Introduction

Oscillation is one of the most common motions in nature, and you have already met dozens of examples without necessarily naming them as such: a cradle rocking, a child's swing, the pendulum of a clock, a guitar or violin string vibrating after being plucked, the needle of a sewing machine moving up and down, the prongs of a struck tuning fork, a spring bouncing after being stretched and released. What all of these share is that the motion repeats itself after a fixed interval of time -- this is called periodic motion. In most of the examples above, the motion is additionally "to and fro" or "up and down" about some central position, which makes it oscillatory motion -- a periodic motion in which the object repeatedly returns through the same path.

This chapter's central claim is that, for a very large and important class of oscillatory motions, the displacement, velocity and acceleration can all be written as sine or cosine functions of time. Functions of this type are called harmonic functions, and a motion whose oscillation is described by them is called harmonic motion. Simple harmonic motion (S.H.M.) is the simplest such motion, and it turns out to be the natural building block for describing almost every real oscillator you will meet -- a spring-mass system, a swinging pendulum, a vibrating tuning fork, an oscillating magnet, even (approximately) the atoms in a solid vibrating about their lattice positions.

By the end of this chapter you should be able to: use the correct vocabulary to describe an oscillation (period, frequency, amplitude, phase); recognise and derive the differential equation that defines S.H.M. and solve it for displacement, velocity and acceleration; represent S.H.M. graphically; work out how the energy of an oscillator is shared between kinetic and potential forms, and show that the total is conserved; extend the ideas of linear S.H.M. to angular S.H.M. (a twisting/torsional oscillator and a vibrating magnet); and finally understand what happens when an oscillation is damped (loses energy to its surroundings) or driven by an external periodic force, including the special, high-amplitude case called resonance.