Foucault Pendulum Vector Diagrams - Evaluation of Surface Velocity Vectors

Evaluation of Surface Velocity Vectors

The surface velocity vectors underneath the swing of the pendulum can be separated into three components (x, y, and z) for the 3-dimensional system in order to evaluate the vectors on opposite sides of the pendulum. The evaluation is to identify whether the vectors on each side of the pendulum swing are 1) balanced in the same direction, 2) acting in the same plane, or 3) unbalanced or in opposing directions. If the vector components on opposite sides of the pendulum swing are balanced in the same direction or act in the same plane of the pendulum then the rotation of the Earth will not be observable in relation to the swing of the pendulum. If the plane of the pendulum swing establishes the x-y plane then the z-component determines when the Earth's rotation will be observable and only if the z-component is not balanced in the same direction on each side. The magnitude of the opposing component is proportional to how long it takes for one full turn of the Earth to be observed in relation to the plane of the pendulum. The length of time is a minimum of one day at the poles, increases from the pole to the equator, and is not visible at the equator (infinitely long).

For any two points in the pendulum swing that are equidistant from the center of the swing there are two related points projected onto the surface of the Earth. These points can be used to determine the corresponding surface velocity components that are in opposition and not acting in the same plane of the swing. The magnitude of the difference between these two points (for a given latitude of the center-point) is a relative measure of the time to observe one full rotation. The ratio of the velocity vector difference to the corresponding points at the North Pole with the same equidistance from the center of the swing and the same projection to the surface can then be determined. The inverse ratio will determine the time observed for one full rotation of the pendulum swing in comparison to the duration at the pole of one day.

From the diagrams two points of the pendulum swing can be chosen to project straight down to two points on opposite sides of the Earth (180° apart). This makes it easy to obtain the velocity vector difference and then the time observed for a full rotation from the inverse ratio.

The examples show that the Earth turns underneath the plane of the pendulum swing and how this change in relationship can be interpreted at different latitudes.

• For the North Pole pendulum (Figure 1) the velocity vector by inspection is 1 EVU on one side of the swing (as projected to the equator) and 1 EVU in the opposite direction on the other side of the swing. The difference between these two points is 2 EVU for the North Pole pendulum.

• Using the same projection for the equatorial pendulum with longitudinal swing (Figure 2A) the velocity vector is 0 EVU on one side of the swing (for the North Pole) and 0 EVU on the other side of the swing (for the South Pole). The difference between these two points is 0 EVU for this arrangement. The time to observe a full rotation is infinitely long since the ratio divides by zero. For any two equidistant points the difference between the two vectors is zero, meaning the vectors are balanced in the same direction on each side of the pendulum swing.

• Using the same projection for the equatorial pendulum with latitudinal swing (Figure 2B) the velocity vector is 1 EVU on each side of the swing and are in opposite directions. Even though the difference in the velocity vectors is 2 EVU, these vectors are acting in the same plane as the pendulum, therefore, cannot be observed by the pendulum swing. The z-component determines when the Earth's rotation will be observable and these are both zero.

• Using the same projection for the 45° North pendulum with longitudinal swing (Figure 3A) the velocity vector is 0.707 EVU on one side of the (corresponding to 45° North on the opposite side of the world from the center point) and 0.707 EVU in the opposite direction on the other side of the swing (corresponding to 45° South). The difference between these two vectors is 1.414 EVU.

• Using the same projection for the 45° North pendulum with longitudinal swing (Figure 3B) the velocity vector is 1 EVU but the z-component is only 0.707 EVU since the x-y plane is at 45°on one side of the swing (corresponding to the equator). On the other side of the swing the velocity vector is 0.707 EVU in the opposite direction (corresponding to the equator with a tilt of 45° to the x-y plane). The difference between these two vectors is 1.414 EVU.

• The ratio of the velocity vectors for 45° to that of the pole is 1.414 ÷ 2.0 which equals 0.707. The time to observe the full rotation is then the inverse, or 1.414 days for a pendulum at 45° on the Earth.