[Edit] The dynamics of the Foucault pendulum

Change of direction of the plane of swing of the pendulum in angle per sidereal day as a function of latitude. The pendulum rotates in the anticlockwise (positive) direction on the southern hemisphere and in the clockwise (negative) direction on the northern hemisphere. The only points where the pendulum returns to its original orientation after one day are the poles and the equator.

From the perspective of an inertial frame outside of Earth, 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 force acting on the pendulum is parallel to the plane of oscillation at all times. But the plane satisfies the constraint that it contains the plumb line. Thus the plane of oscillation undergoes parallel transport. The difference between initial and final orientations is as given by the Gauss-Bonnet theorem. 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, which is the magnitude of the projection of the angular velocity of Earth onto the normal direction to Earth.

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 explained 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

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

Animation of a Foucault pendulum showing the direction of rotation on the southern hemisphere. The rate of rotation is greatly exaggerated. A real Foucault pendulum, released from rest, does not pass directly over its equilibrium position as the one in the animation does.

Using Newton's laws of motion this leads to the system of equations

Switching to complex coordinates z = x + iy the equations read

To first order in Ω / ω this equation has the solution

If we measure time in days, then Ω = 2π and we see that the pendulum rotates by an angle of during one day.

[Edit] 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 φ 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 φ.

Similarly, consider a non-spinning perfectly balanced bicycle wheel mounted on a disk so that its axis of rotation makes an angle φ 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 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.