Sci/Hist: The science of the pendulum

Some history:

Galileo Galilei said later that while he was sitting in a church in Pisa in 1581, someone pulled a hanging lamp to one side, lit, and released it (it may have been a censer). Galileo noticed that even when the swinging item slowed down, it still took just as long to go from side to side. He used his pulse, he said, to time the swings, and so discovered the pendulum principle.

There are problems with this story: Leonardo da Vinci knew about the pendulum in 1494, and according to Arthur Koestler’s The Sleepwalkers, the candelabra on show in the cathedral at Pisa, the one Galileo is supposed to have observed, was only installed several years after 1581.

That aside, surely, his pulse would have been racing too much to get reliable results, but who cares? Galileo still discovered the pendulum, and if he didn’t discover it, then it must have been a friend of his, because it was certainly Galileo who first told people of the idea. We know he later used a pendulum to time experiments, but this was a friend’s invention, the pulsilogium of Sanctorius (Santorio Santori was known to his learned friends as Sanctorius, which was the same thing, but in Latin).

Sanctorius was a friend of Galileo (whose name is never put into Latin), but Santori’s main claim to fame was that he designed a small pendulum to measure patient’s pulses, and this may be the origin of the legend, which never seems to appear as a direct quote from Galileo. Like Joule’s honeymoon thermometer and James Watt’s boiling kettle (both in chapter 6), the swinging lamp must be a myth. And by the way, there is no evidence that Galileo ever dropped weights from the leaning tower of Pisa, either!

The pulsilogium was a lead bullet, hanging from a silk cord on a small frame. By varying the length of the cord, Santori could synchronise the little pendulum with a person’s pulse (which makes sense when you look at the equation below). It wasn’t easy, but Santori managed to find that people’s pulse rates varied with their mood or excitement.

By the late 1600s, people knew this neat formula for the period of a pendulum (T):

The value l is the pendulum length in metres, and g is the acceleration due to gravity, which is usually about 9.8 metres per second per second, or 9.8ms-2. Notice the about, in that, as I will explain at the end.

Making a simple pendulum is easy. Just attach a weight to a bit of string, set it swinging, and that’s it. Making a good simple pendulum is a bit harder. You need to take some care, mainly in getting a firm, non-slipping grip on the top of the string, but when you do, if your maths skills are up to it, a good watch and a few minutes can establish the truth of the equation.

Depending on how your local mathematics curriculum is organised, you may need some help with the mathematics: ask around about ‘square root’ and ‘pi’.

That simple equation sums up all you need to know about a pendulum. The value l is the length in metres, and g is the acceleration due to gravity, which is usually about 9.8 metres per second per second, (or 9.8 ms^-2 in physics). Don’t worry too much about the ms^-2 right now: you can read more in the notes at the end. Just take note of the about, because the variation in g in different parts of the world led to a string of interesting enquiries about the shape of the world (which is an oblate spheroid).

A good pendulum needs strong cord and a compact mass like a builder’s plumb bob, which can be bought from a hardware store, or borrowed from a local DIYer (that’s a do-it-yourselfer). The top clamp needs to be very firm.

Why the pendulum mattered:

The variation in g led to a lot of interesting discoveries, and even gave us rubber! Huygens’ scheme might have worked, and the length of a second-pendulum, around 99.4 cm, would have been near enough to the metre or the yard, but there was a snag: the value of g varies from place to place, and sorting that out had some surprising ramifications. Here are some examples, mainly drawn from sources in the USA:

In 1672 Jean Richer reported that the period of a pendulum varied with latitude, and Isaac Newton jumped in to say that Richer’s observed variation was due to an equatorial bulge, and that offended the French. By this time, nobody thought the world was a perfect sphere any more, but France and England disagreed on the real shape, and national honour was at stake.

How g varies with latitude and altitude, mainly in North America.

Newton said the Earth was an oblate (squashed) spheroid, a bit like a pumpkin. If this were so, argued Newton, something called the precession of the equinoxes could be explained. Against that, French scientists had taken some sloppy measurements, getting results which suggested an Earth more like an on-end watermelon than a pumpkin.

Richer’s pendulum clock had been accurate in Paris but by then, astronomers had a very reliable ‘clock’, based on the way the four main moons of Jupiter disappeared behind the planet, or passed in front of it. So Richer knew his clock lost two and a half minutes each day at Cayenne in French Guiana, closer to the equator. Scientists rushed urgently to different places, mainly in South America, a mere 63 years after Richer’s observations. Newton had died in 1727, but the French still wanted to show up that insolent upstart!

The Académie française, the French Academy, France’s main scientific body, sent scientists out to measure a degree of latitude in both Lapland and Central America. We will ignore the Lapland group for now, but the scientists who went to South America included Pierre Bouguer and Charles La Condamine.

Pierre Bouguer’s name lives on today because geophysicists talk about Bouguer anomalies, commemorating his pioneering work in the Americas. When the acceleration due to gravity is measured very accurately, small local fluctuations can indicate equally local deposits of high or low density mineral ores—these fluctuations are Bouguer anomalies, so geophysicists care about him.

Bouguer spent a much of his life studying gravitational effects, and in 1740, he tried to estimate the value of G, the universal gravitational constant (we’ll come to that next), using a mountain as an attracting mass. That method can only be as accurate as the information the enquirer has about the interior of the mountain, and there were other problems which he could not have known about. We will ignore those here, but the key word is isostasy. Look it up!

There is always more to the story than the simple bit.

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