Il Pendolo cicloidale

Alla questione del  pendolo cicloidale  è dedicata la seguente prova per gli esami di stato della scuola secondaria superiore tratta da ‘PROPOSTE PER LE TERZE PROVE’ Modelli e materiali per la definizione di prove pluridisciplinari . Osservatorio Nazionale sugli Esami di Stato. 1999- Franco Angeli- Collana del CEDE- Centro Europeo dell’educazione.

Premessa al Testo e Indicazioni di Lavoro

Ti è noto certamente che Galileo condusse osservazioni sulle oscillazioni del pendolo semplice. Meno conosciuta è invece l’invenzione del pendolo cicloidale da parte dello scienziato olandese Christian Huyghens il quale, costruendolo, giunse alla scoperta di importanti leggi matematiche.

Il testo che segue, tratto da un’opera divulgativa di uno scienziato moderno, riferisce come Huyghens giunse alla sua scoperta. (…).

I quesiti in B sono posti in italiano e ad essi dovrai rispondere in italiano. I quesiti focalizzano aspetti  più specificamente matematici e fisici degli argomenti trattati nel testo e ad essi dovrai rispondere utilizzando i procedimenti e i termini propri di questa area scientifica.

Huyghens was a scientist of international reputation who is recalled for the priciple that bears his name in the wave theory of light, the observation of the rings of Saturn, and the effective invention of the pendulum cock. It was in connection with his search for improvements in horology that he mafde his most important mathematical diacovery. He knew that the oscillations  of a simple pendulum are not strictly isochronous, but depend upon the magnitude of the swing. To phrase it differently, if an object is placed on the side of a smooth hemispherical bowl and relesed, the time it takes to reach the lowest point will be almost, but not quite, independent of the height from which it is released. Now it happened that Huyghens invented the pendulum clock at just about the time of the Pascal cycloid contest(…), in 1658, and it occurred to him to consider what would happen if one were to replace the hemispherical bowl by  one whose cross section is an inverted cycloid arch(…). He was delighted to find that for such a bowl the object will reach the lowest point in exactly the same time, no matter from what height on the inner surface of the bowl the object is released. That is, the cycloid is the truly isochronous curve…

But a big question remained. How does one get a pendulum to oscillate in a cycloidal, rather  than circular arc? Here Huyghens made a further beautiful discovery. If one suspends from a point P at the cusp between two inverted(…) cycloidal semiarches PQ and Pr a pendulum  the lenght  of which is equal  to the lenght of one of the semiarches, the pendulum bob will swing is an arc that is an arch of a cycloid QSR of exactly the same size and shape as the cycoid of which  arcs PQ  and PR are parts. In other words, if the pendulum of the clock oscillates between Cycloidal jaws, it will be truly is isochronous.

… Huyghens in this investigation had made a discovery of capital mathematical significance, which he formulated in the following  theorem: the involut (…) of a cycloid is a similar cycloid, or inversely, the evolute of a cycloid is a similar cycloid.

Carl B. Boyer, A History of the Mathematics, John Wiley & Sons, 1968, pp. 410-412