Velocity And Acceleration In Spherical Coordinates Pdf

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In mathematics , a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuthal angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system. The radial distance is also called the radius or radial coordinate.

In-orbit rendezvous is a key enabling technology for many space missions. Implementing it employing only bearing measurements would simplify the relative navigation hardware currently required, increasing robustness and reliability by reducing complexity, launch mass and cost. The problem of bearings-only navigation has been studied intensively by the Naval and Military communities.

Derivation of the Equations of Atmospheric Motion in Oblate Spheroidal Coordinates

System Simulation and Analysis. Plant Modeling for Control Design. High Performance Computing. This worksheet is intended as a brief introduction to dynamics in spherical coordinates.

Definition and Sketch. Consider a point P on the surface of a sphere such that its spherical coordinates form a right handed triple in 3 dimensional space, as illustrated in the sketch below. These coordinates are usually referred to as the radius, polar angle or co-latitude and azimuth or longitude respectively, and follow the so-called American convention.

For a fuller definition see Maple's help pages at sphericalcoordinatesDefinition. The transformation equations from spherical to Cartesian coordinates are:.

The transformation equations from Cartesian to spherical coordinates are:. Denoting vectors by bold face type, let r be the vector joining the centre of the sphere to P and be its unit vector.

Let v and a be the velocity and acceleration respectively of P. In what follows, it is to be understood that r , v and a , together with their magnitudes in the directions of the unit vectors, are functions of time. In order to present equations that are easier to read the usual functional notation t will be omitted.

The velocity of P is given by. Expanding the second term on the rhs of the above,. Substituting this in equation a. It will be noted that this equation contains no derivatives of the unit vectors. The two rightmost terms are tangential velocities and are much as might be expected from experience with polar coordinates. The acceleration, a , of the point P is obtained by differentiating v w.

The result is:. Again, this equation contains no derivatives of the unit vectors: indeed none are present in any of the higher derivatives of r. The three double differentials w. The terms in and are accelerations toward the centre and the polar axis OZ respectively. The terms with a coefficient of 2 are Coriolis accelerations. The Coriolis term in the coordinate is as might be expected from experience with polar coordinates.

Force and Moment. If m is the mass of a particle at the point P , the force, F , needed to accelerate it is,. The moment or torque needed to produce the acceleration is obtained from. The components of the moment may be thought of as right-handed clockwise rotation vectors with their axes along the angular unit vectors.

The velocity and acceleration shown above are now written as Maple input so that they can be readily copied and pasted into a worksheet. The unit vectors , are inconvenient in a worksheet and will now be written as and respectively, which is the format used in Maple's Vector Calculus package. A derivation of the above table of partial derivatives is given below. It is not a proof; it is but one of several ways of demonstrating the validity of the table.

Ox , Oy and Oz represent the Cartesian coordinates of the point P. It will be seen from the sketch that Ox and Oy can be resolved to form Oq , which itself resolves into components in the directions of the unit vectors and. The components of Ox and Oy perpendicular to Oq form the vector. Let xyz represent the unit vectors in Cartesian coordinates. Let it be called c2s. The orthogonality of c2s is now tested and its determinant is obtained.

Since the transformation matrix, c2s , is orthogonal, the spherical coordinates are orthogonal; and since they were defined as such, this acts as a check on the validity of the transformation matrix. Again, they were defined as such, and this acts as a further check on the validity of the transformation matrix. In passing, it is worth noting that the transpose of c2s is the transformation matrix from spherical to Cartesian coordinates.

The unit vectors in spherical coordinates are given by,. An inert matrix of the partial derivatives of the unit vectors in spherical coordinates is now constructed, thus:. This inert matrix is equated to the Jacobian of the unit vectors in spherical coordinates i. Noting that the magnitudes of the Cartesian vectors in 7. Legal Notice: The copyright for this application is owned by the author s. Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material.

This application is intended for non-commercial, non-profit use only. If you wish to use this application in for-profit activities, please make a donation to Cancer Research. App Preview: Dynamics in Spherical Coordinates You can switch back to the summary page for this application by clicking here. Learn about Maple Download Application. Application Details Author: J. Community Rating:. Tell others about this application! Maple MapleSim Maple T.

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Spherical coordinate system

By changing the display options, we can see that the basis vectors are tangent to the corresponding coordinate lines. Spherical coordinates are defined with respect to a set of Cartesian coordinates, and can be converted to and from these coordinates using the atan2 function as follows. To convert from Cartesian coordinates, we use the same projection and read off the expressions for the spherical coordinates. If the spherical coordinates change with time then this causes the spherical basis vectors to rotate with the following angular velocity. The rotation of the basis vectors caused by changing coordinates can be directly computed, giving the time derivatives below. This gives:.

System Simulation and Analysis. Plant Modeling for Control Design. High Performance Computing. This worksheet is intended as a brief introduction to dynamics in spherical coordinates. Definition and Sketch.

Curl The curl is evaluated in exactly the same way from the expression above, but it ends up being much more complex:. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. This is even known as inclination and this is represented by The divergence of function f in Spherical coordinates is, The curl of a vector is the vector operator which says about the revolution of the vector. If the particle P moves along a space curve, its position can be written as. The basic content of the divergence theorem is the following: given that the divergence is a measure of the net outflow of flux from a volume element, the sum of the net outflows from all volume elements of a 3-D region as calculated from the divergence must be equal to the total outflow from the region as calculated from the flux through the closed surface bounding the region.


The velocity and acceleration of a particle may be expressed in spherical coordinates by taking into account the associated rates of change in the unit vectors.


Divergence And Curl In Polar Coordinates

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These statements are based on the observations that Earth is nearly a sphere and that the depth of the atmosphere or at least the depth over which the bulk of its mass lies is small in comparison to Earth's mean radius. From at least a theoretical point of view, however, it would be desirable to derive the equations of atmospheric motion taking into account the earth's equational bulge caused by the earth's rotation , and then to make a systematic comparison with the spherical formulation. Although all of the global models currently used to simulate the atmospheric general circulation and climate are based on the spherical approximation, this is probably not the cause of the models' major errors.

Theoretical Mechanics

If you make a print-out, you should be aware that some printers apparently do not print Greek letter symbols in boldface, even though they appear in boldface on screen. You should be on the look-out for this. If in doubt, look at what appears on the screen.

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 - Он японец, а не китаец. Беккер поднял. Лейтенант листал паспорт умершего. - Я бы предпочел, чтобы вы ни к чему не прикасались, - попросил. Ничего не трогайте. Ничего не читайте. - Энсей Танкадо… родился в январе… - Пожалуйста, - вежливо сказал Беккер.

Она была установлена на каждом терминале в Третьем узле. Поскольку компьютеры находились во включенном состоянии круглые сутки, замок позволял криптографам покидать рабочее место, зная, что никто не будет рыться в их файлах. Сьюзан ввела личный код из пяти знаков, и экран потемнел. Он будет оставаться в таком состоянии, пока она не вернется и вновь не введет пароль. Затем Сьюзан сунула ноги в туфли и последовала за коммандером.

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Я все. - Довольно, Грег, - тихо сказал Стратмор. Хейл крепче обхватил Сьюзан и шепнул ей на ухо: - Стратмор столкнул его вниз, клянусь. - Она не клюнет на твою тактику разделяй и властвуй, - сказал Стратмор, подходя еще ближе.

Хейл влез в ее компьютер, когда она выходила. Именно он и подал ручную команду на отзыв Следопыта. Вопрос насколько. уступил место другому - с какой целью?. У Хейла не было мотивов для вторжения в ее компьютер.

А ведь у нас в Третьем узле нет друг от друга секретов. Один за всех и все за одного. Сьюзан отпила глоток чая и промолчала. Хейл пожал плечами и направился к буфету.

Это на нижнем этаже. Возле фреоновых помп. Сьюзан повернулась и направилась к двери, но на полпути оглянулась. - Коммандер, - сказала.  - Это еще не конец.

Тонкие губы Клушара изогнулись в понимающей улыбке.

Пестрое сборище пьяных и накачавшихся наркотиками молодых людей разразилось истерическим хохотом. Двухцветный встал и с презрением посмотрел на Беккера. - Чего вы от меня хотите. Беккер задумался: Я бы хотел, чтобы ты как следует вымыл голову, научился говорить по-человечески и нашел себе работу. Но решил, что хочет от этого парня слишком многого.

Она с самого начала возражала против его кандидатуры, но АНБ посчитало, что другого выхода. Хейл появился в порядке возмещения ущерба. После фиаско Попрыгунчика.

 У Танкадо наверняка была при себе копия ключа в тот момент, когда его настигла смерть. И я меньше всего хотел, чтобы кто-нибудь в севильском морге завладел ею.

3 Response
  1. Brezovlousby1975

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  2. Miguelina H.

    in two dimensions and cylindrical and spherical coordinates in three dimensions. We shall see that when calculating the velocity and acceleration. Over an.

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