The
Foucault pendulum ( "foo-KOH"), or
Foucault's pendulum, named after the French
physicist
Léon Foucault, was
conceived as an experiment to demonstrate the
rotation of the Earth.
The experiment
The experimental apparatus consists of a tall
pendulum free to
oscillate
in any vertical plane. The direction along which the pendulum
swings rotates with time because of Earth's daily rotation.
The first
public exhibition of a Foucault pendulum took place in February
1851 in the Meridian Room of the Paris Observatory. A few weeks later, Foucault made his most
famous pendulum when he suspended a 28-kg bob with a 67-metre wire from the dome of the
Panthéon in Paris. The
plane of the pendulum's swing rotated clockwise 11° per hour,
making a full circle in 32.7 hours.
A Foucault pendulum at the north
pole.
The pendulum swings in the same plane as the Earth rotates
beneath it.
In 1851 it was well known that Earth rotated: observational
evidence included Earth's measured polar flattening and equatorial
bulge. However, Foucault's pendulum was the first dynamic proof of
the rotation in an easy-to-see experiment, and it created a
sensation in the academic world and society at large.
At either
the North
Pole or South
Pole, the plane of oscillation of a pendulum remains
fixed with respect to the fixed stars
while Earth rotates underneath it, taking one sidereal day to complete a rotation. So
relative to Earth, the plane of oscillation of a pendulum at the
North or South Pole undergoes a full clockwise or counterclockwise
rotation during one day, respectively.When a Foucault pendulum is
suspended on the
equator, the plane of
oscillation remains fixed relative to Earth. At other latitudes,
the plane of oscillation precesses relative to Earth, but slower
than at the pole; the angular speed,
α (measured in
clockwise
degrees per
sidereal day), is proportional to the
sine of the
latitude,
φ:
- \alpha=360^\circ \sin\varphi.
Here, latitudes north and south of the equator are defined as
positive and negative, respectively. For example, a Foucault
pendulum at 30° south latitude, viewed from above by an earthbound
observer, rotates counterclockwise 180° in one day.
In order to demonstrate the rotation of the Earth without the
philosophical complication of the latitudinal dependence, Foucault
used a
gyroscope in an 1852 experiment.
The gyroscope's spinning rotor tracks the stars directly. Its axis
of rotation is observed to return to its original orientation with
respect to the earth after one day whatever the latitude, not
subject to the unbalanced Coriolis forces acting on the pendulum as
a result of its geometric asymmetry.
A Foucault pendulum requires care to set up because imprecise
construction can cause additional veering which masks the
terrestrial effect. The initial launch of the pendulum is critical;
the traditional way to do this is to use a flame to burn through a
thread which temporarily holds the bob in its starting position,
thus avoiding unwanted sideways motion.
Air resistance damps the oscillation, so
Foucault pendulums in museums often incorporate an electromagnetic
or other drive to keep the bob swinging; others are restarted
regularly. In the latter case, a launching ceremony may be
performed as an added show.
Foucault pendulum precession as a form of parallel
transport
Parallel transport of a vector around
a closed loop on the sphere.
The angle by which it twists, \alpha, is proportional to the
area inside the loop.
From the perspective of an inertial frame moving in tandem with
Earth, but not sharing its rotation, the suspension point of the
pendulum traces out a circular path during one
sidereal day. At the latitude of Paris a full
precession cycle takes 32 hours, so after one sidereal day, when
the Earth is back in the same orientation as one sidereal day
before, the oscillation plane has turned 90 degrees. If the plane
of swing was north-south at the outset, it is east-west one
sidereal day later. This implies that there has been exchange of
momentum; the Earth and the pendulum bob have exchanged momentum.
(The Earth is so much heavier than the pendulum bob that the
Earth's change of momentum is totally unnoticeable. Nonetheless,
since the pendulum bob's plane of swing has shifted the
conservation laws imply that there must have been exchange.)
Rather than tracking the change of momentum the precession of the
oscillation plane can efficiently be described as a case of
parallel transport. For that it
is assumed that the precession rate is proportional to the
projection of the
angular velocity of Earth onto the
normal direction to Earth, which implies that
the plane of oscillation will undergo parallel transport. The
difference between initial and final orientations is , in which
case the
Gauss-Bonnet theorem
applies.
α is also called the
holonomy or
geometric
phase of the pendulum. Thus, when analyzing earthbound motions,
the Earth frame is not an
inertial
frame, but rather rotates about the local vertical at an
effective rate of radians per day.
From the perspective of an Earth-bound coordinate system with its
x-axis pointing east and its
y-axis pointing
north, the precession of the pendulum is described by the
Coriolis force. Consider a planar pendulum
with natural frequency
ω in the
small angle approximation. There
are two forces acting on the pendulum bob: the restoring force
provided by gravity and the wire, and the
Coriolis force. The Coriolis force at
latitude φ is horizontal in the
small angle approximation and is given by\begin{align}F_{c,x}
&= 2 m \Omega \dfrac{dy}{dt} \sin(\varphi)\\F_{c,y} &= - 2
m \Omega \dfrac{dx}{dt} \sin(\varphi)\end{align}where Ω is the
rotational frequency of Earth,
F_{c}_{,x} is the component of
the Coriolis force in the
x-direction and
F_{c}_{,y} is the component of
the Coriolis force in the
y-direction.
The restoring force, in the small angle approximation, is given
by\begin{align}F_{g,x} &= - m \omega^2 x \\F_{g,y} &= - m
\omega^2 y.\end{align}
Using
Newton's laws of
motion this leads to the system of
equations\begin{align}\dfrac{d^2x}{dt^2} &= -\omega^2 x + 2
\Omega \dfrac{dy}{dt} \sin(\varphi)\\\dfrac{d^2y}{dt^2} &=
-\omega^2 y - 2 \Omega \dfrac{dx}{dt} \sin(\varphi)
\,.\end{align}
Switching to complex coordinates , the equations read
- \frac{d^2z}{dt^2} + 2i\Omega \frac{dz}{dt}
\sin(\varphi)+\omega^2 z=0 \,.
To first order in Ω/
ω this equation has the solution
- z=e^{-i\Omega \sin(\varphi) t}\left(c_1 e^{i\omega t}+c_2
e^{-i\omega t}\right) \,.
If we measure time in days, then and we see that the pendulum
rotates by an angle of −2π sin(
φ) during one day.
Related physical systems
The device described by
Wheatstone.
There are many physical systems that precess in a similar manner to
a Foucault pendulum. In 1851,
Charles
Wheatstone described an apparatus that consists of a vibrating
spring that is mounted on top of a disk so that it makes a fixed
angle \phi with the disk. The spring is struck so that it
oscillates in a plane. When the disk is turned, the plane of
oscillation changes just like the one of a Foucault pendulum at
latitude \phi.
Similarly, consider a non-spinning perfectly balanced bicycle wheel
mounted on a disk so that its axis of rotation makes an angle \phi
with the disk. When the disk undergoes a full clockwise revolution,
the bicycle wheel will not return to its original position, but
will have undergone a net rotation of 2\pi\, \sin(\phi).
Another system behaving like a Foucault pendulum is a
South Pointing Chariot that is run
along a circle of fixed latitude on a globe. If the globe is not
rotating in an inertial frame, the pointer on top of the chariot
will indicate the direction of swing of a Foucault Pendulum that is
traversing this latitude.
In physics, these systems are referred to as
geometric phases. Mathematically they are
understood through
parallel
transport.
The animation describes the motion of
a Foucault Pendulum at a latitude of 30°N.
The plane of oscillation rotates by an angle of -180° during
one day, so after two days the plane returns to its original
orientation.
Foucault pendulums around the world
There are numerous Foucault pendulums around the world, mainly at
universities, science museums and planetariums.
A
particularly famous and prominent one is located at the United Nations in Manhattan.
The experiment has even been carried out at the South Pole.
See also
References
- Classical dynamics of particles and systems, 4ed,
Marion Thornton (ISBN 0-03-097302-3 ), P.398-401.
- Frank Wilczek and Alfred Shapere,
"Geometric Phases in Physics", World Scientific, 1989
External links
Derivations
Visualisations, video imaging and models
History
Notable
Educational supplies