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Rotating particle (complex numbers)

  1. May 13, 2014 #1
    This problem is from Boas Mathermatical Methods 3ed. Section 16, problem 1.

    Show that if the line through the origin and the point z is rotated 90° about the origin, it becomes the line through the origin and the point iz.

    Use this idea in the following problem: Let z = ae^iωt be the displacement of a particle from the origin at time t. Show that the particle travels in a circle of radius a at velocity v = aω and with acceleration v^2 / a directed towards the centre of the circle.




    3. The attempt at a solution

    I can show the first part: adding [itex]\pi[/itex] / 2 to the argument of z gives ae[itex]^{i\theta}[/itex]e[itex]^{\pi / 2}[/itex] which is just iz.

    I'm not sure how to set up the second part of the problem, though. How exactly should I use this result?

    Thanks
     
  2. jcsd
  3. May 14, 2014 #2

    ehild

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    In the problem, the displacement vector is represented by the complex number z=aeiωt. What is the distance of the point from the origin?
    You can also show, that adding iθ to the argument of a complex number z, it gets rotated by the angle θ.
    The velocity is the time derivative of the displacement. What is dz/dt in this case? What is the magnitude of the velocity?
    The acceleration is the time derivative of velocity, second derivative of displacement. What is its magnitude? What is its direction?

    ehild
     
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