Arc length and radius of a helix

Question

I have a cylinder of diameter $7.5\operatorname{cm}$, I want to make a helix with angle $19^o$ from horizontal plane.

What will be the profile of the helix on the helix plane? Will it be circular and if yes, then what will be its radius?

Answer 1

A circular helix has the following parametric representation $$ x=r\cos t,\; y=r\sin t,\; z=ct $$ Our goal is to figure out $r,c$.

r

As the helix lies on the cylinder, they both have the same radius, therefore $r={7.5\over2}$cm.

c

Suppose $\cdot$ represents differentiation with respect to $t$. The tangent vector to the helix is given by $$ (\dot x,\dot y,\dot z)=(-r\sin t,r\cos t,c) $$ If angle between the tangent and $z$-axis is $90^\circ-19^\circ=71^\circ$, we have $$\begin{align} &(\dot x,\dot y,\dot z)\cdot(0,0,1)=|(\dot x,\dot y,\dot z)|\cdot|(0,0,1)|\cos71^\circ\\ \implies&\dot z=\sqrt{\dot x^2+\dot y^2+\dot z^2}\cos71^\circ\\ \implies& c=\sqrt{r^2+c^2}\cos71^\circ\\ \implies& c=r\cot71^\circ\approx1.29\text{cm} \end{align}$$