User:John R. Brews/Coriolis force

The Coriolis force is a force experienced by a object traversing a path in a rotating framework that is proportional to its speed and also to the sine of the angle between its direction of movement and the axis of rotation. It is one of three such inertial forces that appear in an accelerating frame of reference due to the acceleration of the frame, the other two being the centrifugal force and the Euler force. The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist Gaspard-Gustave Coriolis in connection with the theory of water wheels, and also in the tidal equations of Pierre-Simon Laplace in 1778.

Although sometimes referred to as an apparent force, it can have very real effects.

Reference frames
Newton's laws of motion are expressed for observations made in an inertial frame of reference, that is, in any frame of reference that is in straight-line motion at constant speed relative to the "fixed stars", an historical reference taken today to refer to the entire universe. However, everyday experience does not take place in such a reference frame. For example, we live upon planet Earth, which rotates about its axis (an accelerated motion), orbits the Sun (another accelerated motion), and moves with the Milky Way (still another accelerated motion).

The question then arises as to how to connect experiences in accelerating frames with Newton's laws that are not formulated for such situations. The answer lies in the introduction of inertial forces, which are forces observed in the accelerating reference frame, due to its motion, but are not forces recognized in an inertial frame. These inertial forces are included in Newton's laws of motion, and with their inclusion Newton's laws work just as they would in an inertial frame. Coriolis force is one of these inertial forces, the other two being the centrifugal force and the Euler force.

These motions are slight, but the Coriolis force does affect aiming artillery pieces and plotting transoceanic air flights. The way Coriolis forces work is illustrated below by a few examples.

Carousel
The rotating carousel is perhaps the most common example used to illustrate the effects of rotation upon the formulation of Newton's laws of motion and the introduction of inertial forces into these laws. A simple case is playing catch on the carousel. Standing on the rim of the carousel, a ball is tossed to another player standing at a different position on the rim.

As shown in the figure, from the viewpoint of a stationary observer, the tossed ball is a free body (apart from downward gravity) so its horizontal motion is that of a free body: a straight line at constant speed. The catcher of the ball rotates during the flight of the ball, so the tosser must anticipate just where the catcher will be and tosses the ball toward that future position, rather than directly at the catcher.

From the viewpoint of an observer at the center of the carousel but rotating with it, both the person tossing the ball and the catcher are standing at fixed locations, as they both turn with the carousel. However, the tosser cannot toss the ball directly toward the catcher, because the ball veers off to the right, assuming the carousel in counter-clockwise rotation. To the rotating observer, the curved path means that Newton's laws require a force to cause the ball to curve. The intuitive reaction is that the ball is pushed to the right during flight, and so this force must be countered by throwing the ball somewhat to the left.

For both the rotating and the fixed observer, the distance of the ball from the center of the carousel at any moment in time is exactly the same, as shown by the arrows in the figure that indicate the moment of closest approach of the ball to the center.

How can these observations be formalized?

Inertial frame
The basic question is how to determine the angle at which the ball must be launched so it will arrive at the rim of the carousel exactly when the catcher arrives there. The answer is pretty straightforward. If the carousel rotates with angular velocity &omega;, then in time t the catcher rotates an angle &omega;t. If the horizontal rate of travel of the ball is v, it will travel a distance vt. The actual distance it must cover is chord length of the carousel subtending the angle &theta; + &omega;t, where &theta; is the angle between the thrower and the catcher at the beginning of the throw. For a carousel of radius R, this chord is the distance:
 * $$d = 2 R \sin \left(\frac{\theta + \omega t}{2} \right) \ . $$

The time allotted is now d/v. The distance d is then:


 * $$ d = 2 R \sin \left( \frac{\theta +\omega d/v}{2} \right ) \, $$

an implicit equation for d.

The angle &phi; at which the ball must be tossed is now:


 * $$ \varphi = \frac{\pi-\theta - \omega d/v}{2} \, $$

where, assuming the catcher is always on the right of the diameter drawn through the tosser, &phi; is the angle measured to the right of this diameter at which the ball must be tossed.

For example, if d = 2R, the relation must hold:


 * $$\frac{\theta +\omega 2R/v}{2} = \pi/2 \, $$

or


 * $$\theta = \pi-\omega 2R/v \ . $$

As it must, for d = 2R, the angle &phi; = 0 and the tosser must throw the ball directly across the carousel.

Rotating frame
The description in the rotating frame is more complex than in the inertial frame. One approach to the analysis of this case is to observe that the ball traces a circular path from the tosser to the catcher. Choosing some point as center for a prospective path, a circle can be drawn that includes both the tosser and the catcher and with the same radius as the carousel. To travel this path requires a centripetal force v2/R, and the speed v is set by the way the ball is tossed. Consequently, the time of travel between toss and catch can be found, and the center of the proposed circle is set to make this time the actual time observed.

Apparent motion of stationary objects
Though centrifugal force adequately describes the force on objects at rest relative to a steadily rotating frame of reference, the fictitious force on objects moving in the rotating frame includes the Coriolis force. In the figure, the vector Ω represents the rotation of the frame at angular rate ω; the vector v shows the velocity tangential to the circular motion as seen in the rotating frame. The vector Ω × v is found using the right-hand rule for vector cross products. It is related to the negative of the Coriolis force (the Coriolis force is −2 m Ω × v).

To deal with motion directly in a rotating frame of reference by applying Newton's laws, it is necessary to take these pseudo-forces into account. For example, consider the apparent revolution of a stationary object (such as a distant star or planet), which is in motion as viewed from the rotating frame:

A body that is stationary relative to a non-rotating inertial frame appears to be rotating when viewed from a frame rotating at angular rate Ω. Therefore, application of Newton's laws to what looks like circular motion in the rotating frame at a radius r, requires an inward centripetal force of −m Ω2 r to account for the apparent circular motion. This centripetal force in the rotating frame is provided as a net fictitious force that is the sum of the radially outward centrifugal force m Ω2 r and the Coriolis force −2m Ω × vrot. To evaluate the Coriolis force, we need the velocity as seen in the rotating frame, vrot. As explained in the Derivation section, this velocity is given by −Ω × r. This rotational motion, following the fixed stars, diminishes the apparent centrifugal force (the Eötvös effect), because the Coriolis force points inward when $$2m \boldsymbol{\Omega}\times \left[ d \mathbf{r}/dt \right]$$ is evaluated for $$\left[ d \mathbf{r}/dt \right]$$ in the direction of vrot. This inward force has the value −2m Ω2 r. The combination of the centrifugal and Coriolis force is then m Ω2 r − 2m Ω2 r = −m Ω2 r, exactly the centripetal force required by Newton's laws for circular motion.

Velocity
In a rotating frame of reference bodies can have the same positions as in a nonrotating frame, but they will move differently due to the rotation of the frame. Consequently, the time derivatives of any position vector r depending on time (velocity dr/dt and acceleration d2r/dt2) will differ according to the rotation. When time derivative [dr/dt] is evaluated from a reference frame with a coincident origin at r=0 but rotating with the absolute angular velocity Ω:

$$\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t} = \left[\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t}\right] + \boldsymbol{\Omega} \times \boldsymbol{r}\ ,$$

where × denotes the vector cross product and square brackets '[…]' denote evaluation in the rotating frame of reference. In other words, the apparent velocity in the rotating frame is altered by the amount of the apparent rotation $$\boldsymbol{\Omega} \times \boldsymbol{r}$$ at each point, which is perpendicular to both the vector from the origin 'r' and the axis of rotation 'Ω' and directly proportional in magnitude to each of them. The vector 'Ω' has magnitude 'Ω' equal to the rate of rotation and is directed along the axis of rotation according to the right-hand rule. The angle must be measured in radians per unit of time.

Acceleration
Newton's law of motion for a particle of mass m can be written in vector form as
 * $$\boldsymbol{F} = m\boldsymbol{a}\ ,$$

where 'F' is the vector sum of the physical forces applied to the particle and 'a' is the absolute acceleration of the particle, given by:


 * $$ \boldsymbol{a}=\frac{\operatorname{d}^2\boldsymbol{r}}{\operatorname{d}t^2} \, $$

where 'r' is the position vector of the particle. The differentiations are performed in the inertial frame.

By twice applying the transformation above from the inertial to the rotating frame, the absolute acceleration of the particle can be written as:


 * $$\begin{align}

\boldsymbol{a} &=\frac{\operatorname{d}^2\boldsymbol{r}}{\operatorname{d}t^2} = \frac{\operatorname{d}}{\operatorname{d}t}\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t} = \frac{\operatorname{d}}{\operatorname{d}t} \left( \left[\frac{\operatorname{d}\boldsymbol{r}}{\operatorname{d}t}\right] + \boldsymbol{\Omega} \times \boldsymbol{r}\ \right) \\ &= \left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] + \frac{\operatorname{d} \boldsymbol{\Omega}}{\operatorname{d}t}\times\boldsymbol{r} + 2 \boldsymbol{\Omega}\times \left[ \frac{\operatorname{d} \boldsymbol{r}}{\operatorname{d}t} \right] + \boldsymbol{\Omega}\times ( \boldsymbol{\Omega} \times \boldsymbol{r}) \. \end{align} $$

Force
The apparent acceleration in the rotating frame is [d2r/dt2]. An observer unaware of the rotation would expect this to be zero in the absence of outside forces. However Newton's first law applies only in the inertial frame, to the absolute acceleration d2r/dt2. Therefore the observer perceives the extra terms as accelerations imposed by fictitious forces. These terms in the apparent acceleration are independent of mass; so it appears that each of these fictitious forces, like gravity, pulls on an object in proportion to its mass. When these forces are added, the equation of motion has the form:


 * $$\boldsymbol{F} - m\frac{\operatorname{d} \boldsymbol{\Omega}}{\operatorname{d}t}\times\boldsymbol{r} - 2m \boldsymbol{\Omega}\times \left[ \frac{\operatorname{d} \mathbf{r}}{\operatorname{d}t} \right] - m\boldsymbol{\Omega}\times (\boldsymbol{\Omega}\times \boldsymbol{r}) $$&ensp;$$ = m\left[ \frac{\operatorname{d}^2 \boldsymbol{r}}{\operatorname{d}t^2} \right] \ ,$$

which, from a formal mathematical standpoint, is the same result as simply moving the extra acceleration terms to the left hand side (the force side) of the equation. From the viewpoint of the rotating frame, however, the terms on the force side all result from forces really experienced as forces. The terms on the force side of the equation can be recognized as the Euler force $$m \operatorname{d}\boldsymbol{\Omega}/\operatorname{d}t \times\boldsymbol{r}$$, the Coriolis force $$2m \boldsymbol{\Omega}\times \left[ \operatorname{d} \boldsymbol{r}/\operatorname{d}t \right]$$, and the centrifugal force $$m\boldsymbol{\Omega}\times (\boldsymbol{\Omega}\times \boldsymbol{r})$$, respectively. The centrifugal force points directly away from the axis of rotation of the rotating reference frame, with magnitude mΩ2r.

Notice that for a non-rotating inertial frame of reference $$(\boldsymbol\Omega=0)$$ the centrifugal force and all other fictitious forces disappear.

Examples
Below several examples illustrate both the inertial and rotating frames of reference, and the role of centrifugal force and its relation to Coriolis force in rotating frameworks. For more examples see Fictitious force, rotating bucket and rotating spheres.

Straight-line motions
The principles described above are illustrated by considering two example motions: (i) a straight-line motion as seen in a stationary frame, and (ii) a straight-line motion as seen in a rotating frame.

In the first case, straight line motion as seen in a stationary frame, no external force is required according to Newton's law of inertia. However, in the rotating frame, this straight-line motion appears curved; if the laws of Newtonian mechanics were to apply in this rotating frame, an external force would be required in order to account for the curved motion. These points of view can be reconciled by introducing pseudo-forces, or fictitious forces, in the rotating frame. A centrifugal force is attributed to every mass in the rotating frame. A non-zero Coriolis force also acts upon the mass if its trajectory is at an angle to the axis of rotation. In order for the rotating observer to agree with the stationary observer that no net external force acts on the mass, the sum of the centrifugal force and Coriolis force must be exactly equal to the force required by the curved motion. In this way the laws of Newtonian mechanics are now applicable in both the stationary and rotating frames of reference.

In the second case, straight-line motion as seen in a rotating frame, the roles of the observers are reversed. This time the rotating observer sees straight-line motion, and so finds no net force is required to support the motion. On the other hand, the stationary observer sees a curved motion, and so believes a external force is required. Again, these viewpoints are reconciled via fictitious forces acting on all masses in the rotating frame. If these forces are unopposed, they cause curved motion. Hence, straight-line motion observed in the rotating frame must find an external influence at work that exactly balances these fictitious forces. Thus, the two observers are in agreement once again, but in this case an external force is required for the observed motion.