Simple harmonic motion acceleration

The first term again represents drag in steady motion at the instantaneous velocity, with Cd an empirical function of Re as in Chapter 5. The other terms represent contributions from added mass and history, with empirical coefficients, Aa and Ah, to account for differences from creeping flow. From measurements of the drag on a sphere executing simple harmonic motion in a liquid, Aa and Ah appeared to depend only on the acceleration modulus according to ... 

Succession of vibrational states For a simple harmonic motion the acceleration is related to the displacement from the equhibiium position by the equation... 

In Eq. (4-31), the first three terms describe a simple damped harmonic oscillator the first term is due to molecular accelerations, the second is due to viscous drag, and the third is due to the restoring force. Qq is the oscillator frequency, which is of order 10 sec", and p is a viscous damping coefficient. The crucial term producing the dynamic glass transition is, of course, the fourth term, which has the form of a memory integral, in which molecular motions produce a delayed response. The kernel m(t — t ) is determined self-consistently by the time-dependent structure. One simple choice relating m(s) to the structure is ... 

Then I started worrying about how to reach that point with a class of students. I saw that the same graphical methods could be used for the first order equation for exponential decay, and for the second order equations for uniformly accelerated motion and harmonic oscillations. This was obvious to me because 1 had been lucky enough to be taught at Cambridge by the great Douglas Hartree, whose lectures inspired us with the notion that very simple step-by-step arithmetic methods could both solve difficult problems and illuminate their inner stracture. 

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