Foucault pendulum

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For the novel Foucault's Pendulum by Umberto Eco, See Foucault's Pendulum (book).
Foucault's Pendulum in the Panthéon, Paris
Foucault's Pendulum in the Panthéon, Paris

A Foucault pendulum, or Foucault's pendulum, named after the French physicist Léon Foucault, was conceived as an experiment to demonstrate the rotation of the Earth; its action is a result of the Coriolis effect. It is a tall pendulum free to oscillate in any vertical plane and ideally should include some sort of motor so that it can run continuously rather than have its motion damped by air resistance. The first Foucault pendulum exhibited to the public was in February 1851 in the Meridian Room of the Paris Observatory, although Vincenzo Viviani had already experimented with a similar device in 1661. 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. In 1851 it was well known that the Earth moved: experimental evidence included the aberration of starlight, stellar parallax, and the Earth's measured polar flattening and equatorial bulge. However Foucault's pendulum was the first dynamical proof of the rotation in an easy-to-see experiment, and it created a justified sensation in both the learned and everyday worlds.

A Foucault pendulum at the north pole. The pendulum swings in the same plane as the Earth rotates beneath it.
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A Foucault pendulum at the north pole. The pendulum swings in the same plane as the Earth rotates beneath it.

At either the North Pole or South Pole, the plane of oscillation of a pendulum remains pointing in the same direction with respect to the fixed stars, while the Earth rotates underneath it, taking one sidereal day to complete a rotation. When a Foucault pendulum is suspended somewhere on the equator, then the plane of oscillation of the Foucault pendulum is at all times co-rotating with the rotation of the Earth. What happens at other latitudes is an intermediate between these two effects.

At the equator the equilibrium position of the pendulum is in a direction that is perpendicular to the Earth's axis of rotation. Because of that, the plane of oscillation is co-rotating with the Earth. Away from the equator the co-rotating with the Earth is diminished. Between the poles and the equator the plane of oscillation is rotating both with respect to the stars and with respect to the Earth. The direction of the plane of oscillation of a pendulum with respect to the Earth rotates with an angular speed proportional to the sine of its latitude; thus one at 45° rotates once every 1.4 days and one at 30° every 2 days.

n = degrees per day

φ = Latitude

n = 360^\circ \sin \phi

Foucault pendulum at the Musée des arts et métiers (Paris); pegs are placed around and are knocked down as the pendulum swing plane veers. This is the original bob from the 1851 Panthéon pendulum
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Foucault pendulum at the Musée des arts et métiers (Paris); pegs are placed around and are knocked down as the pendulum swing plane veers. This is the original bob from the 1851 Panthéon pendulum
 
 
 
 
 
 
Foucault pendulum at the Franklin Institute (Philadelphia)
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Foucault pendulum at the Franklin Institute (Philadelphia)

Many people found the sine factor difficult to understand, which prompted Foucault to conceive the gyroscope in 1852. The gyroscope's spinning rotor tracks the stars directly. Its axis of rotation turns once per day whatever the latitude, unaffected by any sine factor.

A Foucault pendulum is tricky 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, without imparting any unwanted sideways motion, is to use a flame to burn through a thread which is temporarily holding the bob in its starting position. Air resistance damps the oscillation, so Foucault pendulums in museums usually incorporate an electromagnetic or other drive to keep the bob swinging.

The Foucault pendulum that hangs in the rotunda of the Lexington Public Library in Lexington, Kentucky in the United States is the largest ceiling clock in the world.

Contents


  • 1 The dynamics of the Foucault pendulum
    • 1.1 The forces that are exerted on the pendulum bob
    • 1.2 Comparison with driving on a banked circuit
    • 1.3 Method of calculation
  • 2 Advanced reading
  • 3 See also
  • 4 External links
  • 5 References

The dynamics of the Foucault pendulum

Precession in degrees per hour of the plane of swing of the pendulum. Red line: precession with respect to the Earth. Blue line: precession with respect to the fixed stars.
Precession in degrees per hour of the plane of swing of the pendulum. Red line: precession with respect to the Earth. Blue line: precession with respect to the fixed stars.

The diagram shows the precession of the plane of swing of a Foucault pendulum as a function of latitude. The horizontal axis is the latitude: from 90 degrees latitude to 0 degrees latitude. The vertical axis shows the rate of precession in degrees per hour; positive for clockwise precession, negative for counterclockwise precession.

The red line shows the precession with respect to the Earth of a Foucault pendulum located anywhere on the northern hemisphere. At the north pole the pendulum precesses (with respect to the Earth) through an entire circle in one sidereal day.

The blue line marks the precession of the plane of swing of a Foucault pendulum with respect to the fixed stars. The distance between the red line and the blue line is a constant 15 degrees per hour, which is the rotation rate of the Earth.

A Foucault pendulum located on the northern hemisphere at 30 degrees latitude will take two days to precess through an entire circle with respect to the Earth, precessing clockwise with respect to the Earth at a rate of 7.5 degrees per hour. In those two days the pendulum also precesses counterclockwise through a full circle with respect to the fixed stars at a rate of 7.5 degrees per hour.

The precession of the plane of swing at 30 degrees latitude. The view is from very high above the north pole. The oval represents a circle that appears distorted because it is viewed at a 60 degrees angle. The line inside the oval is the trace of the plane of swing over the surface that the pendulum is suspended above. On the left the view as seen from a non-rotating point of view, on the right the co-rotating-with-the-Earth point of view. ( Double sized version of this animation )
The precession of the plane of swing at 30 degrees latitude. The view is from very high above the north pole. The oval represents a circle that appears distorted because it is viewed at a 60 degrees angle. The line inside the oval is the trace of the plane of swing over the surface that the pendulum is suspended above. On the left the view as seen from a non-rotating point of view, on the right the co-rotating-with-the-Earth point of view. ( Double sized version of this animation )


The forces that are exerted on the pendulum bob

The Earth, with its equatorial bulge exaggerated. The red arrow represents the force of gravity. The green arrow represents the force that the pendulum wire exerts on a non-swinging pendulum bob. The blue arrow shows the resultant force for object that are constrained to move parallel to the surface: the poleward force. In the case of the actual Earth, at 45 degrees latitude the angle between the red and the green vector is about a tenth of a degree.
The Earth, with its equatorial bulge exaggerated. The red arrow represents the force of gravity. The green arrow represents the force that the pendulum wire exerts on a non-swinging pendulum bob. The blue arrow shows the resultant force for object that are constrained to move parallel to the surface: the poleward force. In the case of the actual Earth, at 45 degrees latitude the angle between the red and the green vector is about a tenth of a degree.

The reason for precession of the plane of swing of the Foucault pendulem with respect to the surface of the Earth is straightforward: the earth turns under the pendulum. Only the local vertical component of the Earth's angular velocity vector contributes to the precession, however. The pendulum only "sees" the component of rotation about an axis which is vertical at its location (the rate of precession relative to the Earth is equal to half the planetary vorticity). This means that at locations other than the poles, a Foucault pendulum must also precess with respect to the fixed stars. The rate of this precession can be calculated from the difference between the rate of precession relative to the Earth and the rate of rotation of the Earth itself. This indirect approach to the calculation sidesteps the question of what force actually causes the precession relative to the fixed stars. This section is an explanation of that force.

The origin of the force that is involved in the precession of the Foucault pendulum with respect to the fixed stars is the centripetal force that keeps objects on the Earth's surface moving in circular paths. The solid Earth is ductile and the Earth's bulk is redistributed in such a way that there is a dynamic equilibrium between the Newtonian gravitational force, the normal force, and the centripetal force, resulting in the equatorial bulge. Mass that is resting on the surface of the Earth, such as the water of the oceans, and the air of the atmosphere, is subject to a poleward force, because there is a slight angle between the force of gravity and the normal force. This poleward force is precisely the amount of force that is required to circumnavigate the Earth's axis at a rate of one revolution per day. That is why at each latitude the water and air remain where they are. If the poleward force were not present, the water and air would all flow toward the equator.

This poleward force is exerted only on objects that have all of their weight resting on the Earth. The poleward force is exerted only on mass that experiences a normal force. In the case of the Foucault pendulum the normal force is exerted by the wire.

To give an idea of the magnitude of the poleward force: Let a floating airship with a weight of 1000 kilogram be located at 45 degrees latitude. Since the airship is floating there is negligable friction. An object located at 45 degrees latitude is circumnavigating the Earth's axis at a velocity of about 328 m/s. At 45 degrees latitude the radial distance to the Earth's axis is about 4500 kilometers. That corresponds to a centripetal acceleration of 0.024 m/s2

This means that the required center-seeking force to maintain the 1000 kilogram airship in circumnavigating motion at the same latitude is about 2.4 kilogram of force. That required force can be decomposed in a component perpendicular to the surface that the airship is floating above, and a component parallel to the surface that the airship is floating above. At 45 degrees latitude these components are equal in magnitude: about 1.7 kilogram of force.

In summary: at 45 degrees latitude, an object with a weight of a 1000 kilogram that is buoyant, is subject to a poleward force of 1.7 kilogram of force. Of course, this poleward force does not apply in the case of an object in free fall.

Comparison with driving on a banked circuit

Precession of a pendulum with respect to the Earth. (Pendulum located on the Northern hemisphere) Eight successive swings, with the precession very much exaggerated.
Precession of a pendulum with respect to the Earth. (Pendulum located on the Northern hemisphere) Eight successive swings, with the precession very much exaggerated.

The dynamics of the motion of the pendulum bob can be understood by comparison with the dynamics of a car following a banked curve on circuit. If the roadsurface of that circuit is extremely slippery, then there is only one velocity with which a car can negotiate the banked curve: only one velocity that allows the car to keep following the curve. If there is no sideways grip at all, then a car going too fast will swing wide, and a car going too slow will slump down. The car that is "speeding" swings wide because there is not enough center-seeking force, and the car that is too slow slumps down because there is a surplus of center-seeking force then.
There is another aspect of driving on a banked circuit that applies in the case of the Foucault pendulum. When a driver steers the car from high up the incline to below, the car will pick up extra speed. Conversely, when a driver steers up the bank, the car loses speed, because climbing up the incline takes energy.

Overall, the pendulum is circumnavigating the Earth's axis. The diagram shows the precession (exaggerated) of the pendulum, with respect to the point it is suspended above. When the pendulum swings from south to north the poleward force is doing work, increasing the angular velocity of the pendulum, so the pendulum deviates to the right. When the pendulum swings from east to west the pendulum is going "too slow", so there is a surplus of poleward force then, pulling the pendulum bob to the north. Remarkably, although the mechanism is different for the north-south and the east-west directions, the rate of the precession is the same for all directions.

Method of calculation

The Earth, with its equatorial bulge rather exaggerated. The red arrow represents the force of gravity. The green arrow represents the normal force. The blue arrow shows the resultant force: the poleward force.
The Earth, with its equatorial bulge rather exaggerated. The red arrow represents the force of gravity. The green arrow represents the normal force. The blue arrow shows the resultant force: the poleward force.

In performing calculations on the Foucault pendulum, the most efficient approach is to use a co-rotating coordinate system. The equation of motion for motion with respect to a rotating coordinate system has a centrifugal term and a coriolis term.

In the case of the Earth, there is an equality that allows a very efficient calculation. Because the Earth's equatorial bulge is in dynamic equilibrium, at every latitude the poleward force provides precisely the amount of center-seeking force that is required. In other words: at every latitude the vector of the center-seeking force is equal in magnitude to the vector of the centrifugal term of the equation of motion.

In the case of buoyant/suspended objects, that are free to move over the surface of the Earth, the poleward force must be taken into account. In calculations, the effects of the poleward force are taken into account simply by omitting the centrifugal term from the equation. It is common to describe the precession of the pendulum as being 'due to the coriolis effect'. This is connected to the fact that the equation of motion for a Foucault pendulum only contains the coriolis term; because of the presence of the poleward force, the centrifugal term is omitted.

By contrast, in the case of ballistic motion, the objects are not buoyant or suspended. Then there is no normal force and hence no resultant poleward force, and then both the centrifugal term and the coriolis term are incorporated into the calculations.

Advanced reading

  • Classical dynamics of particles and systems, 4ed, Marion Thornton (ISBN 0030973023 ), P.398-401.

See also

External links

References