Transcription of Other Coordinate Systems - MIT OpenCourseWare
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Other Coordinate Systems - MIT OpenCourseWare
S. Widnall, J. Peraire Dynamics Fall 2008. Version 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 ). 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 xed line, such as the x axis. Thus, the trajectory of a particle will be determined if we know r and as a function of t, r(t), (t). The directions of increasing r and are de ned by the orthogonal unit vectors er and e . The position vector of a particle has a magnitude equal to the radial distance, and a direction determined by er.
the origin, and apply a standard change of basis procedure. This will give, for a generic vector A, ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎝ A r ⎠ = ⎝ cos θ sin θ ⎠ ⎝ A x ⎠ and ⎝ A x ⎠ = ⎝ cos θ − sin θ ⎠ ⎝ A r ⎠ . A θ − sin θ cos θ A y A y sin θ cos θ A θ Example Circular motion
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