Simple Nature - Light and Matter

Three functions with thesame curvature at x=0.c / The amplitude would usuallybe defined as the distancefrom equilibrium to one extremeof the motion, i.e., half the totaltravel.Physically, k is a measure of stiffness. For example, the heavy-dutysprings in a car’s shock absorbers would have a high value of k. Itis often referred to as the spring constant, but we’re only using aspring as an example here. As shown in figure b, any two functionsthat have U(0) = 0, dU/ dx = 0, and d 2 U/ dx 2 = k, with the samevalue of k, are virtually indistinguishable for small values of x, so ifwe want to analyze small oscillations, it doesn’t even matter whichfunction we assume. For simplicity, we’ll just use U(x) = (1/2)kx 2from now on.Now we’re ready to analyze the mass-on-a-spring system, whilekeeping in mind that it’s really only a representative example ofa whole class of similar oscillating systems. We expect that themotion is going to repeat itself over and over again, and since we’renot going to include frictional heating in our model, that repetitionshould go on forever without dying out. The most interesting thingto know about the motion would be the period, T , which is theamount of time required for one complete cycle of the motion. Wemight expect that the period would depend on the spring constant,k, the mass, m, and and the amplitude, A, defined in figure c. 11In examples like the brachistochrone and the Apollo 11 mission,it was generally necessary to use numerical techniques to determinethe amount of time required for a certain motion. Once again, let’sdust off the time3 function from page 93 and modify it for ourpurposes. For flexibility, we’ll define the function U(x) as a separatePython function. We really want to calculate the time required forthe mass to come back to its starting point, but that would beawkward to set up, since our function works by dividing up thedistance to be traveled into tiny segments. By symmetry, the timerequired to go from one end to the other equals the time requiredto come back to the start, so we’ll just calculate the time for half acycle and then double it when we return the result at the end of thefunction. The test at lines 16-19 is necessary because otherwise atthe very end of the motion we can end up trying to take the squareroot of a negative number due to rounding errors.11 Many kinds of oscillations are possible, so there is no standard definition ofthe amplitude. For a pendulum, the natural definition would be in terms of anangle. For a radio transmitter, we’d use some kind of electrical units.114 Chapter 2 Conservation of Energy