Pendulum Clocks

The Workings:

Pendulum clocks, featured in today’s divertimento, were creations of science and technology. Galileo discovered the isochronous properties of the pendulum in the early 17th century, and fifty years later, Christian Huygens followed up Galilean studies by designing a pendulum clock, which was built by the clockmaker Salomon Coster. In connection with the design of the pendulum clock, Huygens posed and solved the problem of the tautochrone (or isochrone) curve: that by which a body descending by gravity to its lowest point takes the same time to reach it regardless of its initial height. Huygens identified the tautochrone as the cycloid, the curve described by a point on a circumference that rolls without sliding. At the end of the 17th century, the cycloid was also identified as the brachistochrone curve: the curve along which a body falling by gravity descends at maximum speed between two non-vertical points. The brachistochrone problem was proposed by Johann Bernoulli in 1696, one of whose aims was to find out whether or not Isaac Newton was able to solve it. Newton was up to the task and solved the problem in just one afternoon, despite having returned home tired after a busy day of minting coins at the English Treasury, which he then managed. Newton published the solution anonymously in the journal of the Royal Society of London, but the mere eighty words he used were enough for Bernoulli to guess who the author was: “Tanquam ex ungue leonem”, he said: You will know the lion by its claws.

The Fun:

A watchmaker has two misadjusted pendulum clocks. One is one minute ahead a day and the other one is one and a half minutes behind a day. If they are set correctly at 12 midnight today, when will they match again and what time will it really be?

 

 

 

 

Solution:

We encourage the readers to try to solve the divertimento for themselves. Whether you succeed or not, you can always consult the solution in this link.