1. ## 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)

2. Originally Posted by Haytham
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 $\displaystyle z= h- \frac{hx}{R}$. Because of the circular symmetry of the cone, the entire cone, has equation $\displaystyle z= h(1- \frac{\sqrt{x^2+ y^2}}{R})$. Because of the circular symmetry, this is simpler in cylindrical coordinates: $\displaystyle z= h(1- \frac{r}{R})$
That also gives $\displaystyle r= R(1- \frac{z}{h})$.

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 $\displaystyle \theta$ is the angle, in radians, of rotation, $\displaystyle z= \frac{\theta}{2\pi}$ and then $\displaystyle r= R(1- \frac{\theta}{2h\pi})$. Also, z= 0 when $\displaystyle \theta= 0$ and z= h when $\displaystyle \theta= 2h\pi$
Writing x, y, and z in terms of theta, $\displaystyle x= r cos(\theta)= R(1- \frac{\theta}{2h\pi})cos(\theta)$, $\displaystyle y= r sin(\theta)= R(1- \frac{\theta}{2h\pi}sin(\theta)$, $\displaystyle z= \frac{\theta}{2\pi}$.

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

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)

3. thanks

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

does this make sense, because r tends to zero?