every 1/2 second.
The applet should render properly using any Java-enabled browser, such as Netscape 3.0.
Macintosh users with Netscape 3.0 may experience difficulties. Sometimes the applet fails
to correctly re-display after you have scrolled through the page. If so, simply click
Reload.
If you receive a message informing you that your browser "does not understand the applet
tag," examine the Security Preferences Menu of your browser. There should be a checkbox
which you can use to enable Java; absence of such a box probably indicates you are using
an older web-browser and you must upgrade it to a newer version in order to run the applet.
Please take a moment to send me an e-mail
reporting any difficulties you may have experienced with the applet. The applet
sometimes misbehaves when run under certain versions of Netscape.
Questions to Investigate:
- What do you expect should happen when the damping coefficient c is zero and
the other coefficients are not?
- An interesting situation also occurs when k=0 and the other coefficients are
not zero. How do such solutions behave in the limit as t goes to infinity? What happens
if y'(0)=0 ? If y'(0) is non-zero, what role does it play in determining the
limiting behaviour?
- There are values of m, c, k for which the differential equation has
solutions of the straight line form y=at+b. Can you make the applet graph such
a solution (both with and without setting y'=0)? Why are such solutions possible?
- Are there overdamped solutions which cross the t -axis?
- We did not give the applet the capability of considering the cases of c or
k being less than zero, although there is no mathematical reason not to allow
these cases. What do you think would happen? Physically, would it be sensible to let
c be less than zero? Why did we not allow the applet to display the case m=0 ?