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Quick question with spherical coordinates and vectors

  1. Sep 27, 2011 #1
    So heres the question:
    An ant crawls on the surface of a ball of radius b in such a manner that the ants motion is given in spherical coordinates by the equations:
    r = b, [itex]\phi[/itex] = [itex]\omega[/itex]t and [itex]\vartheta[/itex] = [itex]\pi[/itex] / 2 [1 + [itex]\frac{1}{4}[/itex] cos (4[itex]\omega[/itex]t).

    Find the speed as a function at time t and the radial acceleration of the ant.

    I found the speed, doing [itex]\left|v\right|[/itex] = b[itex]\omega[/itex][cos[itex]^{2}[/itex]([itex]\frac{\pi}{8}[/itex]cos 4[itex]\omega[/itex]t) + [itex]\frac{\pi^{2}}{4}[/itex] sin[itex]^{2}[/itex] 4[itex]\omega[/itex]t] [itex]^{1/2}[/itex]

    Now I don't even know where to begin to take the derivative of that, lol. I know i derive the actual vector v, not the magnitude of it. But how do i derive e[itex]_{\phi}[/itex] and e[itex]_{\vartheta}[/itex]?

    I got for velocity

    v = [itex]\widehat{e}[/itex][itex]_{\phi}[/itex]b[itex]\omega[/itex]cos [[itex]\frac{\pi}{8}[/itex]cos 4[itex]\omega[/itex]t] - [itex]\widehat{e}[/itex][itex]_{\vartheta}[/itex]b[itex]\omega[/itex] [itex]\frac{\pi}{2}[/itex]sin (4[itex]\omega[/itex]t)

    Any help on deriving that to find acceleration would be awesome :s Maybe I'm missing a rule with [itex]\widehat{e}[/itex][itex]_{\phi}[/itex], but I'm getting stuck.

    Thanks :)
  2. jcsd
  3. Sep 28, 2011 #2
    Any help would be greatly appreciated.
  4. Sep 28, 2011 #3


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    See: http://en.wikipedia.org/wiki/Spherical_coordinate_system,


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