Foucault Pendulum Vector Diagrams - Relative Motion of The Plane of The Pendulum Swing To The Surface of The Earth

Relative Motion of The Plane of The Pendulum Swing To The Surface of The Earth

In order to observe the rotation of the Earth in relation to the plane of the pendulum swing there must be a basic difference in the two types of motion that are being compared. This basic difference is then manifested as (1) being able to 'observe' the change in position of the Earth in relation to the pendulum swing and (2) the time to observe a complete 'relative rotation' decreases with the sine of the latitude (decreases with an increase in the angle of alignment with the Earth's axis of rotation).

At the North Pole:

The central axis of the pendulum aligns with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth.

A pendulum bob at rest at the North Pole still has spin on the bob.

If a pendulum bob is hanging vertically at the North Pole and held in place, the bob is stationary but is rotating (spinning) with the Earth. Once the bob is released (but not swinging) it will continue to rotate (spin) unless one stops the rotation (spin) by forcing a spot on the side of the bob to always line up or point to one star. If the rotation of the bob is stopped then the connection wire will twist one turn every day unless there is a connection that is free to rotate (spin) (at either end of the wire or the support structure). If the bob at the North Pole is allowed to continue to rotate (spin) then it will and the wire won't twist one turn in one day.

The Foucault pendulum connection is constructed such that the pendulum is free to swing in any direction.

This is not the same thing as the support connection being free to rotate (spin). The swing of the pendulum is different than the rotation (spin) of the bob.

If a pendulum is hanging at the North Pole, before the bob is released, the bob is stationary but is rotating (spinning) with the Earth. Once the bob is released it will continue to rotate (spin) unless one stops the rotation (spinning) by forcing a spot on the side of the bob to always line up or point to one star.

If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then the bob will revolve about the central axis of the pendulum along with the rotation of the Earth and has an angular velocity equal to that of the Earth's angular velocity.

Before the bob is released there is a force that is exerted through the holding point of the bob that causes the bob to revolve about the pendulum axis and rotate (turn) with the Earth. This is because the holding point is attached to the surface of the Earth just like the structure of the pendulum is attached to the Earth.

Once the bob is displaced from the central axis of the pendulum and then released there no longer is a force acting on the bob that causes it to revolve about the central axis of the pendulum and rotate (turn) with the Earth.

As observed from an end-view of the swinging bob, the swing of the bob will always line up or swing towards one star (just like the axis of the Earth points at one star for the time periods considered) as the bob swings through the central axis of the pendulum. There can be a slight ellipsoid swing if the initial conditions of angular motion are not cancelled but there is no longer a force acting on the bob causing it to have an angular velocity after the bob is released. The plane of the swing of the pendulum bob is now independent of the surface of the earth which was imparting a force to the bob before it was released (through the holding point). As noted previously, the bob is still spinning with the Earth (a spot of the bob will spin with the Earth), even though the bob is no longer turning with the Earth. Thus the Earth continues to turn underneath the swing of the pendulum while the swing of the pendulum remains in a fixed plane that doesn't rotate (turn).

The point of significance is that the force imparting an angular velocity to the pre-released bob is no longer acting on the swinging bob. At the North Pole, this force takes one day for the direction of the force to complete a full circle since it takes one day for the Earth to rotate.

At the equator:

The central axis of the pendulum is perpendicular with the axis of rotation of the Earth. The central axis of the pendulum is always determined by the force of gravity directed towards the center of the Earth.

A pendulum bob at rest at the Equator is still rotating with the Earth and there is no spin on the bob.

The pendulum is moving with the rotation of the Earth when located at the equator, as is the support structure, so one can't see the rotation of the Earth in relation to the pendulum. The observation of the relative motion of the Earth in relation to the pendulum depends on the location of the surface of the Earth where the initial conditions are established.

If a pendulum bob is hanging vertically at the Equator and held in place, the bob is stationary relative to the Earth and is rotating (turning) with the Earth. Once the bob is released (but not swinging) it continues to rotate (turn) with the Earth.

If the bob is displaced from the central axis of the pendulum in preparation for swinging and held in place, then the bob is still rotating (turning) with the Earth with the same angular velocity equal to that of the Earth's angular velocity. This is the same angular velocity when at rest. Since the central axis of the pendulum is perpendicular with the axis of rotation of the Earth this is not the same as the North Pole where the central axis is aligned with the axis of the Earth. The bob is not revolving about the axis of the pendulum when held in place.

Before the bob is released there is a force that is exerted through the holding point of the bob that causes the bob to rotate (turn) with the Earth. This is because the holding point is attached to the surface of the Earth just like the structure of the pendulum is attached to the Earth.

Once the bob is displaced from the central axis of the pendulum and then released there is still the same force acting on the bob that causes it to rotate (turn) with the Earth.

As observed from an end-view of the swinging bob, the swing of the bob will not line up or swing towards one star as the bob swings through the central axis of the pendulum. There will not be a slight ellipsoid swing in relation to the Earth since the initial conditions of angular motion are not changed and there is still a force acting on the bob (transmitted through the support structure, pendulum wire, and gravity) causing it to have an angular velocity after the bob is released. The plane of the swing of the pendulum bob is independent of the surface of the earth but is not independent of the pendulum system which is still imparting the same force to the bob as before it was released through the single support point of the pendulum. As noted previously, there is no spin on the bob (a spot of the bob does not change with respect to the Earth) and the bob is not revolving about the axis of the pendulum. Thus the Earth continues to turn underneath the swing of the pendulum and the swing of the pendulum continues to turn with the Earth since there is still a force acting on the bob of the pendulum swing.

The point of significance is that the same forces imparting an angular velocity to the pre-released bob are still acting on the swinging bob. At the Equator, the relative motion of the Earth is not observable because there is no change in the force imparting an angular velocity to the bob. This is because the central axis of the pendulum is perpendicular with the axis of rotation of the Earth.

For a separate, imaginative arrangement, if one could imagine a large pendulum structure that is mounted at the North Pole and free to not rotate with Earth (e.g., mounted on a platform that is free of the rotation (spin) of the Earth) but has long arms that allows the pendulum to swing at the Equator then the Earth's surface would move underneath the pendulum. The Earth doesn't rotate (turn) under the pendulum swing like at the North Pole but the equatorial plane rotates perpendicular to the pendulum swing. This is a very large pendulum and an idealized situation.

At intermediate latitudes:

The rotation of the Earth is observable in relation to the plane of the pendulum swing but the time to observe a full rotation depends on the latitude of the location. The time to observe a full rotation is equal to one day at the North Pole with the time increasing with decreasing latitude and not observable at the Equator (infinite length of time).

The time increases because the central axis of the pendulum is aligned with the axis of rotation of the Earth at the North Pole and then the angle of misalignment increases as the latitude decreases to the point of perpendicularity at the Equator.

The angular velocity that is imparted to the pendulum bob about the axis of the pendulum prior to release decreases with the cosine of the degree of misalignment of the central axis of the pendulum in comparison to the axis of rotation of the Earth (zero degrees of misalignment at the North Pole, cosine of zero degrees equals 1; 90 degrees of misalignment at the Equator, cosine of 90 degrees equals 0).

This is equivalent to stating that the angular velocity that is imparted to the pendulum bob prior to release decreases with the sine of the latitude of the location (the sine of 90 degrees latitude equals 1; the sine of zero degrees latitude equals 0).

When the bob is released there is no longer a force acting on the bob causing it to revolve about the central axis of the pendulum. That force that is no longer applied is less than that applied at the North Pole where axis are fully aligned.

The time to observe a complete rotation of the Earth is inversely proportional to the angular velocity that is not imparted to the pendulum bob.

The statements above are thus equivalent to the inverse sine law for the observed time for a full rotation of the pendulum in relation to the rotation of the Earth.

Final Note: There is only one point of connection to the Earth for the swinging pendulum and that point of connection doesn't move in relation to the Earth.