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The inextensible string

Published online by Cambridge University Press:  31 October 2008

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An object to which we were all introduced at an early stage in mechanics is the inextensible string. This appears frequently without causing much trouble, but there is one type of problem which, in my opinion, stands apart from the rest, and which certainly caused me a lot of trouble. Such a problem is when impulses are given to a system which includes an inextensible string, as, for example, a system consisting of two rigid parts joined by a string. If an impulse is applied to one of these parts, an impulsive tension (T) may be set up in the string, which, in turn, gives an impulse to the other part. One new quantity, T, has appeared, and one equation in addition to the ordinary dynamical equations is thus required before the problem of finding the change in motion of the system can be solved. It is at this stage that opinions can differ, for this extra equation depends essentially upon what concept of an inextensible string is being adopted, and there is more than one. The usual procedure is to employ a “geometrical equation” based upon the argument that the two ends must have equal component velocities in the line of the string as long as the string is taut. This seems almost obvious when described in such general terms, and is followed by such eminent writers as Routh and Loney, amongst others. I suggest, however, that this method implies a concept which leads to results contrary to common-sense and to everyday experience. This is best illustrated by a simple example.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1943

References

page 13 note 1 See for example the worked exercise (170) on p. 149 of his Elementary Rigid Dynamics (1882).

page 13 note 2 Loney devotes two sections to methods involving the geometrical equation in Dynamics of a Particle and of Rigid Bodies (1919), p. 180.

page 14 note 1 Routh's exercise, cited above, gives a similar considerable loss of energy.