Buquoy ’ s problem

We analyse the one-dimensional motion of a uniform thin fibre which is pulled upwards from a horizontal plane by a constant vertical force exerted against the homogeneous gravitational field. The solution of the equation of motion which describes this variable mass problem is discussed and the character of the resulting damped oscillations is described. 1. Historical background and formulation of the problem The problem we will discuss here is associated with the name of count Jiřı́ František August Buquoy (in German transcription Graf Georg von Buquoy), Czech aristocrat, mathematician and gifted inventor (1781–1851). After his studies in Prague and Vienna where he was educated in mathematics, natural science, philosophy, law and economy, he devoted his time from 1803 to taking care of the large family possessions and to his investigations. In 1810 he constructed a steam engine and did his best to apply it in practice. Above all, he was engaged in glass works. On the basis of many experiments he succeeded in inventing an original process technology (now forgotten) of a black opaque glass called hyalite (1817). Buquoy was the first who investigated mechanical systems with a varying mass. In 1812 he explicitly formulated the correct dynamical equation of motion for the case when the mass of a moving object is changing (see [1], p 66). He subsequently suggested several nice concrete examples of such a motion [2] to which he applied his new generalized dynamical equation, attempting also to solve the corresponding differential equations. In August 1815 he presented his results at the Paris Academy of Sciences (Institut National des Sciences et des Arts, Première Classe) to Laplace, Poisson, Ampère, Delambre, Arago, Cauchy, Fourier and others [3, 4]. Nevertheless, apart from a single short article [5] by Poisson, his ideas did not attract attention, and they gradually became forgotten. Buquoy’s general equation of motion and other explicit examples were later formulated independently by various authors [6–8]. The pioneering work of Buquoy on systems of non-constant mass was rediscovered and his 0143-0807/05/061037+09$30.00 c © 2005 IOP Publishing Ltd Printed in the UK 1037 1038 V Šı́ma and J Podolský Figure 1. A schematic picture of the vertically moving fibre. The actual position of its end above the horizontal plane is represented by x. role in the history of physics was recognized only recently by Mikhailov [9–11]. Since then it has begun to appear in some new textbooks on mechanics [12, 13]. The first explicit example of a system with a varying mass suggested by Buquoy in 1814 (see [2] on p 34) is the following: Consider an ideally flexible fibre lying reeled on a horizontal plane. Determine its motion when a constant vertical force (directed upward) is exerted on the end of the fibre. We wish to analyse this problem here and to demonstrate that it exhibits some surprising properties which may be of pedagogical interest for undergraduate students. (Let us note that the original solution proposed in [2] was not correct. The particular case of the problem was correctly solved in [12].) Denoting the position of the end of the fibre above the horizontal plane by x > 0 (see figure 1), we make the following natural simplifying assumptions: • the vertical gravitational field is homogeneous, • the fibre is thin and its linear density η is constant, • the fibre reels off without friction at the origin of the x-axis during the upward motion, • the fibre ‘smoothly disappears’ at the origin of the x-axis during the motion downward, i.e., the part that has already landed on the plane does not move. The problem could be modelled experimentally as the vertical motion of a balloon with a heavy rope hanging down. The constant upward force would be the buoyant force exerted on the balloon in air. The mass of the balloon and the friction forces would have to be neglected. 2. The equation of motion The motion of the fibre is, of course, determined by Newton’s law—the rate of change of the momentum p of the moving part of the fibre is due to the resultant force, ṗ = F −mg, (1) whereF > 0 is the vertically upward oriented constant force, andmg is the oppositely exerting weight of the reeled fibre. Supposing x to be the height of the end of the fibre above the plane then p = mẋ, and the mass is given by m = ηx where η is a constant linear density of the fibre (we emphasize that only x > 0 has a physical meaning). Therefore, the equation of motion has the form (xẋ)̇ = F η − gx, (2) Buquoy’s problem 1039