We began with things that swing, things that spin, and things seen from somewhere that moves. We asked three questions: when is motion simple, what is the right point of view, and can intuition be trusted? Here is what the chapter’s fourteen questions showed.
1. When is motion simple?
Simple harmonic motion, with a single frequency independent of amplitude, is the ideal. Driving it at its natural frequency produced resonance (Section 11.1), with beats just off resonance and unlimited growth exactly on it. Two coupled oscillators turned out to be two independent simple oscillators in disguise (Section 11.2), but with frequencies in the irrational ratio \(\sqrt3\), so the combined motion never repeats. The real pendulum is not simple at all (Section 11.3): its period grows with amplitude, through an elliptic integral that Gauss’s arithmetic–geometric mean computes with astonishing speed. And the tautochrone (Section 11.4) showed how to make motion exactly simple again, by bending the track into a cycloid.
2. What is the right point of view?
Almost every question was solved by choosing the right variable or the right frame.
Coordinates. Arc length made the tautochrone simple; the combinations \(x \pm y\) decoupled the springs; the fraction of the rope removed the stretching from the ant’s problem (Section 11.12); writing time as a function of distance made the snowploughs’ equations linear (Section 11.13).
Frames. In the accelerating train (Section 11.5), a fictitious force tilted gravity, and the bead oscillated about a new lowest point. Einstein turned this observation into general relativity.
Points to take moments about. For the spinning marble (Section 11.6), angular momentum about the contact point was conserved, because friction acts there, giving the final speed without knowing the friction.
Equivalent systems. A rod behaves exactly like three particles (Section 11.7), with the weights of Simpson’s rule; and the equilibria of a solid are where its normals pass through the centre of mass (Section 11.8).
3. Can intuition be trusted?
Often not. The boundary of the region a cannon can reach is not a trajectory but their envelope (Section 11.9). A ball thrown steeply enough comes back towards you before it lands (Section 11.10). A fielder can catch a ball without predicting where it will land, by keeping its image rising steadily (Section 11.11). An ant walking slowly along a rope stretched a hundred thousand times faster still reaches the end (Section 11.12). A hanging chain is not a parabola, but a chain carrying a bridge is (Section 11.14). In each case, a short calculation replaced a confident but wrong intuition with a precise statement, and usually with an explanation.
4. A toolkit for oscillations and frames
| Method | Where you used it |
|---|---|
| Particular plus complementary solution | Resonance (11.1) |
| Decoupling by eigenvectors | Normal modes (11.2) |
| Energy integral and substitution | Pendulum (11.3) |
| The natural variable | Tautochrone (11.4); ant (11.12) |
| Fictitious forces | Train (11.5) |
| Angular momentum about a point | Backspin (11.6) |
| Matching moments | Simpson’s rod (11.7) |
| Counting stationary points | Tilted equilibria (11.8) |
| Discriminants and envelopes | Safety parabola (11.9) |
| Sign of a derivative | Coming back (11.10) |
| Comparing coefficients | Fielders (11.11) |
| Integrating factors | Ant (11.12); snowplough (11.13) |
| Resolving at joints | Bridge (11.14) |
Where next
The next two chapters turn to probability and statistics, starting with Section 12.1.
Normal modes are eigenvectors (Section 7.2), and the quantum harmonic oscillator of Section 8.6 is the oscillator of this chapter in quantum mechanics.
Limit cycles (Section 8.4) are oscillations that sustain themselves, and chaos (Section 8.7) is what nonlinear oscillators can do when driven.
A. P. French, Vibrations and Waves: the classic introduction to oscillations, resonance and normal modes.
David Morin, Introduction to Classical Mechanics: rotation, frames and many surprising problems.
Steven Strogatz, Sync: how oscillators fall into step, from fireflies to bridges.
George Simmons, Differential Equations with Applications and Historical Notes: the tautochrone, the brachistochrone and the snowplough, with their history.
Coleman joked that a physicist’s career consists of treating the harmonic oscillator in ever more abstract ways. This chapter has tried to show why: the oscillator is the simplest motion there is, and almost everything looks like one if you choose the right point of view. Learning to find that point of view, a variable, a frame or a combination, is the most useful skill that mechanics teaches.