LECTURE NOTES FOR PHYSICS 307: CLASSICAL MECHANICS TEXT: THORNTON AND MARION'S CLASSICAL MECHANICS OF PARTICLES AND SYSTEMS
SOME USEFUL REFERENCE STUFF: Greek alphabet, metric prefixes, conversion factors
ASIGNMENTS: (Subject to change: check back often.) HW #5: Due Monday, October 6: Problems 3.1, 2, 3, 4a, 6. HW #6: Due Friday, October 24: Problems 3.4b, 7, 10, 22. HW #7: Due Friday, October 31: Problems 4.4, 5, 6, plus spreadsheet exercises. (Download the spreadsheets ODEDrivenSHO.xls, ODECubicPendulum.xls, and ODEPlanarPendulum.xls. You will need to enable the macros.) "Dailies": Tuesday, Oct. 7: Sketch the Lissajous Figure for a 2D oscillator for which the two frequencies differ by a factor of 1.5. Thursday, Oct. 9: Sketch the phase diagram of a ball dropped from a height, bouncing several (three) times, which loses energy with each bounce. Tuesday, Oct.14: Show that the critical damping solution works.
WEEK 7: GO TO LECTURE 11, 12
LECTURE 11: SECTION 3.5 (Return to top.)
DAMPING We shall return to the one-dimensional case, but add complexity this time by introducing damping ( friction). We shall do this by adding an extra term to Hooke's law that is a drag term, linear in velocity. We choose the linear term for two reasons. First, as long as the medium in which the pendulum is swinging or the mass is oscillating is not turbulent, this is a good approximation. Second, it makes the mathematics so much easier. If we have to include quadratic drag, we should probably solve things numerically rather than trying to solve than exactly. For the linear term, Hooke's law now becomes
F kx bx˙ 1/2 This leads to a second order differential equation in x. After defining to new quantities, β = b/2 m, and 0=(k/m) as we defined it earlier, and assuming a solution of an exponential form, x=e-ωt, this reduces to a quadratic equation in ω. The solutions to this differential equation depend on the relative sizes of β2 and ω2.
UNDERDAMPED OSCILLATORS The first family of solutions occurs when β2 is less than ω2. In this case the solutions look like sine or cosine functions modulated by a dying exponential.
VISUALIZING THE UNDERDAMPED OSCILLATOR If we visualize this solution using a graph of position vs time, we get a sine wave whose amplitude grows exponentially smaller.
t i1t i1t x(t) e A1e A2e t , Ae cos1t 2 2 where 1 0 .
Using the phase diagram, we get an ellipse that spirals in towards the origin. How quickly does this oscillator decay? One way we can quantify this is with a dimensionless quantity that the book calls the "decrement". Decrement e T Equivalently, we can use the "logarithmic decrement", or βT, which is also dimensionless.
LECTURE 12: SECTION 3.5 (Return to top.)
OVERDAMPING 2 2 On the other hand, when β is greater than ω0 , which is the limit as the logarithmic decrement, βT gets large, the system no longer looks like a decaying sine wave, and passes through the equilibrium position no more than once. Mathematically, the solutions no longer have a sine or cosine component, just exponential ones.
t 2t 2t x(t) e A1e A2e , where 2 2 2 0 .
It's pretty hard to describe the behavior of such an overdamped systems in these notes, so I will simply direct you to page 123 for the graphs of what x vs t might look like, and page 122 for the phase diagrams. Notice that the phase diagrams no longer look like ellipses at all. If you are trying to imagine a system that behave like this, look no further than the outer doors on the restrooms on the second floor of Bewkes.
CRITICAL DAMPING Critical damping is the border case between the underdamping and overdamping. Physically, it is almost identical with the overdamped systems; mathematically it looks quite different. Here's the equation for position as a function of time
t x(t) (A Bt)e , where 0
As for visualizing this case, both the graph of x vs t and the phase diagram are practically indistinguishable (at least to me) from the overdamped cases on page 122 and 123.
(p.s. The spreadsheet I used in class to illustrate these three damping cases was ODEDrivenSHO.xls. To play around with it yourself, save it to disk, double click the file, and then enable the macros when Excel asks.)
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