Math Science Chemistry Economics Biology News Search

Everyone knows that the pendulum, and its fundamental properties, were studied for the first time by Galileo Galilei (1564-1642), who discovered one of the most important characteristics of this physical object: the pendulum isochronisms. Galileo discovered this property observing that the pendulum does not change its period, even if its amplitude decreases. This behaviour depends only on the constant of gravity and on the pendulum length, and not on its amplitude. Galileo discovered that this property was confirmed for small angles, and extended isochronisms property, erroneously, even to bigger angles (but we can understand him, if we think that he measured the time with his heartbeats). This property is valid with an acceptable approximation only for little angles because when we resolve the equation for the motion of the pendulum, we consider the space that the pendulum makes similar to a chord of circumference and not like the corresponding arc. Obviously only for little chord we can approximate the pendulum’s motion in the right way.

In spite of this, he could determine the characteristic period of every pendulum with generic length l, with an acceptable approximation for that time; so employing pendulums that make oscillations of little amplitude, he allowed the craftsmanship of the first chronometers. Here’s a reconstruction of the first pendulum used as a chronometer:

Then comes Christian Huygens (1629-1695), who built in 1656 the first clock with a pendulum. He realized that to make oscillation more regular, the pendulum should describe a cycloid arc. Modern watches are made on this property.

Then other discoveries came about the pendulum properties, for example the property concerning the pendulum oscillation plane: the pendulum keeps its plane of oscillation, in an inertial system.

But the Earth is not an inertial system, because it rotates around its axis, and so the plan of oscillation rotates respect to the Earth of an angle that depends, for every unit of time, by the latitude of the place where the experiment takes place (the longer is the distance from Earth axis the longer is the rotation of the pendulum).

The first who made this experiment with adequate instruments was Leon Foucault (1819-1868); in 1851 in a very famous experiment in Paris under Pantheon’s dome, with a 30 kg copper sphere, holed by a steel cable 68 meters long, he demonstrated the Earth rotation in respect the “fixed star”. The pendulum oscillation plane rotated completely in 32 hours. The experiment was confirmed later.

But there is not only the story of the pendulum in itself; its understanding allows us to study a lot of harmonic motions, essential for modern science. And the story of harmonic motion starts from Kepler. His first law is:

So we all know that, if a and b are ellipse semi axis, we can individualize the position of a planet moving in its elliptical orbit with the coordinate x and y in this way (equation that describes an ellipse centred in the point (0;0) in the Cartesian axis).

The importance of harmonic motion is evident even in the scientific revolution carried out by Max Planck, studying the characteristic of the blackbody: But what is the link between the blackbody radiation problem and the harmonic motion? In few words (I hope you will prefer in this way) a body, for example a metal, can be considered as a reticule of ions in equilibrium, while the electrons are around the metal (the electronic cloud). Ions are free to vibrate around the position of equilibrium and, as we know from electromagnetic theory, an oscillating electric charge generates an electromagnetic wave (the frequency of the wave is proportional to the charge orbital frequency). Since every charge has a different velocity respect to other charges, the wave frequency is the reason for which a continuous electromagnetic spectrum is originated. But Raleigh and Jeans’ blackbody radiation experiment did not confirm the theory. For this reason Planck tried to search for an explanation to this phenomenon. He proposed that for every frequency (and when we speak about frequency we must talk about harmonic motion) there must be a well-defined quantity of energy. He proposed this well known relation between Energy and frequency:

*E=nhf*

where h is the Planck constant, f the frequency of the vibrating body, and n an integer number called quantic number. The energy was in this way quantized, and theory was confirmed by experiment; this equation had a very important role in modern physics, and a lot of physical phenomena were explained with this new revolutionary theory (we can easily remember the Einstein and his explanation of photoelectric effect).

But harmonic motion is quite important even to describe the system periodicity. if we took for example the equation that describes the energy of the pendulum (with a elasticity constant k) in a system without any kind of friction we have

The graphic will be similar to the previous one

If we are choose a point (x;p) we obtain that after a period T (the period of the system) we will go back to the same point. Even in this case the study of harmonic motion allows us to understand better the problems of the system periodicity.

This little excursus permits us to understand how the pendulum and the harmonic motion influenced the History of Sciences.

- Feynman, Leighton, Sands, The Feynman Lectures on Physics, Addison Wesley Company, Reading, MA, 1963
- Aleksandrov, Kolmogorov et al., Le Matematiche, Bollati Boringhieri, Bologna,2000
- G. Zampieri, Notes from his lessons, Liceo Scientifico Alvise Cornaro, Padova 2002
- R. Courant, H. Robbins, What is mathematics? An Elementary Approach to Ideas and Methods, Oxford University Press, 2nd edition revised by Ian Stewart, N.Y., 1996
- F. Ayres, Calcolo differenziale ed Integrale, McGraw-Hill , Milano, 1994
- W. W. Sawyer, Che cos’è il calcolo infinitesimale, BMS Boringhieri, Bologna, 1978
- Lamberti, Mereu, Nanni, Corso di Matematica 1-2-3, Etas Libri, Torino,2000
- Peter J. Nolan, Complementi di Fisica, Fisica Moderna, Zanichelli, Bologna,1996

- The drawings of the ellipses are made by myself using Microsoft Excel 2002
- www.pcangelo.eng.unipr.it
- www.culture.fr
- www.matematicamente.it
- www.vialattea.net
- www.the.planet.saturn.com

I would like to thank prof Giuseppe Zampieri for the aid he gave me in developing this work, for the lectures he kept about this subject and for the very great patience he had with me.