After viewing the question and answer in Does there exist a tool to construct a perfect sine wave? . I have a question of interest and for fun:-
“How to choose the size (in term of radius R) of tube tool; and what is relation between the angle of the cut to the given amplitude and the wavelength?”
That is, describe exactly the tool needed to sketch the sine wave curve on the following given graph paper in specific scale.
The wavelength is $2a$ and the amplitude is $1$. Assume negligible thickness of the tube.
As the tube rolls over once, it should cover the distance of the wavelength. So, $$ 2\pi r = 2a $$ hence, $r=a/\pi$ The angle which the cut has to make with the vertical axis of the tube can be determined by trigonometry. The $tan$ of that angle will be the ratio of the amplitude of the sinecurve to the radius of the tube. Hence, angle = $\arctan{(1/r)}$.
