Curvature — Question 9

Question 9 – Solution

Strategy. Use the helix curvature formula and then translate parameter change into vertical rise.

Step 1: Solve the design equation For a helix of radius $a$, $$\kappa=\frac{a}{a^2+b^2}.$$ With $a=3$ and $\kappa=1/5$, $$\begin{align*} \frac 3{9+b^2}&=\frac 15,\\ 15&=9+b^2,\\ b^2&=6. \end{align*}$$ Since $b>0$, $\boxed{b=\sqrt 6}$.

Step 2: Rise per revolution A full revolution changes $t$ by $2\pi$. Since $z=bt$, $$\Delta z=b(2\pi)=\boxed{2\pi\sqrt 6}.$$

Step 3: Verification Substituting $b^2=6$ into the curvature formula gives $$\kappa=\frac 3{3^2+(\sqrt 6)^2} =\frac 3{9+6}=\boxed{\frac 15}.$$ The positive root also makes the helix rise as $t$ increases, satisfying the stated orientation.