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Unformatted text preview: S. Widnall, J. Peraire 16.07 Dynamics Fall 2008 Version 2.0 Lecture L5- Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates in two dimensions and cylindrical and spherical coordinates in three dimensions. We shall see that these systems are particularly useful for certain classes of problems. Polar Coordinates ( r − θ ) In polar coordinates, the position of a particle A , is determined by the value of the radial distance to the origin, r , and the angle that the radial line makes with an arbitrary fixed line, such as the x axis. Thus, the trajectory of a particle will be determined if we know r and θ as a function of t , i.e. r ( t ) , θ ( t ). The directions of increasing r and θ are defined by the orthogonal unit vectors e r and e θ . The position vector of a particle has a magnitude equal to the radial distance, and a direction determined by e r . Thus, r = r e r . (1) Since the vectors e r and e θ are clearly different from point to point, their variation will have to be considered when calculating the velocity and acceleration. Over an infinitesimal interval of time dt , the coordinates of point A will change from ( r, θ ), to ( r + dr , θ + dθ ) as shown in the diagram. 1 We note that the vectors e r and e θ do not change when the coordinate r changes. Thus, d e r /dr = and d e θ /dr = . On the other hand, when θ changes to θ + dθ , the vectors e r and e θ are rotated by an angle dθ . From the diagram, we see that d e r = dθ e θ , and that d e θ = − dθ e r . This is because their magnitudes in the limit are equal to the unit vector as radius times dθ in radians. Dividing through by dθ , we have, d e r d e θ = e θ , and = − e r . dθ dθ Multiplying these expressions by dθ/dt ≡ θ ˙ , we obtain, d e r dθ d e r d e θ dθ dt ≡ dt = θ ˙ e θ , and dt = − θ ˙ e r . (2) Note Alternative calculation of the unit vector derivatives An alternative, more mathematical, approach to obtaining the derivatives of the unit vectors is to express e r and e θ in terms of their cartesian components along i and j . We have that e r = cos θ i + sin θ j e θ = − sin θ i + cos θ j . Therefore, when we differentiate we obtain, d e r d e r = 0 , = − sin θ i + cos θ j ≡ e θ dr dθ d e θ d e θ = 0 , = − cos θ i − sin θ j ≡ − e r . dr dθ Velocity vector We can now derive expression (1) with respect to time and write v = r ˙ = r ˙ e r + r e ˙ r , or, using expression (2), we have v = r ˙ e r + rθ ˙ e θ . (3) Here, v r = r ˙ is the radial velocity component, and v θ = rθ ˙ is the circumferential velocity component. We also have that v = v r 2 + v θ 2 . The radial component is the rate at which r changes magnitude, or stretches, and the circumferential component, is the rate at which r changes direction, or swings....
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This note was uploaded on 11/06/2011 for the course AERO 112 taught by Professor Widnall during the Fall '09 term at MIT.
- Fall '09