Karlstads University, faculty of science (FYGC04) Analytic Mechanic Coriolis Effect & Foucault Pendulum teacher: Jürgen Fuchs Examiner : Igor Buchberger Arjun Lama llama82@live.se 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
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
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
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 Figure 1 the coordinate system K rotates about the system k with the angular velocity ω Where, refers to K and v to K. Denoting the change per unit time as it is observed from. by (1) : Time derivative as observed from K 5
: 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
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 10-4 s -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. Northward motion Eastward motion Downward motion Northward 0-1 0 deflection Eastward deflection 1 0 1 Downward deflection 0-1 0 7
Foucault s pendulum Foucault s pendulum is an ordinary simple pendulum which illustrates the effect due to the Coriolis force. The pendulum is suspended freely to swing in any direction. The perfect symmetrical form of the pendulum gives an equal period of oscillation in all directions. It is long and heavy, so that it will swing freely for many hours despite air resistance. Due to the rotation of the Earth, the plane of oscillation of the pendulum will rotate with respect to the surface beneath it. The observer standing on the Earth resides in the reference frame, with the swinging pendulum oscillating in a rotating frame. From the pendulum's point of view, it keeps oscillating in the same plane, but the Earth spins below it. The deflection from its original plane of oscillation as far as the observer is concerned is due to the Coriolis force given by Where, v is the horizontal velocity of the pendulum bob and Ω is the angular velocity of the Earth. It will act to divert the direction of the pendulum bob as it swings (see figure below), rotating the plane of oscillation clockwise in the northern hemisphere and anti-clockwise in the southern. Figure 3 the changing trajectory of the pendulum bob with each swing (angle of deviation greatly exaggerated) Motion of Foucault s pendulum: Let s now analyze in detail the Foucault pendulum motion, I follow [2]. For the inertial frame, there are just two forces on the bob, the tension T in the wire and the weight mg. whereas in the rotating frame of the earth, there are also the centrifugal and Coriolis forces, so the equation of motion in the earth's frame is 8
For the small angles approximation allows, the z component of the tension T is well approximated by the magnitude; that is, and Putting these two approximations together, gives We now need to examine the x and y components of the equation of motion (6).This requires, identifying the x and y components of T. one can easily see from the figure 4, Figure 4 Foucault pendulum comprises a bob of mass m. suspended by a light wire of length L from the point P on a high ceiling. The tension force on the bob is shown as T and its x and y components are Tx and Ty. For small oscillations the angle β is very small. The x and y components of g are of course 0, and the components of are given below ( ) Where denotes the colatitude of the location of the experiment. Putting these components in equation (5) and dropping a term involving z, which is negligible compared to or for small oscillations, gives The factor is just where is the natural frequency of the pendulum, and is just, the z component of the earth's angular velocity. Thus these two equations of motion can be rewritten as The above coupled equations can be solved by defining a complex number. This complex number contains the same information as the position in the x-y plane. If we multiply equation (9) by i and add it to the (8), we get the single differential equation. This is a second-order, linear, homogeneous differential equation and so has exactly two independent solutions. By ansatz, 9
Substituting that in the equation (10), ( ) This gives us the required two independent solutions and the general solution to the equation of motion (10): ( ) ( ) One can determine the values of with the initial conditions. if provided initial conditions one should be able to see the motion of Foucault pendulum similar to figure 3. Impact due to the Coriolis force: The better way to illustrate the impact due to the Coriolis force is by considering an example: Let a body be dropped from rest at a height (h) =100m above the surface of the earth and at a latitude 40 North, if we want to calculate the lateral displacement of the point of impact due to the Coriolis force. If the body has mass m, a Coriolis force of is seen to acts on the body in the rotating frame of the earth. We choose a frame with origin at the point on the earth's surface below the starting point of the body, with z-axis pointing south, y-axis pointing east and z-axis pointing vertically up as in the figure below : then the equation of motion : Figure 5 a frame with origin at the point on the earth's surface below the starting point of the body, with x- axis pointing south, y-axis pointing east and z-axis pointing vertically up. From the above, expressions for can be obtained, which are readily integrated to give, These results are then used in the expressions for x, y and z. as the time of the drop of the body is short compared with the period of rotation of the earth, we can ignore terms of order and write the following. Integrating the above twice and using the initial conditions, we obtain The last equation gives the time of arrival of the body at the earth's surface z = 0: 10
Then the lateral displacement of the body at impact is The impact is very small that s why that Coriolis Effect could only be seen for the large scale motions. Summary: Coriolis force can be interpreted as the difference between centrifugal and gravitational force. One aspect of Coriolis force can be seen to arise from movement towards the axis of rotation to point where the speed relative to the inertial frame is smaller, or away from the axis of rotation to a point where the speed relative to the inertial frame is larger. Reference: [1] From Newton s Laws to Deterministic Chaos fifth edition, Prof. Dr. Florian Scheck. Springer Heidelberg Dordrecht London New York. [2] Classical mechanics Taylor, John R. (John Robert), University Science Books. [3] An article The Coriolis Effect a conflict between common sense and mathematics by Anders Persson, The Swedish Meteorological and Hydrological Institute, Norrköping. 11