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Math of a spiral with two constants

  1. Jan 17, 2016 #1
    1. The problem statement, all variables and given/known data
    What type of spiral has constant angular velocity and constant magnitude of velocity? Is there a description of the mathematics that describe the spiral?

    2. Relevant equations
    To be determined.

    3. The attempt at a solution
    The closest type I can find is the Archimedean spiral, which is the locus of points corresponding to the locations over time of a point that moves away from a fixed point with a constant speed along a line that rotates with constant angular velocity.

    The spiral that I am interested in is the locus of points corresponding to the locations over time of a point that (a) moves away from a fixed point along a line that rotates with a constant angular velocity, and (b) moves with a constant magnitude of velocity. Unlike the Archimedean spiral, the moving point does not move at a constant speed along the rotating line. Instead, it moves at a constant total magnitude of velocity, with the result that its speed along the rotating line decreases as the point moves farther from the fixed center (because the moving point's total magnitude of velocity is constant, and its magnitude of tangential velocity increases as it moves farther from the fixed center).

    Thanks.
     
  2. jcsd
  3. Jan 17, 2016 #2

    Simon Bridge

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    The next step is to describe it mathematically ... easiest done parametrically in polar coordinates, since you are including the physics terms "angular velocity" - we shall interpret that as the rate of change of the angle with some arbitrary parameter we'll call "t".

    So you are describing something like: ##(r,\theta) = (ut,\omega t)##, in polar coordinates, parameterised by ##t##: ##u## and ##\omega## are constants defining the spiral. Notice that u=0 describes a point at the origin. If you put r = c+ut then u=0 is a circle radius c.

    Now we need to find r as a function of angle:
    So ##r=(u/\omega)\theta## would be an Archimedean spiral with ##a=0## and ##b=u/\omega##, see:
    https://en.wikipedia.org/wiki/Archimedean_spiral

    ... for this one you want to do some calculus to get the equation.
    Follow the same process and it may match with a standard equation with a name.

    (A quick zip through .. if I put u=const and allow angular velocity to vary so that the speed is a constant, the shape I get is on the list of spiral types in wikipedia.)
     
    Last edited: Jan 17, 2016
  4. Jan 18, 2016 #3
    Thanks, I'll have a look.
     
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