A simple energy-conserving model exhibiting elastic restitution

Extract

From the date of my own A level studies (many moons ago!) I have been dissatisfied with the presentation of the law of elastic restitution. The title Newton’s ‘Experimental’ Law implies that it is a purely empirical phenomenon, yet if there is a wide range of conditions in which it is approximately valid (and it isn’t worth using at all if this isn’t true), then there must be some reason for it. Equally unsatisfactorily, the law implies loss of energy in most circumstances. Even in my school days I could see that the usual argument about conversion into heat, sound etc didn’t really hold water, as this process couldn’t conceivably be instantaneous. A recent article by O’Connor [1] stimulated me to crystallise my thoughts on the subject [2], and I was later able to develop a simple and plausible mechanical model exhibiting restitution, which also conserved energy [3]. My presentation in [3] was mainly focussed on the physical consequences of the model, and solution of the differential equations involved was numerical. Yet, in the simplest case, the model is amenable to fairly elegant exact analytical techniques, which it is the aim of this paper to present. Apart from the interest of the model, the methods used are a nice illustration of the role of normal modes in the solution of simultaneous linear differential equations—a topic approachable by A level students of Further Maths, or by first-year undergraduates.