Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-04T21:12:27.446Z Has data issue: false hasContentIssue false

A simple energy-conserving model exhibiting elastic restitution

Published online by Cambridge University Press:  01 August 2016

Richard Bridges*
Affiliation:
King Edward’s School, Birmingham B15 2UA

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.

Type
Articles
Copyright
Copyright © The Mathematical Association 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. O’Connor, W. J. The famous ‘lost’ energy when two capacitors are joined: a new law?, Phys. Educ. 32 (1997) pp. 8891.CrossRefGoogle Scholar
2. Bridges, R. T. Joining capacitors, Phys. Educ. 32 (1997) p. 217.Google Scholar
3. Bridges, R. T. Energy conservation and restitution in inelastic collisions: a simple model, Phys. Educ. 33 (1998) pp. 311315.CrossRefGoogle Scholar
4. Roper, Tom and Hartley, Ron “… Assume the string is inextensible and elastic…” Math. Gaz. 75 (1991) pp. 1523.CrossRefGoogle Scholar
5. Bridge, N. J. The way balls bounce, Phys. Educ. 33 (1998) pp. 174181.CrossRefGoogle Scholar