Physics
Two Dimensional Polar Coordinates
Two-dimensional polar coordinates are a way of representing points in a plane using a distance and an angle. The distance is measured from the origin, while the angle is measured counterclockwise from the positive x-axis. This system is useful in physics for describing circular motion and other phenomena that involve angles and distances.
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12 Key excerpts on "Two Dimensional Polar Coordinates"
eBook - PDF- P. C. Deshmukh(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
The description of planetary motion becomes extremely complicated. We have learned that physical quantities are represented by scalars, vectors, and tensors. We now proceed to discuss how the components of a vector are described in the plane polar coordinate system. We begin our consideration with 2-dimensional vectors and then extend it to three-dimensions. On a 2-dimensional surface, we know that the position vector is expressed as a linear superposition of the Cartesian base vectors, (e x , e y ): r = xe x + ye y . (2.6a) The same vector can also be expressed as a superposition of any two linearly independent base vectors, which are easily chosen to be orthogonal to each other. Therefore we can use a 43 Mathematical Preliminaries pair of base vectors (e r , e j ) such that e r points along the direction of the position vector itself, and e j is orthogonal to it in the direction in which the angle j, which the position vector r makes with the coordinate X-axis, increases in the anticlockwise sense (Fig. 2.11). The angle j is called the azimuthal angle. The location of the point at a certain distance from the origin, at the tip of the position vector, r = re r , (2.6b) is uniquely specified by the pair (x, y) of the Cartesian coordinates, and also, equivalently, by the pair (r, j) of the plane polar coordinates. Y e j e y e r j e x r j O X Fig. 2.11 Plane polar coordinates of a point on a 2-dimensional surface. An arbitrary point on this surface can be uniquely represented by the pair (r, j) just as well as by its Cartesian coordinates (x, y). Note that while the Cartesian coordinates x and y both have the dimensions of length, only the distance r of the plane polar coordinates has the dimension of length. The angle j is of course dimensionless. Thus j = constant is an infinite line, or, rather, an infinite half-line, since it resides strictly in only one quadrant. Its extension in the diagonally opposite quadrant corresponds to the angle j + p, and not j.
eBook - PDFEngineering Mechanics
Dynamics
- Benson H. Tongue, Daniel T. Kawano(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Assume that the wheel is attached to a car that is traveling to the 52 CH 2 MOTION OF TRANSLATING BODIES 2.3 POLAR AND CYLINDRICAL COORDINATES Learning Objective: Describe a particle’s motion in polar and cylindrical coordinates. The next set of coordinates we will examine are called polar coordinates because they are radially based, like a map of the world would be if it were centered on the North or South Pole, showing lines of latitude and longitude. A typical polar plot is shown in Figure 2.3.1. Rather than determining the position of a body P by going forward some amount and then right some amount, as we did in Figure 2.2.2, the polar approach determines P’s position by the distance (r) it is from the origin and the angle that P’s position vector makes with the positive horizontal axis. The radial unit vector e r always points from the origin to the body. The total position vector is given by r P P O = r e r (2.41) which says that to get to P, you must go a distance r in the e r direction. Unlike the Cartesian case, in which we needed two terms to define position, in the polar case we need just one. You would use this viewpoint, for example, if you were an airport traffic controller. Because you were using radar, the information you would have about an airplane would be its straight-line distance from you (r) and its vertical inclination (). As just mentioned, e r is not oriented permanently vertical or per- manently horizontal as a Cartesian unit vector would be. Rather, it is determined by where P is with respect to the origin O. The unit vector e r always points from the origin to P, and therefore if P moves around O, the orientation of e r will change. This is going to make life difficult when we try to find velocities and accelerations. But before we do that, let’s introduce e , the other unit vector we’ll be using (and which we’ll call the angular unit vector). This vector is found by imagining that the length r is fixed and then increasing .
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
In two dimensions, this description is accomplished with the use of the Cartesian coor- dinate system, in which perpendicular axes intersect at a point defined as the origin O (Fig. 3.1). Cartesian coordinates of a point in space, representing the x and y values of the point, and expressed as ( x , y), are also called rectangular coordinates. Sometimes it is more convenient to represent a point in a plane by its plane polar coordinates ( r , u) as shown in Figure 3.2a. In this polar coordinate system, r is the dis- tance from the origin to the point having Cartesian coordinates ( x, y) and u is the angle between a fixed axis and a line drawn from the origin to the point. The fixed axis is often the positive x axis, and u is usually measured counterclockwise from it. From the right triangle in Figure 3.2b, we find that sin u 5 y/ r and that cos u 5 x/ r. (A review of trigonometric functions is given in Appendix B.4.) Therefore, starting with the plane polar coordinates of any point, we can obtain the Cartesian coordi- nates by using the equations x 5 r cos u (3.1) y 5 r sin u (3.2) Conversely, if we know the Cartesian coordinates, the definitions of trigonometry tell us that the polar coordinates are given by tan u 5 y x (3.3) r 5 Ïx 2 1 y 2 (3.4) Equation 3.4 is the familiar Pythagorean theorem. These four expressions relating the coordinates ( x, y) to the coordinates ( r , u) apply only when u is defined as shown in Figure 3.2a—in other words, when pos- itive u is an angle measured counterclockwise from the positive x axis. (Some sci- entific calculators perform conversions between Cartesian and polar coordinates based on these standard conventions.) If the reference axis for the polar angle u is chosen to be one other than the positive x axis or if the sense of increasing u is cho- sen differently, the expressions relating the two sets of coordinates will be different from those above.
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
In Section 1.9 we learned how to graph points in rectangular coordinates. In this chapter we study a different way of locating points in the plane, called polar coordinates. Using rectangular coordinates is like describing a location in a city by saying that it’s at the corner of 2nd Street and 4th Avenue; these directions would help a taxi driver find the location. But we may also describe this same location “as the crow flies”; we can say that it’s 1.5 miles northeast of City Hall. These directions would help an airplane or hot air balloon pilot find the location. So instead of specifying the location with respect to a grid of streets and avenues, we specify it by giving its distance and direction from a fixed reference point. That’s what we do in the polar coordinate system. In polar coordinates the location of a point is given by an ordered pair of numbers: the distance of the point from the origin (or pole) and the angle from the positive x-axis. Why do we study different coordinate systems? It’s because certain curves are more naturally described in one coordinate system rather than another. For example, in rectangular coordinates lines and parabolas have simple equations, but equations of circles are rather complicated. We’ll see that in polar coordinates circles have very simple equations. 587 Polar Coordinates and Parametric Equations 8 8.1 Polar Coordinates 8.2 Graphs of Polar Equations 8.3 Polar Form of Complex Numbers; De Moivre’s Theorem 8.4 Plane Curves and Parametric Equations FOCUS ON MODELING The Path of a Projectile © gary718/Shutterstock.com Copyright 2017 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
eBook - PDFFundamental Maths
For Engineering and Science
- Mark Breach(Author)
- 2017(Publication Date)
- Red Globe Press(Publisher)
20 Coordinate systems What this chapter is about In this chapter you will investigate spatial relationships in a two-dimensional plane. By the end of this chapter you will be able to calculate distance and orientation in a two-dimensional plane, the areas of straight-sided figures, find the intersection of lines, and calculate parameters and formulae of circles. You will be able to change from Cartesian to polar coordinates and vice versa and to apply Pythagoras’ theorem in three dimensions. 20-01 Cartesian coordinate system applications The Cartesian coordinate system was introduced in Chapter 6 and it is now appropriate to look at some of its implications. The first is that not everyone uses it in the same way as mathematicians. In the Cartesian coordinate system the primary reference direction is the x -axis and orientation is measured anticlockwise from it. Geographers, and particularly land and engineering surveyors, use a system in which the primary direction is north and orientation, such as a compass bearing, is measured clockwise from it. Although the fundamental mathematical relationships are not affected by this difference, their applications may be. 20-02 Distance and orientation between two points Two points in a coordinate frame may be joined by a straight line. The length of that line will be the distance between the two points. The orientation of the line is measured with 209 Figure 20.1 north east α P ( E P , N P ) x -axis y -axis α P ( x P , y P ) respect to the x -direction within the Cartesian coordinate system and with respect to the north direction in the geographical frame. An application of Pythagoras’ theorem can be used to find the length of the line, and the tangent of its orientation relates the sides of the right angled triangles in Figure 20.2.
eBook - PDF- David H. Eberly(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
The ideas we present here for the xy -plane apply directly to the coordinates within the general plane. Whether in two or three dimensions we may choose Cartesian coordinates to represent the particle’s position. However, some problems are better formulated in different coordinate systems. Particle motion is first discussed in Cartesian coordi- nates, but we also look at polar coordinates for 2D motion and at cylindrical or spherical coordinates for 3D motion, because these coordinate systems are the most common ones you will see in applications. Planar Motion in Cartesian Coordinates First let us consider when the particle motion is constrained to be planar. In Cartesian coordinates, the position of the particle at time t is r(t ) = x (t ) ı + y (t ) j (2.1) 16 Chapter 2 Basic Concepts from Physics where ı = (1,0) and j = (0,1). The velocity of the particle at time t is v(t ) = ˙ r = ˙ x ı + ˙ y j (2.2) The dot symbol denotes differentiation with respect to t . The speed of the particle at time t is the length of the velocity vector, |v|. If s (t ) denotes the arc length mea- sured along the curve, the speed is ˙ s = |v|. The quantity ˙ s = ds /dt is intuitively read as “change in distance per change in time,” what you expect for speed. The acceleration of the particle at time t is a(t ) = ˙ v = ¨ r = ¨ x ı + ¨ y j (2.3) At each point on the curve of motion we can define a unit-length tangent vector by normalizing the velocity vector, T(t ) = v |v| = (cos(φ(t )), sin(φ(t ))) (2.4) The right-hand side of equation (2.4) defines φ(t ) and is valid since the tangent vector is unit length. A unit-length normal vector is chosen as N(t ) = (−sin(φ(t )), cos (φ(t ))) (2.5) The normal vector is obtained by rotating the tangent vector π/2 radians counter- clockwise in the plane. A coordinate system at a point on the curve is defined by origin r(t ) and coordinate axis directions T(t ) and N(t ). Figure 2.1 illustrates the coordinate systems at a couple of points on a curve.
eBook - PDFEngineering Dynamics
A Comprehensive Introduction
- N. Jeremy Kasdin, Derek A. Paley(Authors)
- 2011(Publication Date)
- Princeton University Press(Publisher)
10.1.1 Cartesian Coordinates in Three Dimensions Recall that we can locate a particle in an inertial reference frame most directly by using Cartesian coordinates (x, y, z) I : r P/O = x e x + y e y + z e z , 1 We do not address three-dimensional path coordinates, as this is a more advanced subject. 410 CHAPTER TEN O z r P (a) I θ e y e z e x r P / O O r P (b) I θ ϕ e y e z e x r P / O Figure 10.1 (a) Cylindrical (r, θ, z) I and (b) spherical (r, θ, φ) I coordinates locate point P in reference frame I in three dimensions. where I = (O, e x , e y , e z ) . The inertial velocity and acceleration of P are found from vector differentiation and Definition 3.3: I v P/O = I d dt ( r P/O ) = ˙ x e x + ˙ y e y + ˙ z e z I a P/O = I d dt I v P/O = ¨ x e x + ¨ y e y + ¨ z e z . Finding the equations of motion in Cartesian coordinates involves applying Newton’s second law to each of the three directions, just as in Chapter 3. What about other coordinate systems? As in the planar case, it is often convenient to use another coordinate system to describe the position of P in I (usually because of constraints in the problem). The two most common three-dimensional coordinate systems other than Cartesian coordinates are the cylindrical coordinates shown in Figure 10.1a and the spherical coordinates shown in Figure 10.1b. 10.1.2 Cylindrical Coordinates Cylindrical coordinates (r, θ, z) I are a simple generalization of polar coordinates. All previous results for polar coordinates apply to the components of motion in the plane defined by e x and e y ; we simply add the third Cartesian coordinate z in the e z direction and apply Newton’s second law in that direction.
eBook - PDFCalculus in 3D
Geometry, Vectors, and Multivariate Calculus
- Zbigniew Nitecki(Author)
- 2018(Publication Date)
- American Mathematical Society(Publisher)
Cylindrical Coordinates a (vertical) plane, which can be regarded as the ?? -plane rotated so that the ? -axis moves 𝜃 radians counterclockwise (in the horizontal plane); we take as our coordinates the angle 𝜃 together with the coordinates of ? in this plane, which equal the distance ? of the point from the ? -axis and its (signed) distance ? from the ?? -plane. We can think of this as a hybrid: combine the polar coordinates (?, 𝜃) of the projection ? ?? with the vertical rectangular coordinate ? of ? to obtain the cylindrical coordinates (?, 𝜃, ?) of ? . Even though in principle ? could be taken as negative, in this system it is customary to confine ourselves to ? ≥ 0 . The relation between the cylindrical coordinates (?, 𝜃, ?) and the rectangular coordinates (?, ?, ?) of a point ? is essentially given by Equation (1.3): ? = ? cos 𝜃, ? = ? sin 𝜃, ? = ?. (1.4) We have included the last relation to stress the fact that this coordinate is the same in both systems. The inverse relations are given by ? 2 = ? 2 + ? 2 , cos 𝜃 = ? ? , sin 𝜃 = ? ? (1.5) and, for cylindrical coordinates, the trivial relation ? = ? . The name “cylindrical coordinates” comes from the geometric fact that the locus of the equation ? = ? (which in polar coordinates gives a circle of radius ? about the origin) gives a vertical cylinder whose axis of symmetry is the ? -axis, with radius ? . Cylindrical coordinates carry the ambiguities of polar coordinates: a point on the ? -axis has ? = 0 and 𝜃 arbitrary, while a point off the ? -axis has 𝜃 determined up to adding even multiples of 𝜋 (since ? is taken to be positive). Spherical Coordinates. Another coordinate system in space, which is particularly useful in problems involving rotations around various axes through the origin (for ex-ample, astronomical observations, where the origin is at the center of the earth) is the system of spherical coordinates . Here, a point ? is located relative to the origin ?
- Kuldip S. Rattan, Nathan W. Klingbeil, Craig M. Baudendistel(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
Two-Dimensional Vectors in Engineering CHAPTER 4 The applications of two-dimensional vectors in engineering are introduced in this chapter. Vectors play a very important role in engineering. The quantities such as displacement (position), velocity, acceleration, forces, electric and magnetic fields, and momentum have not only a magnitude but also a direction associated with them. To describe the displacement of an object from its initial point, both the distance and direction are needed. A vector is a convenient way to represent both magnitude and direction and can be described in either a Cartesian or a polar coordinate system (rectangular or polar forms). For example, an automobile traveling north at 65 mph can be represented by a two-dimensional vector in polar coordinates with a magnitude (speed) of 65 mph and a direction along the positive y-axis. It can also be represented by a vector in Cartesian coordinates with an x-component of zero and a y-component of 65 mph. The tip of the one-link and two-link planar robots introduced in Chapter 3 will be represented in this chapter using vectors both in Cartesian and polar coordinates. The concepts of unit vectors, magnitude, and direction of a vector will be introduced. 4.1 INTRODUCTION Graphically, a vector −− → OP or simply P with the initial point O and the final point P can be drawn as shown in Fig. 4.1. The magnitude of the vector is the distance between points O and P (magnitude = P) and the direction is given by the direction of the y Magnitude = P P x O θ Figure 4.1 A representation of a vector. 107 108 Chapter 4 Two-Dimensional Vectors in Engineering arrow or the angle in the counterclockwise direction from the positive x-axis as shown in Fig. 4.1. The arrow above P indicates that P is a vector. In many engineering books, the vectors are also written as a boldface P.
eBook - PDF- D.H. Maling(Author)
- 2013(Publication Date)
- Pergamon(Publisher)
The point £ = 0 m, TV = 0 m is referred to as the false origin of the grid to distinguish it from the point in latitude 49''N, 2''W which is the true origin. They way in which the shift has been applied may be imagined mathematically as the parallel shift of each axis through the defined distances. This is called translation of the axes. Plane polar coordinates Polar coordinates define position by means of one linear measurement and one angular measurement. The pair of orthogonal axes passing through the origin is replaced by a single fine OQ, in Fig. 2.04, passing through the origin O, or pole of the system. The position of any point A may be defined with reference to this pole and the polar axis or initial line, OQ by means of the distance OA = x and the angle QOA = Θ. The line OA is known as the radius vector and the angle θ is the vectorial angle True North G Ν G 34 Coordinate Systems and M a p Projections L 0 Q FIG. 2.04 Plane polar coordinates. which the radius vector makes with the initial line. Hence the position of A may be defined by the coordinates (r, Ö). The order of referring to the radius vector followed by the vectorial angle is standard to all branches of pure and applied mathematics. The vectorial angle may be expressed in sexagesimal (degree) or centisimal (grad) units to plot or locate a point instrumentally.* In the theoretical derivation of map projections, where θ enters directly into an equation and is not introduced as some trigonometric function of the angle, it is necessary to express this angle in absolute angular units, or radians. This is because both elements of the coordinate system must have the character of length. The direction in which the vectorial angle is measured depends upon the purpose for which polar coordinates are used. Usually the mathematician regards + Ö as the anticlockwise angle measured from the initial line. This is the sign convention which is used, for example, in vector algebra.
eBook - PDF- Dara W. Childs, Andrew P. Conkey(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
In planar kinematics, one-coordinate kinematic defi-nitions (using Cartesian, polar-, and path-coordinate systems) are much more commonly used (and appro-priate) than the two-coordinate kinematics results of Equation 2.69. However, the two-coordinate system kine-matic equations are frequently required to define the velocity and acceleration of a particle in three dimensions. 2.9.3 Coordinate System Observers The preceding material of this section concentrated on obtaining definitions for vectors and the derivatives of vectors with respect to coordinate systems. The immedi-ately preceding material concerns choices in stating vec-tor equations in coordinate systems. One can easily get lost in terms of vectors, components, direction cosines, etc., and simply wonder, “What do all these vector expressions mean in terms of something approximating everyday experience?” The concept of “observers” fixed in coordinate systems can be useful in interpreting vec-tor expressions, provides some answers to this question, and is the subject of this section. Figure 2.25 largely repeats the results of Figure 2.24; however, two “observers” have been added to the figure. Observers O and o are riding in the X , Y and x , y 32 Dynamics in Engineering Practice coordinate systems, respectively. Some of the terms in the vector kinematic expressions dotnosp dotnosp dotnosp circumflexnosp dotnospdotnosp dotnospdotnosp dotnospdotnosp circumflexnosp dotnosp circumflexnosp dotnosp r R r R = + + × ( ) = + + × ( ) + × ( ) + × × ( ) o o ρ ω ρ ρ ω ρ ω ρ ω ω ρ , , 2 can be explained in terms of the separate velocities and accelerations witnessed by these observers. Observer O is only assigned to watch (1) point o (the origin of the x , y system located by R o ), (2) point P (located by r ), and (3) θ , which defines the orientation of the x , y system with respect to the X , Y system.
eBook - PDF- A. Douglas Davis(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
6.3 Cylindrica l Pola r Coordinate s We ca n easil y exten d ou r notatio n t o thre e dimension s b y addin g a z-componen t an d creatin g cylindrical pola r coordinate s a s sketche d in Figur e 6.3.1 . æ Figur e 6.3. 1 Cylindrica l pola r coordinates . CHAPTE R 6 / COORDINAT E SYSTEM S 6.4 Spherica l Pola r Coordinate s On e o f th e mos t usefu l coordinat e system s is spherical polar coordinates (ofte n simpl y calle d spherical coordinates), a s show n in Figur e 6.4.1 . Suc h a syste m is usefu l wheneve r ther e is spherica l symmetry—fo r example , describin g th e force s bindin g a n electro n t o it s nucleus , givin g th e locatio n o f th e spac e shuttl e in orbi t aroun d Earth , o r chartin g th e cours e o f th e U.S.S. Enterprise throug h th e galaxy . In Figur e 6.4.1 , poin t Ñ is locate d b y givin g it s radial distance fro m th e origi n r, an d azimuthal angle ö locatin g a plan e whos e angl e of rotatio n is measure d fro m th e x-axis , an d a polar angle 0 , givin g it s angula r locatio n measure d down fro m th e z-axis . Not e tha t 0 ca n var y fro m 0 t o ð , an d ö fro m 0 t o 2ð . Thi s is sketche d in Figur e 6.4.1 . We ca n the n defin e mutuall y perpendicula r uni t vector s f, È, an d a s show n in Figur e 6.4. 1 b y givin g th e direction s tha t Ñ move s a s r, 0, an d ö increase , respectively . Thes e uni t vector s ar e sketche d agai n in Figur e 6.4.2 . Not e tha t f an d è bot h lie in a plan e containin g th e z-axi s an d rotate d a n amoun t ö fro m th e x-axis . Th e uni t vecto r cji lie s in th e xy-plane . Figur e 6.4. 3 Th e locatio n o f a poin t Ñ is give n b y it s radia l distanc e fro m th e z-axis , p , its angula r rotatio n fro m th e x-axis , ö (jus t a s in th e previou s cas e wit h r an d 0), and it s elevatio n abov e th e xy plane , z.
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