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]
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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
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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)
◦
10
◦
20
◦
30
◦
40
◦
50
◦
60
◦
70
◦
80
Rotational velocity (km/h)
1644.4
1569.1
1446.1
1279.1
1073.3
834.9
571.1
289.95
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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
́
́
Figure 1 the coordinate system K´ rotates about the
system k with the angular velocity ω
(1)
́
: Time derivative as observed from K´
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: 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 =
(
)
and centrifugal force =
If ω = 0 than these fictional forces vanish.
(4)
(5)
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.
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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.
Northward
deflection
Eastward deflection
Downward
deflection
Northward motion
0
Eastward motion
-1
Downward motion
0
1
0
1
0
-1
0
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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
̈
̇
( )
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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,
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( )
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 xaxis 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:
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√
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.
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