Foucault pendulum
The Foucault pendulum or Foucault's pendulum is a simple device named after French physicist Léon Foucault and conceived as an experiment to demonstrate the Earth's rotation. The pendulum was introduced in 1851 and was the first experiment to give simple, direct evidence of the earth's rotation. Foucault pendulums today are popular displays in science museums and universities.
Original Foucault pendulum
The first public exhibition of a Foucault pendulum took place in February 1851 in the Meridian of the Paris Observatory. A few weeks later, Foucault made his most famous pendulum when he suspended a brass-coated lead bob with a wire from the dome of the Panthéon, Paris. The period of the pendulum was seconds. Because the latitude of its location was = 48°52' N, the plane of the pendulum's swing made a full circle in approximately = 31.8 hours, rotating clockwise approximately 11.3° per hour.The original bob used in 1851 at the Panthéon was moved in 1855 to the Conservatoire des Arts et Métiers in Paris. A second temporary installation was made for the 50th anniversary in 1902.
During museum reconstruction in the 1990s, the original pendulum was temporarily displayed at the Panthéon, but was later returned to the Musée des Arts et Métiers before it reopened in 2000. On April 6, 2010, the cable suspending the bob in the Musée des Arts et Métiers snapped, causing irreparable damage to the pendulum bob and to the marble flooring of the museum. The original, now damaged pendulum bob is displayed in a separate case adjacent to the current pendulum display.
An exact copy of the original pendulum has been operating under the dome of the Panthéon, Paris since 1995.
Explanation of mechanics
At either the Geographic North Pole or Geographic South Pole, the plane of oscillation of a pendulum remains fixed relative to the distant masses of the universe 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 Pole – viewed from above – undergoes a full clockwise rotation during one day; a pendulum at the South Pole rotates counterclockwise.When a Foucault pendulum is suspended at the equator, the plane of oscillation remains fixed relative to Earth. At other latitudes, the plane of oscillation precesses relative to Earth, but more slowly than at the pole; the angular speed, , is proportional to the sine of the latitude, :
where 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 360° in two days.
To demonstrate rotation directly rather than indirectly via the swinging pendulum, Foucault used a gyroscope in an 1852 experiment. The inner gimbal of the Foucault gyroscope was balanced on knife edge bearings on the outer gimbal and the outer gimbal was suspended by a fine, torsion-free thread in such a manner that the lower pivot point carried almost no weight. The gyro was spun to 9,000–12,000 revolutions per minute with an arrangement of gears before being placed into position, which was sufficient time to balance the gyroscope and carry out 10 minutes of experimentation. The instrument could be observed either with a microscope viewing a tenth of a degree scale or by a long pointer. At least three more copies of a Foucault gyro were made in convenient travelling and demonstration boxes and copies survive in the UK, France, and the US.
A Foucault pendulum requires care to set up because imprecise construction can cause additional veering which masks the terrestrial effect. As observed by later Nobel laureate Heike Kamerlingh Onnes, who developed a fuller theory of Foucault pendulum for his doctoral thesis, geometrical imperfection of the system or elasticity of the support wire may cause an interference between two horizontal modes of oscillation, which caused Onnes' pendulum to go over from linear to elliptic oscillation in an hour. The initial launch of the pendulum is also 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.
Notably, veering of the pendulum was observed already in 1661 by Vincenzo Viviani, a disciple of Galileo, but there is no evidence that he connected the effect with the Earth rotation; rather, he regarded it as a nuisance in his study that should be overcome with suspending the bob on two ropes instead of one.
Air resistance damps the oscillation, so some Foucault pendulums in museums incorporate an electromagnetic or other drive to keep the bob swinging; others are restarted regularly, sometimes with a launching ceremony as an added attraction. Besides air resistance, the other main engineering problem in creating a 1-meter Foucault pendulum nowadays is said to be ensuring there is no preferred direction of swing.
A "pendulum day" is the time needed for the plane of a freely suspended Foucault pendulum to complete an apparent rotation about the local vertical. This is one sidereal day divided by the sine of the latitude.
Precession as a form of parallel transport
In a near-inertial frame moving in tandem with Earth, but not sharing the rotation of the earth about its own axis, the suspension point of the pendulum traces out a circular path during one sidereal day.At the latitude of Paris, 48 degrees 51 minutes north, a full precession cycle takes just under 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 by just over 270 degrees. If the plane of swing was north–south at the outset, it is east–west one sidereal day later.
This also implies that there has been exchange of momentum; the Earth and the pendulum bob have exchanged momentum. The Earth is so much more massive than the pendulum bob that the Earth's change of momentum is unnoticeable. Nonetheless, since the pendulum bob's plane of swing has shifted, the conservation laws imply that an exchange must have occurred.
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 can be demonstrated, by composing the infinitesimal rotations, 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 trace of the plane of oscillation will undergo parallel transport. After 24 hours, the difference between initial and final orientations of the trace in the Earth frame is, which corresponds to the value given by the Gauss–Bonnet theorem. is also called the holonomy or geometric phase of the pendulum. When analyzing earthbound motions, the Earth frame is not an inertial frame, but rotates about the local vertical at an effective rate of radians per day. A simple method employing parallel transport within cones tangent to the Earth's surface can be used to describe the rotation angle of the swing plane of Foucault's pendulum.
From the perspective of an Earth-bound coordinate system with its -axis pointing east and its -axis pointing north, the precession of the pendulum is described by the Coriolis force. Consider a planar pendulum with natural frequency 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 is horizontal in the small angle approximation and is given by
where is the rotational frequency of Earth, is the component of the Coriolis force in the -direction and is the component of the Coriolis force in the -direction.
The restoring force, in the small-angle approximation, is given by
Using Newton's laws of motion this leads to the system of equations
Switching to complex coordinates, the equations read
To first order in this equation has the solution
If time is measured in days, then and the pendulum rotates by an angle of during one day.
Related physical systems
Many physical systems precess in a similar manner to a Foucault pendulum. As early as 1836, the Scottish mathematician Edward Sang contrived and explained the precession of a spinning . In 1851, Charles Wheatstonedescribed an apparatus that consists of a vibrating spring that is mounted on top of a disk so that it makes a fixed angle 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.
Similarly, consider a nonspinning, perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle 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.
Foucault-like precession is observed in a virtual system wherein a massless particle is constrained to remain on a rotating plane that is inclined with respect to the axis of rotation.
Spin of a relativistic particle moving in a circular orbit precesses similar to the swing plane of Foucault pendulum. The relativistic velocity space in Minkowski spacetime can be treated as a sphere S3 in 4-dimensional Euclidean space with imaginary radius and imaginary timelike coordinate. Parallel transport of polarization vectors along such sphere gives rise to Thomas precession, which is analogous to the rotation of the swing plane of Foucault pendulum due to parallel transport along a sphere S2 in 3-dimensional Euclidean space.
In physics, the evolution of such systems is determined by geometric phases. Mathematically they are understood through parallel transport.