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Math Help - Curve length

  1. #1
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    Curve length

    I got a problem with 2 cones

    The 2 cones have heigth h and base radius R.

    Around one of the cones a spiral swirls around it in a way that for each turn around the cone its height increase 1 unit (meter, foot, doesn't matter)

    In the 2nd cone a spiral swirls around with a 30 degree angle with the horizontal.


    The question is what is the lenght of each spiral and how many turns each one gives around the cone.


    thanks for any help (sorry if it isn't clear, I don't have a picture)
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  2. #2
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    Quote Originally Posted by Haytham View Post
    I got a problem with 2 cones

    The 2 cones have heigth h and base radius R.
    Drawing these on "x-z" axes, as seen from the side, this is a triangle with height h and base 2R. The right side line passes through the points x=0, z= h and x= R, z= 0. From that, the equation of the line is z= h- \frac{hx}{R}. Because of the circular symmetry of the cone, the entire cone, has equation z= h(1- \frac{\sqrt{x^2+ y^2}}{R}). Because of the circular symmetry, this is simpler in cylindrical coordinates: z= h(1- \frac{r}{R})
    That also gives r= R(1- \frac{z}{h}).

    Around one of the cones a spiral swirls around it in a way that for each turn around the cone its height increase 1 unit (meter, foot, doesn't matter)
    So if \theta is the angle, in radians, of rotation, z= \frac{\theta}{2\pi} and then r= R(1- \frac{\theta}{2h\pi}). Also, z= 0 when \theta= 0 and z= h when \theta= 2h\pi
    Writing x, y, and z in terms of theta, x= r cos(\theta)= R(1- \frac{\theta}{2h\pi})cos(\theta), y= r sin(\theta)= R(1- \frac{\theta}{2h\pi}sin(\theta), z= \frac{\theta}{2\pi}.

    The arclength is given by \int_{\theta= 0}^{2h\pi}\sqrt{\left(\frac{dx}{d\theta}\right)^2+  \left(\frac{dy}{d\theta}\right)^2+\left(\frac{dz}{  d\theta}\right)^2}d\theta

    In the 2nd cone a spiral swirls around with a 30 degree angle with the horizontal.


    The question is what is the lenght of each spiral and how many turns each one gives around the cone.


    thanks for any help (sorry if it isn't clear, I don't have a picture)
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  3. #3
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    thanks

    the length for the 2nd is similar, but theta gives me infinity

    does this make sense, because r tends to zero?
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