PISA’S LEANING TOWER

Pisa and its sights

In the 11th Century, Pisa was a busy commercial port, and over the next 200 years it became a prosperous centre of art and learning. Today, this capital city of over 100 000 people has an air of living in the past, its splendid buildings recalling the grandeur and glory of bygone days. In the north-western corner of the main city lies the Piazza del Duomo (Cathedral Square). It is here that one finds the famous Leaning Tower. The tower is built of white marble, and was begun in 1174 by Bonanno Pisano, and completed in 1350. The leaning of the tower is caused either by the settling of the subsoil or some defect in the foundations; there are real fears that one day it will topple over. The tower serves as a bell-tower for the adjacent Romanesque style cathedral. In fact, one can view the tower and then enter the cathedral through the facing transept door with its remarkable bronze panels depicting the life of Christ. The cathedral has a beautiful pulpit, and hanging opposite is an interesting 17th Century bronze lamp.
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Galileo

Galileo Galilei (1564 – 1642) was an outstanding astronomer. He was born in Pisa, and quickly devoted himself to science and mathematics, fields in which he possessed strong natural talent. It was while he was still a student that he watched the sacristan set the cathedral lamp swinging, and made his famous observation that the period of oscillation of a pendulum is independent of the size of the oscillation. At age 25, Galileo was appointed professor at the University of Pisa, and is said to have carried out experiments from the Leaning Tower to show that heavy bodies do not fall faster than light ones. After a few years he resigned his chair, and took an professorship at Padua, where he won widespread fame for his discoveries.

Much of Galileo’s work was in astronomy, and with his telescope he observed sunspots, the moon and the satellites of Jupiter. He wrote a book in which he supported the view of Copernicus that the sun is the centre of the solar system. These claims aroused the opposition of the church, and a year later in 1633, Galileo was summoned to appear before the Inquisition, and there forced to recant his scientific findings. It is said that after renouncing his belief that the earth is in orbit about the sun, he whispered in despair ‘nevertheless, it does turn’.

Simple harmonic motion ...

In fact, Galileo’s observation about the oscillation of a pendulum was not quite correct. To understand what is happening here, we need to investigate an event known as simple harmonic motion. This occurs when a particle moves so that its acceleration along its path is directed towards a fixed central point on that path, and the acceleration is proportional to the distance from that fixed point. Examples are a weight hanging from a spring, or the vibrating prongs of a tuning fork.

Let the fixed point be the origin O, and let s denote the distance of the particle from the origin. It is known that if we measure time from when the particle is at O, then the distance from O at time t is given by s = A sin wt, where A and w are constants.

If we take s = A sin wt or s = B cos wt and we increase t by 2/w, we get the same value of s again, since sin(wt + 2) = sin wt, and cos(wt + 2) = cos wt. Hence after successive intervals of time 2/w, the particle is in the same position, and in fact moving with the same velocity. The time interval 2/w is called the period of a complete oscillation.

*For those with calculus*
The relationship between s and t can be written Explain the meaning of (), and show that s = A* sin wt satisfies it.

... and the pendulum

To understand the pendulum, we will need to use the fundamental equation of dynamics: if a force F produces an acceleration a in a body of mass m, then F = m a.

The pendulum consists of a heavy particle or bob B attached to a fixed point C by a weightless string, and swinging in a vertical arc. Let CO denote the vertical position of the string, l the length of the string, and m the mass of the bob.

The force acting vertically downwards on the bob is m g, where g denotes the acceleration due to gravity ( 980 cm/sec²). This is balanced by the tension in the string, but the force acting along the circle, tending to bring the bob back to position O is mg sin . So the acceleration of B along the circle is g sin . If is small, sin , and noting that the arc length OB is s = l, we see that the acceleration of B along the circle is approximately g = ^gs/ _l = (^g/_l)s.

If s is small, the arc OB is approximately a straight line segment. This means that we have the simple harmonic motion situation, with w = (^g/_l). Thus the period of oscillation of the pendulum is T = 2 (^g/_l).

Falling bodies

Investigate A golf ball and a cricket ball are dropped from a height. Which lands first? Carry out some experiments, and see if you can make a conjecture.

Many people agreed with Aristotle who believed that heavy bodies fall faster than light ones. Of course it is true if the light body is a feather! But we assume here that air resistance plays no part. Galileo carried out a number of experiments from the top of the Leaning Tower, and found a formula which shows that a the distance a falling body travels depends only on the time taken.

Equations of motion

v = u + at,
s = ut + ¹/₂.at²,
v² = u² + 2as.

A particle starts at the origin O at time t = 0 with initial velocity u, and moves with constant acceleration a. If s is the distance of the particle from the origin at time t, and v is the velocity at time t, the three equations of motion for a particle moving in a straight line with constant acceleration are as at left.

Further reading ...

Eves, H. (1976), An introduction to the history of mathematics, Holt, Rinehart, Winston.
Italy: A Phaidon cultural guide (1985), Phaidon Press (Oxford).
http://www.pbs.org/wgbh/nova/pisa/galileo.html

Vertical motion under gravity

For a falling body, if the distance s is measured (upwards) from the ground, then the acceleration is a = – g –980 cm/sec². If the body starts from rest, u = 0, and equation s = ut + ¹/₂.at² becomes

s = – ¹/₂.gt².

This was Galileo’s result. Notice that it does not depend on the mass of the body. Try it!

Investigate

(a) On the lower (south) side, the Leaning Tower is 54.5 metres high. An object is dropped from the top. How many seconds does it take to reach the ground?

(b) From what height would an object need to be dropped from in order for it to take one second to reach the ground?

(c) Compare your answers to (a) and (b). Do they surprise you? Draw the graph of s = (+) ¹/₂.gt², plotting distance s against time t. Notice how s grows large as t increases.

Mysteries of the transfinite

How does one tell if two sets contain the same number of elements? The obvious answer is to ‘pair off’ the elements. Mathematically, we put the elements into a one-to-one correspondence. Thus the sets {a, b, c}, {u, v, w} have the same number of elements, since a u, b w, c v say. This number is in fact ‘3’ because we can set up the correspondence with the integers 1, 2, 3.

Galileo observed that the one-to-one correspondence n 2n shows that there are equal numbers of integers (n) and even integers (2n). For, we set up the pairing ..., –1 –2, 0 0, 1 2, 2 4, 3 6, ... . Surprisingly, the even integers form a proper subset of the set of integers. That is, the proper subset contains just as many elements as the set itself. This characteristic of infinite sets was first properly explored by Cantor in the late 19th Century.

Investigate Use the following argument to show that the number of positive fractions is the same as the number of positive integers.
(a) List the positive fractions as in the table.
(b) Do all positive fractions occur here? all positive integers?
(c) Set up an arrow path as shown to contain all the table entries.
(d) Delete numbers like ²/₂, ⁴/₂,... which represent repeat entries.
(e) Observe that the arrow path sets up a correspondence between the table entries and the numbers 1, 2, 3, ... , the order of the terms in the sequence.