Foucault pendulum

Foucault Pendulum

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Foucault pendulum

The Foucault pendulum ( "foo-KOH"), or Foucault's pendulum, named after the French physicist Léon Foucault

Léon Foucault

Jean Bernard L?on Foucault was a France physics best known for the invention of the Foucault pendulum, a device demonstrating the effect of the Earth's rotation....
, was conceived as an experiment to demonstrate the rotation of the Earth.

experimental apparatus consists of a tall pendulum

Pendulum

A pendulum is a weight suspended from a pivot so it can swing freely.When a pendulum is displaced from its resting Mechanical equilibrium, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position....
free to oscillate in any vertical plane. The direction along which the pendulum swings rotates with time because of Earth's daily rotation.

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The experiment

The experimental apparatus consists of a tall pendulum

Pendulum

Paris Observatory

The Paris Observatory is the foremost astronomy observatory of France, and one of the largest astronomical centers in the world....
. A few weeks later, Foucault made his most famous pendulum when he suspended a 28-kg bob

Bob (physics)

A bob is the weight on the end of a pendulum.The use of a weight concentrated in a small, compact object enables the centre of gravity to be positioned close to the physical end of the pendulum, which minimises the length of pendulum required for a given period ....
with a 67-metre wire from the dome of the Panthéon

Panthéon, Paris

The Panth?on is a building in the Latin Quarter in Paris, France. It was originally built as a church dedicated to St. Genevieve, but after many changes now combines liturgical functions with its role as a List of cemeteries....
in Paris

Paris

Paris is the Capital of France and the country's largest city. It is situated on the river Seine, in northern France, at the heart of the ?le-de-France Regions of France ....
. The plane of the pendulum's swing rotated clockwise 11° per hour, making a full circle in 32.7 hours.

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

North Pole

The North Pole, also known as the Geographic North Pole or Terrestrial North Pole is, subject to the caveats explained below, defined as the point in the northern hemisphere where the Earth's axis of rotation meets the Earth's surface....
or South Pole

South Pole

The South Pole, also known as the Geographic South Pole or Terrestrial South Pole, is one of the two points where the Earth's rotation intersects the surface....
, 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

Equator

The equator is the intersection of the Earth's surface with the Plane perpendicular to the Earth's rotation and containing the Earth's center of mass....
, 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, (measured in clockwise degrees

Degree (angle)

A degree , usually denoted by ? , is a measurement of plane angle, representing 1/360 of a Turn ; one degree is equivalent to p/180 radians....
per sidereal day), is proportional to the sine

Siné

Maurice Sinet, known as Sin? is a France cartoonist.As a young man he studied drawing and graphic arts, earning his life as a cabaret singer....
of the latitude

Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
, :

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 named the gyroscope

Gyroscope

A gyroscope is a device for measuring or maintaining orientation , based on the principles of angular momentum. The device is a spinning wheel or disk whose axle is free to take any orientation....
in 1852. 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.

The dynamics of the Foucault pendulum

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. No forces act to make the plane of oscillation of the pendulum rotate - the plane contains the plumb line, so the gravitational force acting on the pendulum is parallel to the plane of oscillation at all times. Thus the plane of oscillation undergoes parallel transport

Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection....
. The difference between initial and final orientations is as given by the Gauss-Bonnet theorem. is also called the holonomy

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported....
or geometric phase

Geometric phase

In mechanics , the Geometric phase, or the Pancharatnam-Berry phase , also known as the Pancharatnam phase or Berry phase, is a Phase acquired over...
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, which is the magnitude

Magnitude (mathematics)

The magnitude of a mathematical object is its size: a property by which it can be larger or smaller than other objects of the same kind; in technical terms, an ordering of the class of objects to which it belongs....
of the projection of the angular velocity

Angular velocity

In physics, the angular velocity is a vector quantity which specifies the angular speed, and axis about which an object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per second, degrees per hour, etc....
of Earth onto the normal

Surface normal

A surface normal, or simply normal, to a Flatness is a vector which is perpendicular to that surface. A normal to a non-flat surface at a Point P on the surface is a vector perpendicular to the Tangent space to that surface at P....
direction to Earth.

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 explained by the Coriolis force. Consider a planar pendulum with natural frequency in the small angle approximation

Small angle approximation

Small-angle approximation is a useful simplification of the laws of trigonometry which is only approximately true for finite angles, but correct in the limit as the angle approaches zero....
. 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

Latitude

Latitude, usually denoted symbolically by the Greek letter phi gives the location of a place on Earth north or south of the equator. Lines of Latitude are the horizontal lines shown running east-to-west on maps ....
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 x-direction and is the component of the Coriolis force in the y-direction.

The restoring force, in the small angle approximation, is given by

Using Newton's laws of motion

Newton's laws of motion

Newton's laws of motion are three physical laws that form the basis for classical mechanics, Direct relationship the forces acting on a Physical body to the motion of the body....
this leads to the system of equations

Switching to complex coordinates the equations read

To first order in this equation has the solution

If we measure time in days, then and we see that the pendulum rotates by an angle of during one day.

Related physical systems

There are many physical systems that precess in a similar manner to a Foucault pendulum. In 1851, Charles Wheatstone

Charles Wheatstone

Knighthood Charles Wheatstone Fellow of the Royal Society , was a United Kingdom scientist and inventor of many scientific breakthroughs of the Victorian era, including the English concertina, the stereoscope , and the Playfair cipher ....
described 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 non-spinning 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 .

Another system behaving like a Foucault pendulum is a South Pointing Chariot

South Pointing Chariot

The South Pointing Chariot is widely regarded as one of the most complex geared mechanisms of the ancient History of China, and was continually used throughout the medieval period as well....
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 phase

Geometric phase

In mechanics , the Geometric phase, or the Pancharatnam-Berry phase , also known as the Pancharatnam phase or Berry phase, is a Phase acquired over...
s. Mathematically they are understood through parallel transport

Parallel transport