Formulate deviation towards the east

The formula of deviation towards the east is a simplified form of the vectorial representation of the inertia of Coriolis which makes it possible to calculate the Déviation towards the east of a body in freefall in the terrestrial atmosphere.

The length of this deviation is given by the approximate formula:

$D\; =\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash \; Omega\; \backslash \; cdot\; T\_0\; \backslash \; cdot\; H\; \backslash \; cdot\; \backslash \; cos\; (L)$

where:

• $\backslash \; Omega\; =\; \backslash \; frac\; \{2\; \backslash \; pi\}\; \{T\}$ represents the angular velocity of rotation of the ground,
• $T\_0=\; \backslash \; sqrt\; \{\backslash \; frac\; \{2h\}\; \{G\}\}$ is the drop time,
• H is the drop height,
• $L$ the latitude to which the experiment is carried out (0 at the equator, $\backslash \; frac\; \{\backslash \; pi\}\; \{2\}$ with the poles).

Rigorous equation

One leaves the vectorial representation of the force of Coriolis, and one replaces the Flight Path Vector by the derivative of the vector position:

$\backslash \; vec\; \{F\_C\}\; =\; -\; 2m\; \backslash \; cdot\; (\backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{K\})\; \backslash \; wedge\; \backslash \; frac\; \{D\; \backslash \; vec\; \{AM\}\}\; \{dt\}$

where:

• $\backslash \; vec\; \{K\}$ is parallel to the axis of rotation of the Earth, directed towards north;
• has is the point of origin, reference frame turning related to the surface of the Earth.

The basic principle of dynamics makes it possible to write acceleration like summons attraction force of the Earth and force of Coriolis:

$\backslash \; frac\; \{d^2\; \backslash \; vec\; \{AM\}\}\; \{dt^2\}\; =\; \backslash \; frac\; \{1\}\; \{m\}\; (m\; \backslash \; vec\; \{G\}\; +\; \backslash \; vec\; \{F\_C\})\; =\; G\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; -\; 2\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; frac\; \{D\; \backslash \; vec\; \{AM\}\}\; \{dt\}$

where $\backslash \; vec\; \{U\}$ is directed according to the downward vertical.

One integrating once to find speed:

$\backslash \; frac\; \{D\; \backslash \; vec\; \{AM\}\}\; \{dt\}\; =\; WP\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; -\; 2\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; vec\; \{AM\}\; +\; \backslash \; vec\; \{0\}$

initial speed being null.

One integrates again to obtain the position of the body according to time:

$\backslash \; vec\; \{AM\}\; =\; \backslash \; frac\; \{1\}\; \{2\}\; gt^2\; \backslash \; cdot\; \backslash \; vec\; \{U\}\; +\; \backslash \; vec\; \{D\}\; (T)$

where the deviation is given by: $\backslash \; vec\; \{D\}\; (T)\; =\; -\; 2\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; int\_0^t\; \backslash \; vec\; \{AM\}\; dt\; +\; \backslash \; vec\; \{0\}$ (because has is the origin).

First order approximation

The deviation towards the east being small in front of the deviation due to gravity, one takes as approximation:

$\backslash \; vec\; \{AM\}\; (T)\; \backslash \; approx\; \backslash \; frac\; \{1\}\; \{2\}\; gt^2\; \backslash \; cdot\; \backslash \; vec\; \{U\}$

from where the result:

$\backslash \; vec\; \{D\}\; (T)\; =\; -2\; \backslash \; Omega\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; int\_0^t\; (\backslash \; frac\; \{1\}\; \{2\}\; gt^2\; \backslash \; cdot\; \backslash \; vec\; \{U\})\; dt\; =\; -2\; \backslash \; Omega\; \backslash \; frac\; \{1\}\; \{2\}\; G\; \backslash \; cdot\; \backslash \; frac\; \{t^3\}\; \{3\}\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; vec\; \{U\}$

who is valid if D is small in front of H , i.e. for T ₀ (drop time) small in front of T = 86  164  S (sidereal period):

$D\; (T\_0)\; =\; -\; \backslash \; Omega\; G\; \backslash \; frac\; \{T\_0^3\}\; \{3\}\; \backslash \; cdot\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; vec\; \{U\}\; =\; \backslash \; frac\; \{1\}\; \{3\}\; \backslash \; Omega\; \backslash \; cdot\; (G\; \backslash \; cdot\; T\_0^2)\; \backslash \; cdot\; T\_0\; \backslash \; cdot\; -\; \backslash \; vec\; \{K\}\; \backslash \; wedge\; \backslash \; vec\; \{U\}$

maybe, in absolute value:

$D\; =\; \backslash \; frac\; \{2\}\; \{3\}\; \backslash \; Omega\; \backslash \; cdot\; T\_0\; \backslash \; cdot\; H\; \backslash \; cdot\; \backslash \; cos\; (L)$

Complements

• Another term, even weaker, gives a deviation towards the south in the northern hemisphere, and towards north in the southern hemisphere: it is worth $-\; \backslash \; frac\; \{1\}\; \{6\}\; G\; \backslash \; omega^2\; \backslash \; cdot\; t^4\; \backslash \; cdot\; sin\; (L)\; \backslash \; cdot\; cos\; (L)$.

• a great question that the theorists were posed: by reducing the ground to a central mass point, which would be the deviation on a fall of R = 6  400 km ? By using the ellipse of Kepler, one finds: D = parameter of the ellipse = R ·(1/17) ² = R /289 is approximately 22 kilometers.
• One can notice the relationship of the equation of the force of Coriolis with that of the Hall effect in electricity.

History

Huygens (1629-1695) which was on the track of the force of Coriolis with its theory of the total Relativité, did not make the weight vis-a-vis Isaac Newton and his concepts of time and absolute space. This last choked by the monumentality of its work any relativistic attempt, and he did not want to handle the inertias. The Force of Coriolis (1792-1843) will be stated only tardily at the 19th century.

Newton made the demonstration of the deviation towards the east in 1679 in the “absolute” reference frame geocentric. To make simple, let us take the equatorial case. He notices, but he was not the first to make it, that the point has had a speed ω·(R + H ), that is to say a speed higher than the speed of the point O located on the ground at the downward vertical of has . This difference in speed corresponds at a low speed towards the east of ω· H , therefore the deviation towards the east is simply ω· H · T ₀.

It was mistaken thus in a factor 2/3. It quickly took it into account by understanding that, for also a long trajectory, it was necessary to take into account the change of management of gravity, on a round ground and not punt: the trajectory is thus not parabolic.

Random links:

Cuadrilla de Scooby | Homogenic | Genest-Saint-Isle | April in right | The Morin company | Yossef-Haïm Brener | Prospérité,_la_Virginie_Occidentale

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org