PH3110

HW6 Problem #1. COMP1 (20 pts)

• For the following exercise you may use the Runge-Kutta tool kit I have prepared in Fortran in a Unix environment. You may use Matlab instead, if you wish, but I will not be able to help you if you get stuck using Matlab.

• Read the file rungekutta.html and download the necessary files to your computer. The files are in directory

http://www.phy.mtu.edu/~jaszczak/Ph3110/RungeKutta/

• Check the derive.for file to see that it has the drag term that you want you want to use. You may edit the drag term and the acceleration calculation as needed in future exercises. Create the int.x executable as directed above.

• Edit parameters in the file in.dat as needed to accomplish the following exercises. Your output data will be in out.dat.

(a) Numerical integration has an inherent limit to its accuracy, but you can set the desired accuracy or tolerance to be as small as you desire (though smaller tolerances will require more computational time). In this part you will study the effects of varying the tolerance parameter on the integration accuracy. Start with tolerance set to 1.0d-5 and plot x(t). Now make tolerance appreciably smaller and appreciably larger and see what effects that has by plotting these two additional cases on the same graph as the original x(t) graph. Set your system to study an underdamped harmonic oscillator system with damping that is linear in the velocity. Use initial conditions x(0)=1.0d0, and v(0)=0.0d0. Before proceeding, make sure you set tolerance back to a sufficiently small value so that you get accurate results. Comment on your observations.

(b) Now change the damping coefficient gamma to investigate no damping, underdamped, critically damped, and overdamped cases. Plot these four cases for x(t) on one graph, and also plot v(t) on one other graph. (Extra Credit: Plot the energy versus time for these four cases on one graph. 5 pts.)

(c) Now investigate changing the initial velocity on the overdamped case. Choose at least three different v(0) values and plot x(t) for these cases on one plot. At At least one curve should have a crossing of the origin. Comment on your observations.

(d) Finally, change gamma for several underdamped cases. Investigate what happens for very lightly damped oscillations as compared to barely underdamped and discuss the results.