If our Foucault pendulum from last week, instead of hanging it from the Kio Towers, we placed it just above the North Pole (or South Pole, it does not matter), its plane of oscillation would take 24 hours to make a complete revolution with respect to the ground, since in reality it would be the Earth that at that time would make a complete revolution under said plane of oscillation. If we were to place it at the equator, it would remain fixed (which is like saying that it would take an infinite time to make an apparent turn), so we can deduce fermianly that, at an intermediate latitude, such as Madrid, this apparent turn would take more than 24 hours (and less than infinite). In fact, it is not difficult to show that this time (T) is inversely proportional to the sine of the latitude angle (L); expressed in hours:
T = 24 / sin L
At the poles the latitude is 90º and therefore the sine is 1, while at the equator it is 0º and its sine is 0. Without resorting to some trigonometric tables, we know that the sine of 40º is somewhat less than that of 45º, which is one of the acute angles of a right isosceles triangle of leg 1 and hypotenuse √2, so sin 45 = √2 / 2 = 0.707… Taking the value 0.7 as the sine of the latitude of Madrid (40º 30 ′ N) , we can affirm that in the capital the oscillation plane of a Foucault pendulum takes about 34 hours to make an apparent turn (it is not a bad approximation, without tables or calculations, although in reality the Foucault pendulum of the Astronomical Observatory of Madrid takes a little more; how much exactly?).
Regarding the period of oscillation (T), it is given by the formula T = 2π√ (l: g), where l is the length of the wire and g is gravity, so the period of oscillation of a pendulum with a cable of 100 meters would be about 20 seconds: in a low amplitude oscillation, the sphere suspended from the long cable would move with unreal slowness.
Of leaning towers and bodies in free fall
And if we talk about pendulums and inclined towers, it is inevitable to think of Galileo, who was the one who discovered the principle of the pendulum (observing the oscillation of a lamp hanging from the ceiling of a church), and of the tower of Pisa, his hometown, whose Tilt supposedly used to carry out various experiments. Supposedly, because his experiments were most likely purely mental.
Thus, to show that Aristotle was wrong in thinking that bodies fell faster the heavier they were, he dropped from the top of the leaning tower (or simply imagined it) two balls, one heavy and the other light, joined together by one rope. Why joined by a rope? I invite my astute readers to discover why this string is the key to the refutation of the Aristotelian thesis.
As an anecdote, it should be noted that the person who did physically carry out Galileo’s experiments on the falling of bodies was the astronomer and Jesuit Giovanni Battista Riccioli (1598-1671), who in 1644 climbed another leaning tower, that of Asinelli, in Bologna, to drop various objects from there and verify Galileo’s laws, taking advantage, in addition, to study to what extent the friction of the air slows down the fall (which makes Riccioli a precursor of aerodynamics).
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn physics’, ‘Damn maths’ or ‘The great game’. He was a screenwriter for ‘La bola de cristal’.