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Analytic Mechanic Coriolis Effect and Foucault Pendulum

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Analytic Mechanic Coriolis Effect and Foucault Pendulum Karlstads University, faculty of science – (FYGC04) Analytic Mechanic Coriolis Effect & Foucault Pendulum teacher: Jürgen Fuchs Examiner : Igor Buchberger Arjun Lama [email protected] 12/27/2012 Abstract In this essay I’ve described the concept and the mathematical framework of the Coriolis force. I have provided some examples related to the Coriolis Effect which might help the reader to understand the Coriolis force. I’ve mostly followed the paper: “The Coriolis Effect – a conflict between common sense and mathematics” by Anders Persson. [3] 2 2013-1-31 Assignment 1 Contents Introduction ............................................................................................................................................ 4 Coriolis and Centrifugal forces ............................................................................................................ 5 Motion relative to the earth: ..................................................................................................................... 6 Foucault’s pendulum ............................................................................................................................... 8 Motion of Foucault’s pendulum: .............................................................................................................. 8 Impact due to the Coriolis force............................................................................................................. 10 Summary: .............................................................................................................................................. 11 Reference: ............................................................................................................................................. 11 3 2013-1-31 Assignment 1 Introduction In early 1800’s, the warships with long ranged canons were introduced in the sea battle. The sailors of warships noticed for the first time a strange physical phenomenon. When they shoot the canon at the opponent’s moving ship, the canon ball appears to curve slightly from its initial path missing the target. In 1835, this conundrum was solved by the French mathematician, mechanical engineer and scientist called Gaspard Gustave Coriolis (21 May 1792 – 19 September 1843). He explained that the occurrence of the phenomenon is due to the target, which moves as a result of being on the rotating earth. In his explanation he produced the mathematical model for the phenomenon showing the presence of supplementary forces namely centrifugal and Coriolis in the rotating frame of reference. So he suggested adjusting the force by deducing the fictional force, centrifugal and Coriolis which are due to the rotational frame. The Coriolis Effect is enigmatic and of difficult comprehension due to its fictitious characteristic. The perplexity occurs while defining the effect; it is totally dependent on the reference frame of the observant. An observer standing in a rotating frame, i.e. in a frame that is spinning with respect to an inertial frame, to properly describe the dynamics might need to introduce a fictitious force, the so called Coriolis force. As we will see, this force depends on the velocity with respect to the rotating frame and is zero in some particular case, in which the velocity has the same direction of the angular velocity of the rotating frame. The Coriolis Effect can then be defined to be the deviation from a straight trajectory in a rotating frame due to the Coriolis force. Let’s start with a simple example; for an observer from a stationary point in space, it is quite obvious that the earth rotates. However as an observer on the earth’s surface, the only indication that the earth rotates is by the observation of the sun during the day and the stars during the night. Every day the sun rises in the east and sets in the west but an observer on the earth’s surface is ignorant of the fact that the earth rotates towards the east. The earth rotates on its axis once each day. A spot on the equator rotates at approximately 1669.8 km/h, whereas at the North Pole (90 degrees north) and South Pole (90 degrees south), the speed is effectively zero. Table1: The rotational velocity of the earth at different latitude. Latitude (degree) Rotational velocity (km/h) 10◦ 1644.4 20◦ 1569.1 30◦ 1446.1 40◦ 1279.1 50◦ 1073.3 60◦ 834.9 70◦ 571.1 80◦ 289.95 4 2013-1-31 Assignment 1 One could ask the question why the rotational velocity of the earth varies in accordance to the latitude. As one travels from the equator towards either pole, the subject travels closer to the earth’s rotational axis experiencing the decrease in rotational velocity. For better understanding one can consider earth as a flat rotating disc, the so called merry goes round. At the edge of merry goes round the subject will experience very high rotational velocity whereas at the center effect is considerably very low. Let’s consider two people sitting opposite sides of the merry goes round and a ball is set to be passed between them. As the ball leaves the thrower’s hand the ball becomes independent of the rotating coordinates of the merry goes round, therefore its trajectory is also independent of the coordinates. In the meantime, the ball seems to deflect for the catcher, who is dependent of the rotating coordinate of merry goes round. This results in the change of the position of ball’s intended path which leads to the failure in catching the ball. As one passes the ball to another, the ball seems to be deflected toward right if the motion of the merry goes round is anticlockwise and vice versa. The person observing above the merry goes round, the path of the ball appears straight however people on the merry goes round, the path of the ball appears to cause deflection to the right at the intended motion. This simple experiment demonstrates how the rotation of the merry goes round appears to cause the deflection in the path of the ball. Analogously this also demonstrates how the air movement on the earth’s surface is deflected by the rotation of the earth. The phenomenon is called Coriolis Effect. “The direction of the speed changes whereas the magnitude of the speed (kinetic energy) remains invariant under the influence of the Coriolis Effect in the body system in a rotating frame”. However, in the above example the Coriolis Effect is insignificant due to the over power of centrifugal force. But still gives an over view about the subject. So far the description above gives an elementary understanding of Coriolis Effect and yet needs a great deal of clarification on the matter of Coriolis Effect. Coriolis and Centrifugal forces It is good to start with the difference between an inertial frame and a non-inertial frame that enables in easy understanding the procedure of derivation of Coriolis and centrifugal forces. Let K be an inertial system (space system) and K´ (body system) be another system that is identical with the K at time (t=0) and which rotates with the angular speed ω =|ω| about the direction, ̂ as shown in fig.1. let The position vector of a mass point be r (t) with respect to K and r´ (t) with respect to K´. The velocities are related by ́ ́ Where, ́ refers to K´ and v to K. Denoting the change per unit time as it is observed from ́ . ́ by ́ (1) Figure 1 the coordinate system K´ rotates about the ́ : Time derivative as observed from K´ system k with the angular velocity ω 5 2013-1-31 Assignment 1 : Time derivative as observed from K Now, Assume ω = constant, from equation (1),It follows that, ( ) ( ) ( ) ( ) At the same time, in the inertial frame K, we have, ( ) ( ) Combining equation (2) & (3) gives ( ) This means that, when observed from non-inertial frame K`, the dynamic of a point particle can be described in terms of the “original force” (F) and two additional fictional forces: Coriolis forces = (4) and centrifugal force = ( ) (5) If ω = 0 than these fictional forces vanish. Notice that the cross product (x) in equation (4) indicates that the Coriolis force is perpendicular to the rotational axis and also to the relative motion of the particle. This justifies the previously mentioned quote “The direction of the speed changes whereas the magnitude of the speed (kinetic energy) remains invariant under the influence of the Coriolis Effect”. Motion relative to the earth: Figure 2: All the perpendicular motion to the earth’s axis deflects whereas the parallel motions remain the same. 6 2013-1-31 Assignment 1 Consider an object on the earth‘s surface moving in an eastward direction (same direction that earth rotates) at a speed (u). As the object rotates faster than the earth, the centrifugal force acting on it increases to ⃗ ⃗ (( ) ⃗ ) ( ⃗ ) . ( ) = total centrifugal force. ⃗ = centrifugal force due to the earth’s rotation. ⃗ ⃗ = deflecting forces. For large scale motions, the last term in the right hand side is much smaller than the other two terms so one can neglect it. The deflection forces act outward. The Coriolis force, that results from zonal (i.e. east west) motion can be divided into components in the vertical -2uωsinφ and meridional direction 2uωcosφ. Where, φ is the latitude. If one considers an object moving along a longitude at a speed (v), than it has one component, vcosφ, which is parallel to the axis of rotation and other component vsinφ is perpendicular to axis of rotation. vsinφ is completely deflected whereas vcosφ remains invariant which yields 2ωvsinφ. This explains why the Coriolis force on a rotating planet varies with the sine of φ, F= -2mωsinφVr, the “sine law”. So for example, at latitude 43º (of central Italy) where 2ωsinφ is approximately equal to 10-4s-1, a motion of 10 m/s would move in an inertia circle of 100 km radius completing an orbit in almost 14 hours. But the Coriolis Effect is only one part of a three dimensional deflective mechanism. We can summarize the three-dimensional Coriolis deflections for different motions in an array where the mathematical terms have, for simplicity, been indicated only by their signs. 0 represents no deflection. 1 means deflection in the indicated direction and -1 in the opposite direction. Table 1the three-dimensional relation between the motion on a rotating planet and the Coriolis deflection.
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