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Foucault Pendulum

Consider a point B on the surface of earth at latitude λ; point B is located on the rotating frame S’, with origin at O’. The frame S’ rotates about the line O’ N, with angular velocity . As seen by an observer at B on S’ rotates about the line O’ N, with angular velocity As seen by an observer at B on S’, the coriolis force acting on some moving particle P is given by,

F_cor = –2 m ω × v’

where v’ is the velocity of the particle relative to frame S’.

The vectors and v’ can be resolved into two perpendicular directions, viz. vertical and horizontal to surface of earth at B. That is, we write

Where, we find that and ω_H = ω cos λ. In the northern hemisphere points upward, while in southern hemisphere it points downward, to the earth’s surface. At any point on globe, the component ω_H is always directed towards north along meridian (or great circle) passing through that point. The coriolis force, therefore, can be written as



Note that on the other hand, need not vanish because horizontal component of v’ is not always along (i.e. along north at B). Thus, we see that there are three contributions to net coriolis force:

1. The vertical component v_v’ of the velocity of the particle is responsible for the part of the coriolis force. Since vectors and v_V’ constitute the meridional plane O’BN, the above part of coriolis force is perpendicular to the plane O’BN; hence, as seen at B, the force acts in horizontal plane, along east-west direction. If v_V’ is downwards, coriolis force acts towards east, if v_V’ is upwards, the force acts towards west. Hence, a body falling vertically from a height is deflected towards west. Hence, a body falling vertically from a height is deflected towards east by coriolis force.

2. The horizontal component v_H’ of the velocity of the particle is responsible for two distinct parts of the coriolis force. The part, acts normal to horizontal plane, i.e. along the vertical direction as viewed by an observer at B. This force either pushes the particle towards earth or away from it, depending upon whether direction of v_H’ is west or east of north respectively. This force must be balanced to maintain a definite height by objects moving fast and horizontally above earth’s surface, as for example jets or missiles.

3. The last part of coriolis force, acts in the horizontal plane. In the northern hemisphere, where points downward, is directed to the right, if we face towards v_H’; in southern hemisphere, if we face towards v_H’, the above coriolis force acts towards left. Thus, for example, a river flowing in northern hemisphere is constantly drifted towards right side (as we look along the direction of flow) slowly cutting the right bank.

Foucault pendulum: One of the interesting consequence of this (last) force is the slow rotation of the plane of oscillations of a simple pendulum about the vertical axis of suspension. This experiment was first performed by Foucault using a heavy 28-kg bob suspended by a nearly 70 m long wire. The upper end of the wire was attached to a rigid support (ceiling of a dome in Paris) in such a way that pendulum could oscillate freely in any direction.

As the Foucault pendulum is set to oscillations, in (say) northern hemisphere, the line of motion of the bob gets slightly drifted right each time bob moves end to end of its amplitudes. This happens because the above horizontal coriolis force act on the bob. After a definite time period T, the plane of oscillations rotates by complete 2 π angle about the axis of suspension, in clockwise direction as seen from above.

It is a complicated calculation to find out T from the coriolis force and consequent dynamics; the value of force varies with velocity of bob as it moves from end to end during one oscillation. However, a qualitative argument, which Incidently gives correct answer too, goes as follows: plane of the velocity of rotation of earth about the axis of suspension is ω_V = ω sin λ at latitude λ. Hence time period of complete rotation is,

The above argument is regarded as a proof that rotation of Foucault pendulum demonstrates spinning motion of earth.