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Tale 1:
Pendulums Measure the Earth
From ancient times, thinkers have speculated about, theorized upon,
calculated, and measured the physical properties of the earth. Early
Greek and Chinese astronomers estimated the circumference of the
earth using “instruments” that were readily at hand. Think about
shadows from the sun and positions of stars. Man-made instruments,
such as the pendulum, arrived on the scene much later. In this first tale
of the pendulum, we discuss two ways in which the pendulum has
contributed to our understanding of the earth.
Beginning in the seventeenth century and carrying right into the
twentieth century, the pendulum became the primary tool for measuring the earth’s density and shape. Today, more sophisticated instruments carry the burden, especially measurements from satellites. But,
as we describe in the first part of our tale, a general understanding of
these fundamental geological properties can be traced to the lowly
pendulum. The second and larger part of this tale describes the dramatic demonstration that the earth rotates. We refer to the well-known
Foucault pendulum, hung for the public in 1851, in the Pantheon,
which is located in Paris, France. This may have been the most public
scientific experiment of all time. Its success as a demonstration was
undoubtedly enhanced by the fact that it was really a validation of an
already generally held belief. People of the time typically believed that
the earth rotated, but to see a man-made structure affected by this
rotation must have been a powerful confirmation. In our story, we will
learn more about the public reception of – and even public misunderstanding about – the Foucault pendulum.
1.1 Shape
By the latter part of the seventeenth century, the daily rotation of the
earth was an accepted fact. In 1673, the Dutch physicist and astronomer
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Seven Tales of the Pendulum
Christiaan Huygens suggested a picture of how this rotation might
affect the earth’s shape. He proposed the idea of centrifugal force, a
“center-fleeing” force. Seated inside an automobile as it turns a corner,
we feel a tendency to fly outward – a centrifugal tendency. The same
effect happens for the earth, especially near the equator. We can think
of the earth being somewhat elastic, and therefore the centrifugal
effect causes a slight bulge at the equator. The bulge makes the effective
radius of the earth at the equator slightly larger than, for example, at
the poles. And this means that the gravitational field is slighter
smaller at the equator. (The centrifugal effect is not just limited to
bulging of the earth. It decreases the effective gravitational field on
any object on the surface of the rotating earth.) In this way, Huygens
reasoned that the earth is fatter at the equator.
In 1687, Newton published his universal law of gravity in his
famous Principia, giving theoretical foundation to these thoughts.1
What emerges is a picture of the earth as an oblate ellipsoid, fatter at
the equator than at the poles. But the picture required much experimental verification, and for more than two centuries, the pendulum typically provided this verification. Another important date
in this story is Christmas Day 1657, when Huygens presented the
world with the pendulum clock. The clock and other specially
adapted pendulums were to become dominant tools in geological
measurements.
It is the relationship among its . . . , its length, and the strength of the
gravitational field that makes the pendulum a useful tool for measuring the local gravitational field. As we noted in the Introduction both
through words and formula, the stronger the field, the shorter is the
period. Or, in a weaker field, a pendulum must be shortened to maintain the same period. The first recorded use of the pendulum to measure local gravity is usually attributed to the measurements of the
French astronomer Jean Richer (1630–96). He made the measurements
in 1672. Richer found that a pendulum clock beating out seconds in
Paris at latitude 49 degrees north lost about 2½ minutes per day when
placed near the equator in Cayenne, now the capital of French Guiana,
1
Newton coined the term “centripetal force” in parallel with the Huygens centrifugal
force. Centripetal force is that force which keeps a moving object in circular motion.
Otherwise, the object would fly off tangentially to the circle.
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at 5 degrees north.2 From this difference, he concluded that Cayenne
was further from the center of the earth than was Paris. Newton, on
accidentally hearing of this result at a meeting of the Royal Society ten
years later, used it to refine his theory of the earth’s oblateness. Richer’s
result was to be only one of many pendulum measurements to contribute to detailed knowledge of the earth’s shape.
Richer’s swinging of pendulums had another unforeseen consequence. His demonstration confirming that pendulums are sensitive to
gravity meant that pendulum clocks kept different time in different
parts of the world. It then followed that, as the clock traveled the
world, it would not be a reliable timekeeper of a standard time such as
Greenwich time. This feature precluded the use of the pendulum clock
for the measurement of longitude. Accurate measurement of longitude would be possible only with precision spring-driven clocks that
were unaffected by local gravitational variations.3 Therefore, Richer’s
result, while helping to determine the shape of the earth, also led to
the eventual rejection of the idea of using a pendulum clock as a reliable timing standard for the measurement of longitude.
While Richer may have been the first to actually make some pendulum measurements, other famous – or almost famous – scientists also
proposed or used the pendulum to measure the gravitational field.
Robert Hooke (1635–1703), well known for the linear law of elasticity,
for his invention of the microscope, a host of other inventions, and his
controversies with Newton, was one of the first to suggest, in 1666, that
the pendulum could be used to measure the gravitational field.
Edmond Halley (1656–1742), the Astronomer Royal (of Halley’s Comet
fame), also appears in this story. In 1676 Halley sailed to St. Helena’s
island, located in the South Atlantic, in order to make a star catalog for
the southern hemisphere. As a friend of Hooke, he was aware of Hooke’s
suggested use of the pendulum to measure gravity, and he did make
2
Independent checks on mechanical clocks typically utilized astronomical measurements. For example, by 1676 John Flamsteed, the first Astronomer Royal, was calibrating his
pendulum clocks using a fixed telescope aimed at the star Sirius to measure the sidereal day
(23 hours, 86 minutes, 4.1 seconds). This is the time taken for the star to reappear at a precise
position in the field of the telescope. Similarly, Richer could have measured the sidereal day
in Paris and near the equator with a telescope.
3
For the story of the initial determination of longitude, see D. Sobel, Longitude: The True
Story of a Lone Genius who Solved the Greatest Scientific Problem of His Time (Penguin, New York, 1996).
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such measurements while on St. Helena. (Certainly Halley is famous
for having his name applied to the comet, yet he probably rendered a
significantly more important service to mankind by encouraging and
supporting Newton in the publication of the latter’s famous Principia.)
In principle then, a comprehensive series of pendulum measurements taken at a variety of locations should yield a map of the strength
of the gravitational field over the earth’s surface. Yet without special
effort the results tend to be inaccurate. For example, it is often difficult
to accurately determine the effective length of the pendulum. There
can be ambiguity in the measurement at the position of the pivot or at
the bob. In 1817, at the suggestion of the German astronomer F. W.
Bessel (1784–1847), Henry Kater (1777–1835), a captain in the British
Army, invented a reversible pendulum that significantly increased the
accuracy of the measurement of the gravitational field. Kater’s pendulum is shown schematically in Figure 1.1.
It consists of a rod with two sharp pivot points (P), whose positions
(h1 and h2) along the rod are adjustable. The period of the pendulum is
determined with the pendulum suspended from one pivot point. Then
the pendulum is reversed to pivot from the other point and a further
measurement of the period is made. In principle, the determination of
the gravitational field g is made by adjusting the pivot points until the
periods of small oscillation about both positions are equal. In practice,
it is difficult to precisely adjust the pivot points, which are usually knife
p1
h1
h
h2
p2
mg
Figure 1.1 The Kater reversing pendulum.
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edges. Instead, counterweights are attached to the rod and are easily
positioned along the rod until the periods are equal. In this way, the
pivot positions are defined by fixed knife edges that provide the possibility of accurate measurement. Once the periods are found to be equal
and measured (T ), the acceleration due to gravity g is readily calculated using a slight variation on the formula given in the Introduction
for the period of a simple pendulum. That is,
T = 2p
h1 + h2
.
g
Most importantly, the sum h1+h2 is just the easily measurable distance
between the knife-edge pivot points. By gradually moving the counterweights, the convergence toward equal periods from both pivots
creates a significantly more accurate process than that from a single
pendulum. As late as 1936, the U.S. National Bureau of Standards used
a modified version of the Kater pendulum to determine the acceleration due to gravity at Washington, D.C. as g = 980.080 ± 0.003 cm/s².
After its invention, many of the pendulum gravity experiments
were carried out using the Kater “reversing” pendulum. One of the
original pendulums, number 10, constructed by a certain Thomas
Jones, rests in the Science Museum in London. The display card reads as
follows:
“This pendulum was taken together with No. 11, which was identical . . . on a voyage lasting from 1828 to 1831. During this time Captain
Henry Foster swung it at twelve locations on the coasts and islands
of the South Atlantic. Subsequently it was used in the Euphrates
Expedition, of 1835–6, then taken to the Antarctic by James Ross
in 1840.”
Meanwhile, some form of the pendulum continued to travel the
world measuring local gravitational fields. Sir Edward Sabine
(1788–1883), an astronomer with Sir William Parry in the search for
the northwest passage (through the Arctic Ocean across the north of
Canada) spent the years from 1821 to 1825 determining measurements
of the gravitational field with the pendulum along the coasts of North
America and Africa, and, of course, in the Arctic. The American philosopher Charles S. Pierce (1839–1914) makes a surprising appearance
in this historical context. Although known for his contributions to
logic and philosophy, Peirce rarely held academic positions in these
branches of learning. Instead, he made his living with the U.S. Coast
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and Geodetic Survey. Between 1873 and 1886, Pierce conducted
pendulum experiments at a score of stations in Europe and North
America in order to improve the determination the earth’s elliptical
shape. However, his relationship with the Survey administration was
fractious, and he resigned in 1891.4
Thanks to these and many other measurements, we now know that
the earth is an oblate ellipsoid, fatter at the equator than at the poles.
Modern measurements find that the equatorial axis (the diameter
through the earth at the equator) is about 21 kilometers, or 0.3 percent,
longer than the polar axis.
1.2 Density
Pendulums were also instrumental in determining the earth’s density.
The average density of the earth is much greater than that of materials
that are typically found on the earth’s surface. But direct sampling of
the earth’s interior would have been an impossible task. Remarkably,
Pierre Bougeur (1698–1758), a French professor of hydrography and
mathematics, used a clever bit of theory and pendulum measurements
to estimate the earth’s density, r, without using a direct measurement.
In 1735, the French Academy of Sciences sent Bouguer to Peru to
measure, among other things, the length of a degree of a meridian arc,
near the equator.5 (A variety of such measurements at different latitudes, along with pendulum measurements, would help to determine
the oblateness, or “bulginess”, of the earth.) While in Peru, Bougeur
made measurements of the oscillations of a pendulum. His pendulum
had beaten out seconds in Paris, but in Quito, Peru (latitude 0.25 degrees
south) the period was different. From his original memoir, it is a little
confusing to determine whether he maintained a constant-length
pendulum or whether, as his data suggest, he modified the length of
the pendulum to maintain a period of a second. At any rate, he used
the pendulum to measure the gravitational field.
Now, for the density calculation, Bouguer had to measure the gravitational field at two different elevations. He measured it close to sea
level, with a resulting gravitational acceleration of gsea, and then on top
4
Pierce’s problems stemmed from a certain indifference to bureaucratic procedures and
complications in his marital situation.
5
A meridian arc is a distance along a north–south line on the earth’s surface.
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of the Cordilleras mountain range, with a gravitational acceleration of
gmount. Bouguer also measured the average density of material on the surface
of the earth on the mountain range. He then developed a mathematical model for the data, using standard gravitational theory and some
simplifying assumptions about the terrain. It is remarkable that using
this minimal data and the mathematical model, he could actually calculate the average density inside the earth.
Even without following the mathematical details, we can gain some
appreciation for the cleverness of Bouguer’s method. Figure 1.2 shows
a part of the earth’s surface, with the mountain range drawn as a simple “bump” of constant height on top of the earth. The letters on the
diagram correspond to various physical quantities. For example, a is the
earth’s radius, rearth is the average density of the earth, rsurface is the average density of the mountain range found by sampling the local soil, h is
the average height of the mountain range, and A is an estimate of the
average area of the mountain range.
All of these ingredients, combine, at least in theory, to give the ratio of
the gravitational acceleration on top of the mountain range to that near
sea level, gmount/ gsea. The mathematics is complicated, but the clever result is
a formula that connects all these known quantities and, with a little algebra, provides an estimate of the earth’s unknown average density rearth:
gmount
2h 3h rsurface
=1− +
.
gsea
a 2a rearth
A
h
rsurface
rearth
a
Figure 1.2 The little “bump” on the earth’s surface represents a whole
mountain range.
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If the two accelerations were the same, then the ratio would just be
one. This formula accounts for the differences through two correction
terms. In the first (negative) term after the numeral one, the gravitational field on the mountains is reduced because the measuring point is
further from the center of the earth, and this correction is called the
free air term; it involves the mountain height h and the earth’s radius a.
In the second (positive) term, the field is enhanced by the gravitational
pull of the mountain range and this correction is called the Bouguer
term. It combines the two previous quantities as well as the density of
the mountain range and the average density of the earth. Every quantity could be estimated or measured, or was known at the time, and
therefore it was now possible to calculate the ratio of the density of the
mountainous material to that of the rest of the earth. Bouguer’s pendulum measurements convinced him that the earth’s mean density
was about 4 times that of the mountains, a ratio not too different from
a modern value of 4.7. In Bouguer’s own words, he had exploded two
current theories:
“Thus it is necessary to admit that the earth is much more
compact below than above, and in the interior than at the
surface. . . . Those physicists who imagined a great void in
the middle of the earth, and who would have us walk on a
kind of very thin crust, can think so no longer. We can
make nearly the same objections to Woodward’s theory6 of
great masses of water in the interior.”
Bouguer’s experiment was the first of many of this type. A common
variant on the mountain range experiment compared the gravitational accelerations at the top and at the bottom of a mine shaft. In
this case, we can think of the extra structure as not just a mountain
range, but a whole spherical shell above a slightly smaller earth. The
radius of this new “earth” is less than that of the ordinary earth by the
depth of the mine shaft. Figure 1.3 shows the various quantities that
go into the calculation. We see that h is the depth of the mine shaft,
6
The person referred to is probably John Woodward (1665–1728) a prominent English
naturalist and geologist. Bouguer’s result showed that the earth’s interior is much denser
than water.
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and a is the radius as measured from the center of the earth to the bottom of the shaft.
Again the formula has two correction terms, although the
Bouguer term differs by a factor of two from the expression used in
the mountain-top experiment:
gtop
2h 3hrsurface
=1− +
.
gbottom
a
arearth
As in the previous case, a little algebra leads to a value of the earth’s
density, r earth. Coal mines were widely available in England, and the
seventh Astronomer Royal and Lucasian professor of mathematics at
Cambridge, George Airy (1801–92), was one of many to attempt this
type of experiment. Unfortunately, his early efforts in 1826 and 1828
in Cornwall were frustrated by floods and fire. But much later, in
1854, he successfully applied his techniques at a Harton coal pit in
Sunderland and obtained a value for the earth’s density of r earth = 6.6
gm/cm³, which may be compared with the presently accepted value
of 5.5 gm/cm3.
Pendulum experiments continued to be improved. The Austrian geographer Robert Von Sterneck (1839–1910) explored gravitational fields
at various depths inside silver mines in Bohemia and in 1887 invented a
four-pendulum device. The older devices were based upon the 1 second
h
a
rsurface
rearth
Figure 1.3 A schematic of the earth with a mine shaft.
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pendulum, with a length of about 1 meter.7 Von Sterneck felt that little
would be sacrificed and much could be gained by going to 1/2 second pendulums, which were only about 1/4 meter in length. These fitted much
better into a partially evacuated container and yielded a definite improvement in accuracy. He placed two pairs of 1/2 second pendulums at right
angles. Each pendulum, in a given pair, oscillated out of phase with its
partner. This arrangement reduced the flexure in the support structure
that ordinarily contributed a surprising amount of error to the measurements. The two mutually perpendicular pairs also provided a check on
each other. Von Sterneck’s values for the mean density of the earth ranged
from 5.0 to 6.3 gm/cm³. The swing of the pendulums in a pair is compared
with a calibrated 1/2 second pendulum clock by means of an arrangement
of lights and mirrors, as observed through a telescope. Because they have
very slightly different periods, the gravity pendulum and the clock pendulum will only come into synchronization once in a while. The number
of counts between such “coincidences” is observed and used in determining a precise value of the gravity pendulum period. Accuracies as high as
2 × 10–7 seconds were claimed for the apparatus.
Finally, in the twentieth century, we note the work of Felix Andries
Vening Meinesz (1887–1966), a Dutch geophysicist who, as part of his
Ph.D. (1915) dissertation, devised a complicated pendulum apparatus.
Somewhat like Von Sterneck’s device, it used the concept of pairs of perpendicularly oriented pendulums swinging out of phase with each other.
As shown in Figure 1.4, the four pendulums are distinguished by the
bright brass pendulum bobs: one each at the ends of the long dimension
of the apparatus and a pair located in the middle on the short axis. It is
clear that the device was a far cry from the original simple pendulum.
With this complex device, Vening Meinesz eliminated a horizontal
acceleration term due to the vibration of peaty subsoil that seemed to
occur in many places where gravity was measured. Vening Meinesz’
apparatus was also fitted for measurements on or under water and
contained machinery that compensated for the motion of sea. Aside
from the interruption caused by World War II, some version of this
device was used on submarines from 1923 until the late 1950s.8
7
Mathematically inclined readers will note that the formula in the Introduction would
give a period of about 2 seconds for a 1 meter pendulum. However, in early pendulum terminology, the period was taken as only one-half of a full oscillation cycle.
8
A famous, but different, approach to measuring the earth’s density is described in the
tale of the twisting pendulum.
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Figure 1.4 The Vening Meinesz pendulum. Four pendulums are arranged
in mutually perpendicular pairs. The four bobs are the bright fixtures near
the bottom of the apparatus. (Courtesy of the Society of Exploration
Geophysicists Geoscience center. Photo © 2004 by Bill Underwood.)
1.3 Rotation – the Foucault pendulum
The Foucault pendulum demonstrates the rotation of the earth. It is
one of the most famous and fascinating experiments of all time. This
fascination extends the pendulum’s reach even into literature:
“At 11:30 PM in the bell tower of the old Saxon church at
Randall’s bridge, a huge statue toppled and smashed. Heavy
blocks of broken marble now lay up against the door barring entry or exit. The solitary window was too narrow for a
man to pass through; the belfry high above led only to the
steep roof which rose beyond the reach of any ladder. When
Detective Inspector C. D. Sloan put his shoulder to the door
(blocked by the broken statue) it opened only a crack.
Through it he could clearly see the room was empty –
except for the bells, the debris of shattered marble . . . and
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the protruding arm of a dead man. How could a murderer
have escaped from this sealed tower? Sloan’s only clues: a
spent match, a black thread (burned at the end), a pendulum hanging in the center of the tower. And an emerald
earring.”
In 1981, novelist Catherine Aird wrote a mystery entitled His Burial Too that
uses the Foucault pendulum as a murder weapon. Aird’s story opens with
the murder of a certain Richard Tindall, owner of an engineering company and father of an attractive daughter – requisite characters for detective stories of a certain vintage. Various suspects are developed who have
stronger or weaker motives for promoting Tindall’s death and thus a
range of possibilities is created. Yet alibis seem airtight and certain aspects
of the execution of the crime remain a mystery for much of the story.
Later in the course of the investigation, the fog of ignorance begins
to dissipate. “From Sloan’s superior: “What’s all this business about
pendulums, then?” Sloan shook his head. “I don’t think the pendulum
would do the trick, Sir. The sculpture was off-line from the window.”
Cold trickled down Sloan’s spine. Then his head came up. . . . There
was such a thing as a pendulum that turned. He’d seen it. In a museum.
. . . It was coming back. What the pendulum needed was a smooth start,
and half an hour later it was off course. . . . The smooth start had been
important. He remembered watching while a man from the museum
started it by burning the anchoring string in two with a match.”
And so the investigation proceeds to its now-inevitable conclusion.
The pendulum oscillation was initiated some 5 hours before the actual
murder, giving the murderer an opportunity to develop an alibi for his
whereabouts at the time of the killing. During the intervening 5 hours
the plane of oscillation rotated about 60 degrees, until it hit a massive
statue that was delicately balanced to fall on the previously tied up Mr.
Tindall. When the statue fell, it killed the victim and sealed the door to
the chamber. The other clues are explainable and point to a suspect
once the basic physics of the Foucault pendulum is understood. The
general positioning of the tools of the murder is illustrated in Figure
1.5. (At least two features of the story seem slightly implausible. First,
it is very unlikely that this particular pendulum could have sustained
its oscillations for 5 hours without some sort of driving force. The second difficulty lies with the fact that the trigger for the destabilization of
the statue was the very small change in the plane of oscillation of the
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DOOR
STATUE
BOB
STRING
TOP
SUPPORT
Figure 1.5 A schematic of the crime scene in Aird’s novel. (Drawing by
Richard Crane. Reprinted with permission from The Physics Teacher, May
1990, p. 265. © 1990, American Association of Physics Teachers.)
pendulum. The murderous statue must have been balanced very precariously indeed!)
Let us get back to reality. What is a Foucault pendulum, what is its
history, and what physical phenomenon does it demonstrate?
In theory, any earth-based pendulum is a Foucault pendulum.
Every pendulum oscillates in a plane, and a realistic Foucault pendulum is a one that is specially suspended so that its plane of oscillation is free to rotate with the rotation of the earth. Another way
to conceptualize the Foucault pendulum is to think of the pendulum as having a suspension mechanism that is fixed relative to the
stars.9 Then the earth rotates under the pendulum’s oscillations.
From the earth’s point of view, the pendulum’s plane of oscillations
seems to rotate, but seen from a distant star, it is really the earth
that rotates, not the pendulum’s oscillations. Figure 1.6 shows a
replica of the Foucault pendulum hanging in the Pantheon, where
it made its public debut.
9
This is a bit of a simplification.
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The inventor, for whom this special pendulum is named, was Jean
Bernard Léon Foucault, the son of a Parisian publisher (see Figure 1.7).
He was born September 18, 1819 and died February 11, 1868.
Foucault received his secondary education at the prestigious College
Stanislas and later enrolled in the Paris medical school. Yet sometime
around 1844 he abandoned medicine to pursue his strong interests in
physical science. Many of his early experiments were in optics. These
included efforts to improve Daguerre’s10 new photographic processes.
He also studied the intensity of sunlight, the phenomenon of the
interference of light that is observed by combining special beams of
light, and the effect of crystals on light. Foucault showed that light
traveled faster in air than in water and also determined the absolute
speed of light in vacuum. Perhaps as result of his pendulum work,
Figure 1.6 A Foucault pendulum (similar to the original) hanging in the
Pantheon, Paris, where the original was first publicly displayed. (Photo by
GLB.)
10
Louis Daguerre (1787–1851) is known for his invention of a precursor to the photograph labeled, not surprisingly, as the daguerreotype.
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Figure 1.7 Leon Foucault, 1819–68, inventor of the Foucault pendulum,
at age 47. Foucault made many contributions to science and technology.
(Image from Wikipedia.)
Foucault invented the precision gyroscope and demonstrated its use as
a compass. In 1855, Foucault was appointed physicist at the Paris observatory. There, he invented techniques that would allow for the manufacture of large mirrors for reflecting telescopes, thereby significantly
increasing the power and utility of the telescope. Remarkably, the electric currents induced by magnets that we now refer to as eddy currents
were, during the nineteenth century, sometimes known as Foucault
currents, since Foucault studied them experimentally. While not a brilliant theoretician, Foucault was an extremely creative experimentalist
and this brief description hardly does justice to his many contributions.
In early 1851, Foucault had an insight that led to his famous discovery.
This bit of serendipity occurred during construction of a pendulum clock
to regulate the drive of a telescope. Foucault had secured a rod in the
chuck of a lathe. Perhaps accidentally, the rod was disturbed so that it
vibrated. Foucault noticed that the plane of oscillation of the vibrating
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rod remained fixed in orientation even as he slowly turned the chuck.
This observation inspired him with the notion that the vibrating motion
of a freely suspended pendulum might somehow be related to the earth’s
rotation. For a pendulum suspended above the north pole, the plane of
vibration would appear to complete a full circle in 24 hours, whereas at
the equator, the plane of vibration would not rotate; the period of rotation then being infinite. So, depending on the latitude, the period of rotation is between 24 hours and infinity. For the latitude of Paris – Foucault’s
city – the plane of a pendulum’s vibration rotates about 0.19 degrees per
minute, completing the entire rotation in about 32 hours. The mathematical relation between the period for a complete rotation of the plane
of vibration and geographical latitude is given by the formula:
Period of Rotation =
24 hours 11
.
sin(latitude )
In January 1851, Foucault made his initial observations with a pendulum suspended in his mother’s basement. It had a length of 2 meters
and a bob mass of 5 kilograms. While he was able to observe the anticipated effect with the basement pendulum, he quickly understood
the benefits of an even longer pendulum. For example, he realized
that the slight change in the point of maximum displacement of the
pendulum after each oscillation would be more noticeable. Longer
pendulums would also diminish the effect of other spurious motions
that mask the Foucault effect. Moreover, a long pendulum with a
massive bob would have more inertia, and therefore likely oscillate
longer before needing another push. A very long pendulum would
certainly provide a more compelling demonstration. His next pendulum was 11 meters long. In February, it was hung in the Paris
Observatory. Meanwhile the prince-president of France, LouisNapoléon Bonaparte, became interested in Foucault’s work. With
princely backing, Foucault was able to construct a mammoth 67
meter, 28 kilogram pendulum under the dome of the Pantheon in
Paris, in March 1851 (see Figure 1.8).
For Foucault’s large Pantheon pendulum, the period of oscillation
was 16.4 seconds. Foucault had a wooden circle with a 6 meter diameter set up, with its axis concentric with the vertical wire of the pendulum. The circumference of the circle was divided into fractions of
11
The ubiquitous sine curve appears once again.
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Figure 1.8 Public display of the Foucault pendulum. (Engraving from
L’Illustration, Paris, 1851.)
a degree. During one of the slow oscillations, the plane of vibration
would turn by about 0.051 degrees, and the incremental change in
position of the pendulum bob at maximum displacement above the
floor was found to be about 2.5 millimeters for a single oscillation.
After a few swings, the cumulative change was obvious. In order to
display the effect, a spike was fixed downward at the bottom of the
bob so that it would cut a groove in a layer of fine sand spread below.
Figure 1.9 (four consecutive bob paths from a computer simulation)
shows a schematic (and much exaggerated) diagram of the pendulum’s motion.
Over the next 150 years, the pendulum’s history would be connected to that of the Pantheon. But, in order to understand this history, we look briefly at the story of the Pantheon itself. A good starting
point is the year a.d. 451. The fearsome Attila the Hun was marching
toward the then-island community of Paris, causing much consternation to the residents. Genevieve, a young shepherd girl, had predicted
the invasion, but also encouraged the inhabitants to prepare a defense
rather than flee. At the last moment, Attila decided not to attack the
city and this seemingly miraculous event led to Genevieve being considered a patroness of the city, and later canonized. In 512 she died and
was buried in the church of the Holy Apostles, which became known
as the church of St. Genevieve. Eventually, that church decayed to
ruins. Much later, in 1744, King Louis XV, in gratitude for his recovery
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40
X
Figure 1.9 Computer simulation of a few oscillations of the pendulum
bob, as projected onto the floor. The earth’s rotation is artificially speeded
up in order to clearly demonstrate the Foucault effect.
from a serious illness, ordered a new church to be built on the original
site. But progress was slow and the building was not completed until
1789, around the beginning of the French Revolution. Instead of keeping to Louis’ plan, in 1791 the new Republican government ordered
that the building be used as a public secular mausoleum – a pantheon –
for the interment of famous Frenchmen.12 In the next few decades, the
building was to alternate several times between secular and sacred
usage. Napoleon Bonaparte returned the building to the Church in
1806, although the crypt continued to be the burial place for famous
Frenchmen. In 1830, at the end of the Bourbon dynasty, it once again
became the national Pantheon under the new king Louis Philippe.
Somewhat later, it reverted to the Church. In 1848, a revolution against
Louis Philippe created the Second Republic, which restored the building as the secular Pantheon. Such was its status at the time of Foucault’s
public demonstration.
12
1934.
The remains of Marie Curie were placed there in 1995, about 60 years after her death in
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The involvement of Louis-Napoléon with Foucault and his pendulum is curious, in that there is a certain parallelism between the two
men. Both of them spent much of their lives as outsiders. Foucault
was not part of the group of theoretical physicists and mathematicians who dominated the French Academy of Sciences. Similarly,
Louis-Napoléon was not part of the French government until 1848
and, for many years, was not even allowed to set foot on French soil.
Louis-Napoléon (1808–73) was a nephew of the famous Napoleon
Bonaparte. Because of deaths in the family, he eventually became next
in line to be emperor, had the Bonapartes stayed in power. However,
the battle of Waterloo changed all that, and Louis-Napoléon grew and
spent most of his first 40 years living abroad. Yet, his desire to return to
France as leader was strong. He attempted two futile coups d’état and
spent six years imprisoned in a French military fortress. During that
time, he developed a strong interest in science – especially physics – an
interest that in later life led to his support of Foucault’s experiment.
Like Foucault, he was self-taught in science. In 1846, Louis-Napoléon
made a daring escape from his prison and returned triumphantly to
his supporters in London. He used the next year or so to build political
capital in preparation for his eventual return to France. This was to
happen shortly after the 1848 revolution. His political fortunes
improved dramatically. In quick succession, he was twice elected to
the National Assembly and then elected as president of the Assembly.
In 1851, he heard of Foucault’s work and, as president, gave orders to
proceed with the hanging of a large pendulum in the Pantheon. With
this construction, both outsiders shared a moment of triumph.
Unfortunately, the display of the Foucault pendulum in the
Pantheon was short-lived as political events intervened. On December
2, 1851, Louis-Napoléon staged a coup d’état, taking over the government.
On December 6 he gave the Pantheon back to the Church, as a way of
gaining clerical support. Of course, the pendulum had to go. One year
later, Louis-Napoléon consolidated his position by having the people
elect him as emperor.13 However, not long after the original Pantheon
demonstration, a more permanent exhibit was established in another
part of Paris, in the Musée des Arts et Métiers in 1855. At this writing, two
of the 1851 pendulums are on display in that location.
13
Louis-Napoléon, or Napoleon III as he was to become, continued to take a personal
interest in Foucault’s work and career.
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While Foucault pendulums were hung in important locations14
elsewhere in France, it was 50 years before another pendulum would
hang in the Pantheon. At the beginning of the twentieth century, the
Astronomical Society of France, presided over by the noted astronomer and mathematician Henri Poincaré, supported having a pendulum mounted in the Pantheon as it was once more a secular, public
building. This was done and the exhibit officially opened on October
22, 1902. The opening was attended by many notables, including the
composer Camille Saint-Saëns and the sculptors Auguste Rodin and
Frédéric Bartholdi (who created the Statue of Liberty). But a year or so
later the pendulum was again taken down, and it was not until the fall
of 1995 that the original was brought out of storage and again hung
from the dome of the Pantheon for about a year. The pendulum that is
currently in the Pantheon was commissioned in 1996 and designed by
Jacques Foiret. The bob is quite massive at 47 kilograms and has been
gilded by electrolysis with 24 carat gold.
1.4 Frames of reference
In making scientific measurements, we often choose a particular frame
of reference. The Foucault pendulum demonstrates this point most
dramatically. We ordinarily speak of a frame of reference as a point of
view; perhaps the perspective of a specific political philosophy or set of
religious beliefs. In the physical sciences, a frame of reference is the
platform on which an observer is located and from which that observer
makes measurements. Experience suggests that observers who are in
different frames of reference may see the same event differently.
A frame in rotation relative to a stationary frame is actually accelerated
relative to the stationary frame. Objects in that frame are constantly
changing direction; hence the acceleration. An observer in the accelerated frame thinks that new forces occur.15 For example, a passenger in
the frame of reference of a car turning a corner experiences a new force
called centrifugal force, or “center-fleeing” force. This car-bound observer
believes that a force is pulling him out of the car, away from the direction
14
One such location was inside Reims Cathedral, where a 40 meter pendulum was hung
in May 1851. This seems somewhat ironic, given the swift removal of the Pantheon pendulum as that building was again converted to a church.
15
This is another example of the equivalence principle.
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of the turn. Yet an observer standing on the ground only sees the passenger accelerate toward the center of the turn, in going around the corner. This latter acceleration is called centripetal,16 or “center-seeking”,
acceleration. Relative to the stars, we all exist on a grand rotating frame
of reference: the earth. Therefore we view things differently from an
observer located on a distant star. On earth, we experience (but hardly
feel) a slight centrifugal force that diminishes our weight – especially at
the equator – whereas the distant observer sees us accelerate, with centripetal acceleration, toward the center of the earth.
To further complicate the issue, if the observer in the rotating frame
is himself moving relative to that frame, he experiences another, less
obvious, force. This new force, called Coriolis force, is essential for our
discussion of the Foucault pendulum. In 1837, the French mathematician Siméon Denis Poisson (1781–1840) published a paper, based upon
the calculations of a one-time student, that described this force. It
turned out that Poisson’s student was Gaspard Gustave Coriolis (1792–
1843), a physicist who specialized in various problems of theoretical
mechanics and published his theory of rotating frames in the context
of their effects upon machine operation. Although Poisson did not
publicly acknowledge Coriolis’ work, justice eventually triumphed
and we now rightly identify this further effect as Coriolis force.
Here is a typical example of Coriolis force behavior. An observer on
a rotating platform (such as a merry-go-round), who attempts to
throw a ball to her friend on the same platform, would observe the ball
to follow a curved trajectory rather than a straight line. During the ball’s
time of flight, the platform has rotated and the friend, who is supposed
to catch the ball, is no longer in the same place. It turns out that the
Coriolis force – the force observed in the rotating frame – is proportional to both the ball’s velocity and the rotation rate of the platform.
Expressed mathematically, the Coriolis force is the simple product of
several factors. That is,
F = 2mv┴X,
where m is the mass of the ball, v┴ is the amount of the ball’s velocity
that is at right angles to the platform’s axis of rotation, and X is the
platform’s rate of rotation.
16
We recall that the terms “centrifugal” and “centripetal” were coined by Huygens and
Newton respectively.
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In the frame of the rotating earth, the Coriolis force on a moving object
is proportional to whatever part of the object’s velocity is perpendicular to
the axis of the earth’s rotation. Depending on their directions, motions in
the plane of the earth’s surface may be affected by the daily rotation of the
earth. At the north and south poles, the earthbound observer will see
deviation to the right or left, respectively. At the equator, only that component of surface motion that is parallel to the equator will be subject to
Coriolis acceleration. Even then, the acceleration will be perpendicular to
the earth’s surface and perhaps less noticeable than at the poles.
A dramatic example of Coriolis “force” is the deviation of artillery
shells from their targets – to the right in northern latitudes and to the
left in southern latitudes. This phenomenon was powerfully demonstrated in World War I. During a naval battle near the Falkland Islands,
located near the southern tip of South America, the British gunners
initially compensated incorrectly for Coriolis force by using adjustments that were valid for the northern hemisphere. However, once
they changed the sign (plus or minus) of the numerical correction to
that required for the southern hemisphere, their fire became accurate.
Modern guns are less susceptible to Coriolis effects because, while their
muzzle velocities are much higher and thus the Coriolis force is
stronger, the projectile’s time of flight is much shorter and therefore
the time during which the force acts – or the earth rotates – is relatively short.
How does Coriolis force affect pendulums? In his 1837 paper, Poisson
actually mentioned the possibility of an earthbound effect upon a pendulum’s plane of oscillation. But since the rotation of the earth is slow
and the velocities of typical pendulums are small, Poisson concluded
that the effect would be barely observable. Certainly this is more or less
true for a single oscillation of the pendulum. Foucault, on the other
hand, noted in his 1851 paper that the Coriolis effect would be cumulative
and therefore the additive result of many oscillations would be quite
observable under the right conditions.
1.5 Foucault pendulums are tricky
Demonstration of the rotation of the earth is an exciting prospect! But
unless you happen to have a very long rope and a very heavy bob, the
creation of a decent Foucault pendulum is not a simple Saturday morning project. Certainly, every freely suspended earthbound pendulum is
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a Foucault pendulum, but not every one of them will manifestly display the desired effect. Many things get in the way; some are annoying,
difficult to analyze, and difficult to remove. Here are a few of these
problems.
1.5.1 Starting the pendulum
It is often observed that the pendulum swing deviates from a desired
straight line. One reason for this is that the person who starts the
pendulum by displacing it from the downward vertical inadvertently
gives a small bit of push at right angles to the direction of the swing.
Even the tiniest push can cause problems. Therefore one early
method for starting the motion removes the human interaction as
follows. First, the pendulum bob is displaced from equilibrium and
the displacement secured with a string. Then a flame is applied to the
string. Once the flame has burnt through the string, the pendulum
naturally moves exactly in the direction of its previous displacement.
(Remember the murder mystery.) But – and this can be problematic
for small pendulums – at the instant the string burns through, the
pendulum is stationary relative to the rotating earth, and therefore,
relative to the stars, the pendulum starts with a small amount of
motion in a direction at right angles to the plane of oscillation – as
measured in the “stellar” frame of reference.
1.5.2 Finite swing size
Up to this point, we have assumed that the swing of the pendulum is
small and that the period is independent of the size of the swing. But
strictly speaking, even for small angles, the period of the pendulum
does increase slightly with swing size. For something as delicate as
the Foucault effect, even the small variation in period with swing is a
problem. The combination of (a) the variation in the period of oscillation, (b) the very small (Foucault) rotation of the plane of oscillation with each swing, and (c) a possible bad start, results in the
formation of a narrow elliptical orbit for the bob. With this elliptical
motion, the pendulum effectively oscillates in two perpendicular
directions, with two differently sized swings. And because of the differences in the sizes of the swings, there are two slightly different periods for these perpendicular oscillations. Looking down at the path of
the bob, we find that these two perpendicular oscillations trace out a
narrow ellipse, somewhat like that of the phase diagram described in
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the Introduction. We can think of this effect as slightly analogous to
two slightly out-of-tune strings on a musical instrument. One hears
a pulsation or “beat” at a frequency that is the difference between the
frequencies of the strings. It is similar for the pendulum. The elliptical motion of the pendulum bob is a precession17 of the plane of oscillation that can be confused with the Foucault effect. It turns out that
the precession rate is proportional to the area of the ellipse, and adds
or subtracts from the Foucault precession. That is, depending on how
the pendulum is started, the direction of the precession can be with
or against the Foucault rotation. For technical reasons, this difficulty
is called the anharmonicity problem. The easiest way to minimize the
problem is to make the pendulum length very long compared to the
size of the swing. This is one reason why most Foucault pendulums
are very long.
1.5.3 Asymmetry
A very likely complication involves some sort of asymmetry in the
configuration of the pendulum or the circumstances of its location.
Asymmetry means that the pendulum does not oscillate equally in
all directions, as required for the Foucault effect. For example, the
pendulum may be suspended from a beam that has more “give” in
one direction or the other. Air currents around the pendulum,
such as those found in public museums, may also cause asymmetry. Long Foucault pendulums are often located in the stairwells of
museums and, interestingly, currents from one-way traffic patterns
of spectators going down the stairs – more than up the stairs – are
sources of asymmetry. Open doors and windows can also cause
problems.
We can think of the situation in the following way. If the pendulum
is “perfect” – totally symmetric – then the period of oscillation will be
the same no matter what direction the oscillation. But if the pendulum
is not “perfect” then, technically, the motion splits into two mutually
perpendicular motions with two different periods. And again there is a
“beat” frequency effect. But this time (unlike the anharmonicity problem), the motion is periodic with the “beat” in a different way. Again,
17
Most authors refer to the gradual rotation of the plane of oscillation (the Foucault
effect) as “precession” of the plane. However, it has also been labeled as “veering” of the
plane of oscillation.
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Y
we can picture the pendulum bob’s motion being projected onto a twodimensional plane, such as the floor of a museum. This is the motion
illustrated by the phase-plane diagram of two mutually perpendicular
oscillations at the slightly differing frequencies, as shown by the computer simulation of Figure 1.10. Typically, asymmetry will cause the pendulum to first display elliptical motion, then circular, then elliptical,
and so forth in a repeated, time-varying, periodic pattern.
For long pendulums with very low frequencies, this effect is less pronounced, since the frequency differences can be made relatively small.
However, it is a serious problem for short pendulums and, typically,
the asymmetry effect overshadows the Foucault effect in a relatively
short time. A quarter century after Foucault’s original demonstration
of the earth’s rotation, the first mathematical analysis of this asymmetry effect was carried out by Heike Kamerlingh Onnes (1853–1926), the
X
Figure 1.10 An accumulation of the pendulum bob’s path as it would
appear from above the pendulum. The elliptical oscillation orbit caused by
asymmetry gradually fattens to become circular. Then the orbit becomes
thinner again, but in the direction perpendicular to the initial oscillation.
The pattern repeats.
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future discoverer of superconductivity,18 as part of his 1879 Ph.D. dissertation (see Figure 1.11).
The problem was suggested to Kamerlingh Onnes by a former teacher,
Gustav Kirchhoff (1824–87), whose laws of electrical circuitry are well
known to undergraduate physics students. Not only did Kamerlingh
Onnes do the analysis, but he also used his theoretical work as the basis
for construction of a short Foucault pendulum with controllably variable asymmetry. Through a series of adjustments, he could equalize the
pendulum motion in the two mutually perpendicular directions, thereby
equalizing the resonant frequencies. Furthermore, through adjustment
of the pivoting arrangement, there was also a capability for deliberately
injecting asymmetry into the apparatus, in order to quantify its effects.
Figure 1.11 H. Kamerlingh Onnes.
18
Superconductivity occurs when electrical current travels in a material without any
resistance. It requires very low temperatures, although continuing research is finding an
increasing number of materials that are superconducting at less frigid temperatures.
Superconductivity is used in many applications, such as in production of the very strong
electromagnets found in medical MRI machines.
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Figure 1.12(a) shows an exploded view of the pivoting arrangement for
the pendulum.
Kamerlingh Onnes also placed the pendulum inside a gas-tight conical container to negate the effects of air drafts. He developed a special
optical system – note the telescope in Figure 1.12(b) – to observe the
pendulum displacement. The observation windows can be seen near
the bottom of the container.
Unlike the typical very long Foucault pendulum, this one is quite
short, so that Kamerlingh Onnes could effectively study the oscillations’ changes provided by his adjustments to the pivoting arrangement, which is shown in the figure at the top of the support.
Kamerlingh Onnes’ careful experimental work in this context was
a precursor to his later and more famous experimental contribution
to low-temperature physics. At this writing, the Kamerlingh Onnes
apparatus is in storage at the Boerhaave Museum in Leiden, near the
university where he did his Nobel prize-winning work in low-temperature physics – the liquefying of helium and the discovery of
superconductivity.
(a)
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(b)
B
b
~
w
c
A
k
z
Figure 1.12 (a) Several views of the support for the pendulum: adjustments
could be made in the pivoting to exaggerate or minimize asymmetry. (b) A
drawing depicting the arrangement of the complete apparatus: the apparatus was less than 2 meters high. (From Kamerlingh Onnes’ thesis.)
Occasionally, there are some very surprising connections
between two areas of scientific endeavor. Kamerlingh Onnes’ work
on superconductivity provides just such a link to one of those
amazing connections; in this case, a connection between superconductivity and the pendulum. To see this connection, we need a little information about superconductivity and something called a
Josephson junction. Superconductivity – discovered at Kammerlingh
Onnes’ laboratory in 1911 – is a low-temperature phenomenon, in
which electric current flows without resistance. It is important for
both pure and applied science and, even today, is an active area of
research. In 1961, Brian Josephson predicted some novel effects that
might be found in a low-temperature device consisting of a junction: a thin piece of an appropriate nonconductor sandwiched
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between two pieces of superconducting material. He predicted the
existence of (a) the spontaneous flow – without any applied voltage –
of electric current across the junction (the d.c. Josephson effect),
and (b) with an appropriate added constant voltage, the spontaneous
creation of an alternating current across the junction, whose frequency depends only upon the size of the applied voltage (the a.c.
Josephson effect). All this may seem a little technical, but these
were novel and unconventional predictions with important implications. After much discussion of the theory and some experimental work, Josephson’s predictions were shown to be correct, and
Brian Josephson was awarded the Nobel Prize in 1973. Today,
Josephson junctions are used in a variety of high tech instruments,
such as sensitive magnetometers.19
Now, here is the interesting coincidence involving pendulums. The
mathematical equations that describe a typical configuration of the
Josephson junction have an exact parallel with the equations of a driven
pendulum that has friction (damping). The correspondence is so precise that a superconducting Josephson junction can, in a certain way,
be thought of as a high-frequency pendulum – oscillating hundreds of
millions of times per second. And parallel with that coincidence is the
amazing fact that the man who was a pioneer in the world of superconducting research, Kamerlingh Onnes, started his own scientific career
with research on the pendulum!
1.5.4 Energy dissipation
Every real pendulum is subject to various frictional forces and a
Foucault pendulum is no exception. A Foucault pendulum will need
to be restarted periodically. However, it is more pleasing to incorporate
some sort of drive mechanism into the pendulum, so that it receives a
pulse of energy for each oscillation. Then the energy loss on each cycle
is smoothly made up by the drive mechanism. The equipment in
Figures 1.13 and 1.14 indicate that Foucault soon solved this problem –
as early as 1855. In fact, his 1855 pendulum had the electromagnetic
drive mechanism illustrated in Figure 1.13.
19
Magnetometers are sensitive instruments for measuring small variations in magnetic
fields. They are commonly used in metal detectors, for oil and gas exploration, and in
satellites.
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Figure 1.13 An engraving of the electrical impulse mechanism used by
Foucault to maintain the pendulum’s motion against frictional losses.
(Bibliothèque de l’Observatoire de Paris.)
This remnant of this original magnetic drive, shown in Figure 1.14,
is now stored at the Paris Observatory. It may be compared with the
engraving in Figure 1.13.
Many modern pendulums also have various mechanisms for giving
the pendulum a push. Perhaps because these mechanisms are hidden
from view, they sometimes – as we will learn – lead the casual observer
to the wrong conclusion.
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Figure 1.14 The surviving remnant of Foucault’s original electromagnetic driver from the impulse mechanism in Figure 1.13. (Bibliothèque de
l’Observatoire de Paris.)
1.5.5 Some modern solutions
We have seen that the main problems associated with building a working Foucault pendulum can be minimized by making a long pendulum
with a massive bob. Foucault’s dramatic 1851 presentation in the
Pantheon worked because his pendulum was long – 67 meters – and
because he used a massive bob of 28 kilograms, giving the pendulum
plenty of inertia, so that it would oscillate for long enough to illustrate
the desired effect. Furthermore, asymmetry in long pendulums is not
a problem. But long heavy pendulums can cause a safety problem.
A month or two after Foucault swung his pendulum in the Pantheon,
the suspending wire broke at the point of suspension. This was very
dangerous because the tension in the wire caused it to act like a dangerous whip on any spectators who might be nearby. As a result of this
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Figure 1.15 Two members of the south pole group with their Foucault
pendulum bob. (Courtesy of John Bird.)
accident, Foucault designed a “parachute” that would support the wire
just below the suspension, in case of breakage near the pivot. The parachute is a kind of cylinder that is clamped to the wire and, in the event
of breakage, falls a short distance to a supporting ring. Foucault’s original parachute, together with the suspension structure, is stored in the
Paris Observatory.
For short pendulums, all of the previously described effects are significant. In 1931, the asymmetry problem was partially solved by a simple ring, now known as the Charron ring. It is often placed just below
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the pivot point, such that it would surround the suspending wire. The
diameter of the ring is made slightly less than the full swing of the
wire, and therefore the wire20 contacts the ring at the extremes of its
swing. This delicate impact arrests the various spurious precessional
motions of the pendulum. However, as is well known, the ring succeeds only partially, in that it does introduce a faster dissipation of the
pendulum’s energy. It now becomes even more important to provide a
source of energy to keep the pendulum in motion.
Various methods have been developed to provide energy to the pendulum. We noted that Foucault himself built an electromagnetic drive
for the 1855 pendulum. A more sophisticated and modern design was
used by the American physicist Richard Crane (1981). He placed a small
permanent magnet in the bob of the pendulum that interacted with a
combination of permanent and time-varying electromagnets fixed in
the rigid structure below the pendulum. The electromagnet gave a
pulse that changed orientation every quarter of an oscillation. In this
way, the magnet in the bob received a push outward when the bob was
in the middle and an inward pull when the bob was at its extreme displacement. With this setup, Crane was able to eliminate spurious precession and, at the same time, give a small periodic push to the
pendulum bob.
A different mechanism that mitigates problems and provides energy
input is parametric pumping. The word parametric refers to some characteristic of the pendulum such as its length, mass, or the operative gravitational field. Rather than pushing the pendulum at the bob, the
pendulum is driven by a periodic vertical movement of the suspension
point. This motion periodically changes the effective strength of the
gravitational field – a parameter of the system – and thereby injects
energy into the motion. The British physicist Brian Pippard has shown
that parametric pumping of the pendulum helps to eliminate the tendency of the Foucault pendulum toward elliptical motion, as well as
keeping it in motion through the periodic infusion of energy. The pendulum installed in the London Science Museum in 1988 was built with
such a mechanism, according to a design by Pippard. (Another variant
of parametric pumping is the effort a child on a swing exerts to keep
the swing moving. The periodic motion of the child changes the
20
In some cases the ring is actually placed below the bob, where it makes its slight contact with a sharp protrusion from the bob.
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effective length – another physical parameter – of the playground
swing or pendulum. Again, the motional energy is transformed into
larger swings.)
We noted that the Foucault effect is most obvious at one of the
earth’s poles. A crude Foucault pendulum was constructed and tested
at the south pole by three adventurous experimenters, Michael Town,
John Bird, and Alan Baker. Their measurement is probably the closest
ever made to one of the earth’s poles. The pendulum was erected in a
six-story staircase of a new station that was under construction near
the pole. Conditions were challenging; the altitude was about 3,300
meters (atmospheric pressure only about 65 percent that at sea level)
and the temperature in the unheated staircase was about −68°C
(−90°F). The pendulum had a length of 33 meters and a 25 kilogram
bob. Interestingly, it is against the Antarctica Treaty to have an open
flame at the south pole, and therefore the authors could not use the
burnt-thread technique to start the pendulum. Instead, they became
adept at dropping the bob so that it would oscillate in a plane. Installing
even a crude pendulum in these conditions was challenging. But once
they overcame the obstacles, these researchers did confirm a period of
about 24 hours as the rotation period of the plane of oscillation (see
Figure 1.15).
1.6 Pendulums go public
Before television, the internet, and other modern media, scientists
often took their science to the public by giving spectacular demonstrations. Following this tradition, in February 1851 Foucault boldly invited
contemporary scientists to a demonstration at the Paris Observatory
with the words “Vous êtes invités à venir tourner la terre . . .” (“You are
invited to come to the turning of the earth . . .”). The much longer
Pantheon demonstration pendulum, rapidly constructed for a March
opening, produced a flood of visitors. Foucault’s pendulum had indeed
gone public.
Reaction to Foucault’s demonstration was widespread. A few examples will suffice. Two pieces from the Proceedings of the Royal Society of London
are instructive. First, the English geologist John Philips (1800–74), a
Fellow of the Royal Society (FRS), communicated work that Thomas
Cooke, an optician in the city of York, had done with his own version
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Figure 1.16 An engraving from L’Illustration, Paris, 1851.
of the pendulum. Cooke used a 16 meter pendulum with two different
bobs, one an oblate spheroid with a mass of about 18 kilograms, and the
other a prolate spheroid with a mass of 44 kilograms. The pendulums
performed as anticipated, except that in some experiments, there was a
small but noticeable elliptical aspect to the path of the pendulum’s
motion and a veering of the elliptical motion. Philips also undertook
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Seven Tales of the Pendulum
experiments in his house with much smaller pendulums and found
that the spurious effects – which we described above – were more
pronounced.
Second, the contemporary British Astronomer Royal, George Airy,
FRS (1801–92), provided Philips and Cooke with a partial explanation
of the so-called clockmaker’s formula that quantifies the rate of turning of the ellipse of a pendulum due to anharmonicity. Also, the same
issue of the Proceedings had a contribution from an English experimentalist and Fellow of the Royal Society, Charles Wheatstone (1802–1875),
who was a prominent British physicist in the field of instrumentation.
(His most famous invention, the electric telegraph with various
improvements, earned him a knighthood in 1868.) Wheatstone emphasized (as we described earlier) that Foucault had come to the idea of his
pendulum’s rotation of the plane of oscillation by noting the effect of
coupling one motion with another. Wheatstone went on to describe
several other instances of this type of interaction and had himself built
a mechanical device that coupled together vibration of a spring, rather
than a pendulum, with rotation of the earth. The point that Wheatstone
made with this device was that gravity was not necessary to achieve
motion analogous to the Foucault effect.
On the other side of the Atlantic Ocean, a veritable ‘pendulum
mania’ occurred. Over 30 large Foucault pendulums were built at
diverse locations along the East Coast of the United States. Scholarly
discussion in American journals was also forthcoming. For example,
the Journal of the Franklin Institute, in Philadelphia, records several contributions on the subject in 1851. Charles Allen provided a geometric analysis
of the pendulum’s motion. In the same volume, J. S. Brown also proposed an analysis. He prefaces his communication with the provocative
statement: “ . . . in all the explanations of Foucault’s pendulum . . . not
one has given . . . a full exposition of its phenomena, and many have
made false statements”. In a footnote, the journal’s editor takes polite
but firm issue with Brown and, in reference to Allen’s contribution, he
states that it “ . . . appears to us to be thoroughly correct”.
Finally, on the rotation of the earth, there is the much quoted
tongue-in-cheek note to the editor of a contemporary issue of Punch
magazine:
“Sir, – Allow me to call your serious and polite attention to
the extraordinary phenomenon, demonstrating the
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rotation of the Earth, which I at this present moment experience, and you yourself or anybody else, I have not the
slightest doubt, would be satisfied of, under similar circumstances. Some sceptical and obstinate individuals may
doubt that the Earth’s motion is visible, but I say from personal observation it’s a positive fact. I don’t care about latitude or longitude, or a vibratory pendulum revolving
round the sine of a tangent on a spherical surface, nor axes,
nor apsides, nor anything of the sort. That is all rubbish.
All I know is, I see the ceiling of this coffee-room going
round. I perceive this distinctly with the naked eye – only
my sight has been sharpened by a slight stimulant. I write
after my sixth go of brandy-and-water, whereof witness my
hand, Swiggins
Goose and Gridiron, May 5th, 1851.”
The Foucault pendulum was certainly science with a high profile!
Even today, the Foucault pendulum continues to provide a dramatic and obvious demonstration of the earth’s rotation; it is found in
many museums, schools, and other public buildings. A representative
list is given in Table 1.1.
Some of the listed pendulums are very long and have massive bobs,
a combination that allows them to partially sustain their motion
against the inevitable loss of energy due to friction in the bearing, air
resistance, and so forth. Other pendulums, often with smaller dimensions, use some sort of drive mechanism to supply energy that compensates for the losses owing to friction. Unfortunately, the indefinitely
sustained oscillations provided by hidden drive mechanisms can sometimes reinforce misconceptions.
Donald Ivey, a professor at the University of Toronto, once told his
physics class a story regarding such a misconception and the Foucault
pendulum. During a visit to New York City, Ivey joined a tour of the
UN building. At the site of its Foucault pendulum, a fairly large crowd
observed the slow back-and-forth motion of the pendulum, hoping to
discern the veering motion of the plane of oscillation. At some point, a
woman within earshot of Ivey stated emphatically that the phenomenon of the ongoing oscillations was an example of perpetual motion. Ivey
began to squirm. As a physicist, he knew that perpetual motion contravenes the physical principles of both conservation of energy and the
Second Law of Thermodynamics. Eventually, Ivey could not contain
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himself. He turned to the woman and quietly explained that there was
no such thing as perpetual motion, and that the pendulum had a hidden drive mechanism. She flatly and noisily disagreed with him. Ivey
then, and perhaps foolishly, revealed himself to be an expert, namely a
professor of physics. At this point, the woman loudly shared with the
crowd her low opinion of experts in general and of university professors in particular. In the face of this verbal assault and possible public
ridicule, Professor Ivey wisely decided to retreat from this field of
“intellectual” combat.
The degree of “disconnect” between the scientist Ivey and the
unknown tourist is not unique. Figure 1.16 illustrates corresponding
attitudes in the nineteenth century as the scientist tries to explain the
pendulum to the girl.
Table 1.1 A partial list of the existing Foucault pendulums.
Location
Pantheon, Paris
UN Building, New York
City
Franklin Institute,
Philadelphia
University of Guelph,
Canada
University of Maryland
Griffith Observatory,
Los Angeles
Yale University
Ryerson Library,
Michigan
Morrison Planetarium
Science Museum,
London
Ottawa High School,
Michigan
Length (m)
Mass (kg)
67
23
47 (1995 bob)
90
26
410
0.83
4.5
14
12
28
100
11
22
12
14
9
20
87
9
9
14
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Quoting from a translation of the original caption:
“This tableau, which appeared soon after . . . (the original
1851 demonstration of the Foucault pendulum) . . . reveals
some of the typically French attitudes toward science and
nature. The “Philosopher” (or scientist) pays no attention
to the beauty of the scene, seeing neither the vegetation
nor the pretty girl (on) the swing. What he observes is
abstracted to a geometrical problem, free from passion
and life. The girl’s response to his mathematical analysis
is ‘True, but who cares?’ To the philosopher, science is
measurement; to the young girl, it is irrelevant because it
omits life.”
Whatever the general perception, we see, even from this first tale, that
the seemingly unremarkable pendulum produces an abundant harvest
of scientific knowledge and cultural impact.