[math-fun] onion of revolution?
Hold one end of a long string of beads and get it twirling around a vertical axis with a vigorous stirring motion (like in some cowboy lasso tricks, but without the noose loop). As the speed increases the middle of the string pulls outward, drawing the lower end up, until the bottom part looks horizontal. Since the top is basically fixed it makes a cusp, and overall the profile the whole string traces in the air is shaped something like an onion, oriented cusp-up, like the Mogul-style onion domes seen on many buildings. At high speeds the string gets sharply bent, like a hairpin. Can you give an expression describing the "onion curve" the string forms at a given speed? What path does the string's bottom end trace as a function of the speed of rotation?
Quoting Marc LeBrun <mlb@fxpt.com>:
Hold one end of a long string of beads and get it twirling around a vertical axis with a vigorous stirring motion (like in some cowboy lasso tricks, but without the noose loop). [...] Can you give an expression describing the "onion curve" the ? string forms at a given speed?
This system can no doubt be modelled to different degrees of approximation, but I seem to recall that the first approximation is a Bessel function, the number of whose nodes increase with the velocity of rotation, or perhaps with the kind of jiggling given to start it all off. There are probably references in places like American Journal of Physics. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
Once you have stopped actively twirling, the equilibrium state minimizes total energy, potential energy plus kinetic energy. This is a standard problem in calculus of variations with one fixed boundary and one free boundary. It gives rise to a differential equation which may or may not have solutions expressible using well known functions, e.g. Bessel functions. This problem is typical of those on the infamous Cambridge Math Tripos examinations, given from the early 19th century to about 1905. Whittaker's _Analytical Dynamics_ will tell you more than you want to know about such problems.
participants (3)
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John McCarthy -
Marc LeBrun -
mcintosh＠servidor.unam.mx