Foucault Pendulum Vector Diagrams - Information and The Pendulum Sine Law

Information and The Pendulum Sine Law

The surface velocity due to the Earth's rotation is a maximum at the Equator and is equal to the circumference (pi × the diameter of the Earth) per 24 hours (or 3.14159 × 12,756 ÷ 24 = 1670 km/h = 1 equatorial velocity unit, EVU). The time of an Earth's rotation is inversely related to the angular velocity and the surface velocity (T = 1 day for 2 pi radians, or at the equator, 1 circumferential unit per 1 EVU = 40,075 km ÷ 1670 km/h ÷ 24 h/day = 1 day).

At a given latitude the surface velocity is equal to pi times the chord length parallel to the equator per 24 hours. This is equivalent to the cosine of the latitude × 1 EVU. At the poles the surface velocity is zero since zero distance is traveled. For a given longitude the surface velocity varies from 1 EVU at the equator to zero at the pole even though the angular velocities are all the same.

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The ratio of the surface velocity at two given latitudes is equal to the ratio of the cosine for the two given latitudes.

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The time to observe one full rotation of the Earth in relation to the plane of a swinging pendulum is one day at the poles (the minimum time) and cannot be observed (infinitely long, the maximum time) at the equator. One of the great insights by Léon Foucault is to deduce that the time to observe a full rotation of the Earth increased by the inverse of the sine of the latitude.

(ORTRP = observed rotation time in relation to the plane of the pendulum)

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The Pendulum Sine Law also defines that the ratio of the observed Earth's rotation time at two separate latitudes in relation to a pendulum swing is equal to the inverse ratio of the sine of the two latitudes.

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The sine of the latitude also indicates the degree of alignment of the pendulum central axis to the Earth's axis of rotation. At the poles the pendulum axis is parallel or aligned to the Earth's axis and the sine of 90° = 1. At the equator the pendulum axis is perpendicular to the Earth's axis and the sine of 0° = 0.

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 in relation to the rotation of the Earth's axis that is imparted to the pendulum bob 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. There are zero degrees of misalignment at the North Pole and the cosine of zero degrees equals 1. There are 90 degrees of misalignment at the Equator and the cosine of 90 degrees equals 0.

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This equation is very similar to the equation for the reduction in surface velocity with longitude stated above. This is equivalent to stating that the angular velocity that is imparted to the pendulum bob 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). The time to observe a complete rotation is inversely proportional to the angular velocity that is imparted to the pendulum bob in comparison to the angular velocity of the Earth. 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.

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.

To approach the Pendulum Sine Law in basic steps:

• The Earth's surface velocity decreases with increasing latitude directly proportional with the cosine of the latitude.
• The degree of alignment of the pendulum axis in comparison to the Earth's axis increases with increasing latitude directly proportional with the sine of the latitude.
• The angular velocity for the Earth is related to the circumferential surface velocity (2 × pi radians per day = 40,075 km per day at the equator).
  • The observed apparent rotation of the pendulum has an angular velocity (e.g., for the set of points at the end of the pendulum swing). This angular velocity is related to the apparent circumferential surface velocity of the pendulum.
• The time of an Earth's rotation is inversely related to the angular velocity (T = 1 day per 2 × pi radians; or as calculated at the equator, 1 circumferential unit per 1 EVU), and inversely related to the circumferential surface velocity of 1 EVU.
  • The time to observe a full rotation of the Earth in relation to the plane of the pendulum is inversely related to the angular velocity and inversely related to the apparent circumferential surface velocity.

If it is proposed that;

• The angular velocity that is observed for an Earth's rotation in relation to the plane of the pendulum is directly related to the degree of alignment of the pendulum axis to the Earth's axis (Coriolis effect).

Then it follows that,

• The angular velocity that is observed for an Earth's rotation in relation to the plane of the pendulum increases with increasing latitude directly proportional with the sine of the latitude.
• The time required to observe an Earth's rotation in relation to the plane of a pendulum decreases with increasing latitude inversely proportional with the sine of the latitude (the Pendulum Sine Law).