In the film Amarcord, Fellini describes his childhood in Rimini. A world of yesterday, populated by characters ranging from the grotesque to the tender, including the school. The art teacher talks about Giotto while dipping a biscuit in a suspicious flask, while the Latin teacher concentrates on keeping the butt of his cigarette from falling. Bongioanni, the science teacher, stands with a heavy stone tied to a string and asks: «What is this?» The answers: «a stone», «it’s for a slingshot», «an elephant’s testicle»… «I’ll tell you: it’s a pendulum. You’ve certainly all seen a grandfather clock.» It may be my impression, but is it not true that Fellini’s gaze seems kinder on this professor who has come down from the platform and opened up the classroom to demonstrations and dialogue?
They say Galileo Galilei was fascinated by the pendulum and its operation from a young age, observing the oscillations of the lamp in the Pisa cathedral. It may be, but I think the pendulum must have been part of the family lexicon from the cradle. His father, a gifted musician (practical and theoretical), studied string and tube acoustics through experimentation and tried to explain the origin of dissonance. Galileo’s contributions to the understanding of pendulums would find their way into clock technology some decades later. During his years in Padua, he installed a 10-metre clock he could see swinging from his window at the university. Some claim that it allowed him to observe the rotation of the Earth, as Léon Foucault would do two centuries later: was it a missed opportunity?
As Galilei states in his Discourses and Mathematical Demonstrations Relating to Two New Sciences: «from such common and even trivial phenomena, you derive facts which are not only striking and new but which are often far removed from what we would have imagined». One of them (quite surprising, when we stop to think about it) is that there is no way to force a pendulum to oscillate at a different rhythm from the one that corresponds to it, determined exclusively by its length. In other words, the pendulum is isochronous: neither its mass nor (to some extent) the amplitude of its oscillation matters. The time it takes to go back and forth, the period, is constant and changes only if we lengthen (longer period) or shorten (shorter period) the string. If we reach this conclusion autonomously after experimenting with a pendulum, we will prove to have reached cognitive maturity, what Piaget calls the stage of formal operations. Did Trump ever do the experiment?
«There is no way to force a pendulum to oscillate at a different rhythm from the one that corresponds to it, determined exclusively by its length»
This led him to think that a metre could be defined as the length of a pendulum whose period was 1 second: verifiable anywhere on the globe. However, when Jean Richer travelled to Guyana in 1671, he had to shorten the pendulum by 3 mm so that the period matched the one measured in Paris. So much for universality! But gravimetry had just been invented: the acceleration of gravity g changes with latitude because of the Earth’s oblate shape, affecting the period of the pendulum. Let alone if we change planet: once on the moon, the Apollo XIV astronauts released a cargo strap, which started to oscillate (it is visible in the videos). From the period, it can be deduced that the value of g certainly coincides with that of the Moon (to further refute all those conspiracy theories).
«You’ve certainly all seen a grandfather clock. And how does it go?» says Bongioanni. The whole class join in, swinging to the rhythm of the pendulum: «Tick-tock, tick-tock, tick-tock».
TRY IT: You can make a pendulum with any object (a sinker, a ball, some plasticine, or a Christmas ball filled with salt or lentils) tied to a light thread. Attach the free end of the string to your hand or to an overhang with a pair of pliers. 1. Measure the length of the pendulum, from the attachment to the centre of the object (L = 32 cm, for instance) and set it to swing by moving it about 20°. 2. With the pendulum swinging, we time 10 complete oscillations (back and forth to the starting position). I obtain, for example, t = 11.4 seconds. So the period T = t ⁄ 10 = 1.14 seconds. 3. Shorten the thread by half and repeat the measurement. Now t’ = 8 s and T’= t’ ⁄ 10 = 0.8 s; the period is shorter. 4. We can see that (T ⁄ T’)2 = 1.432 = 2 and L ⁄ L’ = 2. It looks like the square of the period depends on the length.