Foucault Pendulum Vector Diagrams

Foucault Pendulum Vector Diagrams

Several vector diagrams are often used to demonstrate the physics underlying the Foucault pendulum.

Diagrams are provided to illustrate a pendulum located at the North Pole, equator, and 45 degrees N to show how the rotation of Earth in relation to the pendulum is observed, or not, at these locations. This is not a rigorous evaluation but is intended to convey information regarding the interaction of the two moving objects, the swinging pendulum and the rotating Earth. One of the great insights by Léon Foucault is that the time to observe a full rotation of the Earth increased by the inverse of the sine of the latitude.

In the examples, the pendulums are of great size to aid in the visualization of the pendulum swing in relation to the Earth (shown as blue circles). The pendulum is drawn so that 90 degrees of pendulum arc sweeps out 90 degrees of arc on the surface of the Earth. Views from the side, the front, and above (right, center, left) are provided to aid in the interpretation of the diagrams and arrows are provided to show the direction of the Earth's rotation. The schematic at the bottom of the each figure represents the range of swing of the pendulum as viewed from above and normalized to a standard orientation. The smaller arrows depict the magnitude of the relative velocity vector for the point on the Earth's surface in-line with the pendulum bob projected to the center of the Earth (the magnitude is shown since the scematic is two-dimensional, not three-dimensional). The pendulum bob is always affected by the force of gravity directed towards the center of the Earth. The force associated with the connection of the pendulum to a support structure directs the pendulum bob along the swing of the arc.

The support structure is dependent on the velocity of the surface of the Earth where it is located. The point of connection of the pendulum moves with the surface velocity vectors of the Earth at that latitude. At the equator the support-point moves with the equatorial rotation of the Earth and moves the pendulum swing along with this rotation. At the poles the support-point is located on the axis of the Earth so the support-point rotates but does not have a horizontal velocity component as it does at the equator (and a progressively less horizontal velocity component with increasing latitude). The plane of the pendulum swing, however, is independent of the surface velocity vectors underneath the swing since there is only one point of connection. The point of connection is configured such that the plane of pendulum swing is free to swing in any direction in relation to the structure of the connection point. The pendulum swing at the poles remains aligned toward a star if not forced to rotate with the support. As a result, it is observable that the Earth turns underneath the plane of swing of the pendulum.