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198 5. Some worked examples arising from physical problems

that the relevant calculations in the previous chapters may need to be rehearsed before
embarking on some of the new material presented here. In each group of problems,
every example will be labelled by a suitable name, and the full list of these will appear
in the preamble to that group. The titles of the groups are: 5.1 Mechanical & elec-
trical systems; 5.2 Celestial mechanics; 5.3 Physics of particles & light; 5.4 Semi- and
superconductors; 5.5 Fluid mechanics; 5.6 Extreme thermal processes; 5.7 Chemical &
biochemical reactions. These chosen headings are intended simply to provide a general
guide to the reader; there is no doubt that some examples could be placed in a different
group—or appear in more than one group. Further, many other examples could have
been included (and the author apologises if your favourite has been omitted); the in-
tention in a text such as this is to give only a flavour of what is possible. Nevertheless, it
is hoped that sufficient information is available to encourage the interested researcher
to appreciate the power of the techniques that we have described.
Although the physical basis for each problem will be outlined, the relevant non-
dimensional, scaled equations will usually be the starting point for the analysis. There is
little to be gained by presenting the original physical problem, in all its detail, together
with the non-dimensionalisation, et cetera, if only because of the requirement, for
example, to define all the physical variables in every problem. Further, the reasonable
limitation on space also precludes this. The interested reader should be able to fill in
the details, particularly with the aid of the original reference(s).

5.1 MECHANICAL & ELECTRICAL SYSTEMS
The examples collected under this heading are based on fairly simple mechanical or
physical principles; more advanced and specific topics (such as celestial mechanics)
which might have appeared in this group are considered separately. The examples to
be discussed are: E5.1 Projectile motion with small drag; E5.2 Child’s swing; E5.3
Meniscus on a circular tube; E5.4 Drilling by laser; E5.5 The van der Pol/Rayleigh
oscillator; E5.6 A diode oscillator with a current pump; E5.7 A Klein-Gordon equa-
tion.

E5.1 Projectile motion with small drag
We consider a projectile which is moving in the two-dimensional (x, z)-plane under
the action of gravity (which is constant in the negative z-direction) and of a drag force
proportional to the square of the speed (and acting back along the local direction of
motion). The non-dimensional equations are most conveniently written as

where and the initial conditions are given as