Find an arc length parametrization of a helix of height 20 cm that makes four full

Step 1 of 4

Find a parametrization for the helix.

The circle part with radius 5 cm can be represented by

\(x=5 \cos t, y=5 \sin t\)

for 4 full rotations that would be \(0 \leq t \leq 8 \pi\)

As the full path is traced out, z can vary from 0 to 20 cm if we have

\(z=\frac{t}{8 \pi} \cdot 20=\frac{5}{2 \pi} t\)

so then

\(\begin{aligned}
\mathbf{r}(t) & =\left\langle 5 \cos t, 5 \sin t, \frac{5}{2 \pi} t\right\rangle \\
\mathbf{r}^{\prime}(t) & =\left\langle-5 \sin t, 5 \cos t, \frac{5}{2 \pi}\right\rangle
\end{aligned}\)