Introduction

Look around, and oscillatory motion is everywhere: a pendulum clock ticking away the seconds, a child on a swing going back and forth, a plucked guitar string vibrating, the diaphragm of a loudspeaker moving rapidly in and out, or even the individual atoms of a solid vibrating about their fixed positions in the crystal lattice. All of these share one common feature: the motion repeats itself, over and over again, after a fixed interval of time. Physics calls any such repeating motion PERIODIC MOTION.\n\nAmong all possible kinds of periodic motion, one particular kind turns out to be far more important -- and far more common -- than any other: SIMPLE HARMONIC MOTION (S.H.M.), in which the restoring force pulling the system back towards its mean (equilibrium) position is always directly proportional to the displacement away from that mean position, and always directed opposite to it. S.H.M. matters so much precisely because an enormous range of real oscillating systems -- stretched springs, swinging pendulums, vibrating molecules, plucked strings, even alternating electric currents in a circuit -- behave, at least for sufficiently small oscillations, EXACTLY like a simple harmonic oscillator, even though the underlying physical systems look nothing alike.\n\nThis chapter builds up the whole picture in a fixed order. Section 13.2 starts with periodic motion in general -- period, frequency, and what it means for a function to be periodic. Section 13.3 introduces displacement as a function of time. Sections 13.4-13.5 derive the equation of S.H.M. itself and explain the idea of phase. Section 13.6 works out velocity and acceleration in S.H.M. Sections 13.7-13.8 apply all of this to a real spring, including what happens when two springs are combined in series or in parallel. Section 13.9 looks at the kinetic and potential energy stored in an oscillating system. Sections 13.10-13.11 derive the time period of two of the most familiar oscillating systems of all -- the simple pendulum, and a mass hanging from a loaded spring. Finally, Section 13.12 moves beyond the idealised, undying oscillations considered everywhere else in the chapter, to look at what really happens to a real oscillator over time: free oscillations, damped oscillations, forced oscillations, and the striking phenomenon of resonance.