Length of differential element along a cone.

Question

If I calculate the differential length $ds$(see picture) as $rd\varphi$ I get a different value than if I take the projection of $r$ and the projected angle $d\theta$ to get $Rd\theta$ (where $R$ is the projection of $r$).

Does $d\varphi$ not approach $d\theta$ in the limit or does $ds$ not approach $ds'$? What is wrong with my approach? Please help( I'm sorry for my notations). enter image description here

Quanto · Answer 1 · 2019-09-18T22:26:09.607

1

The problem is that $rd\phi$ Is not equal to $ds$.

This is because the radius vector $r$ and the arc segment $ds$ are not in the same circle of plane as you may have assumed.

edited Sep 18 '19 at 22:26

answered Sep 18 '19 at 21:29

Quanto

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Yes but won't the arc segment of the circle approach the arc length $ds$ as $d\varphi$ approaches zero? – Akshat Sharma Sep 19 '19 at 13:16
They both approach zero in the limit, but their ratio is not 1 – Quanto Sep 19 '19 at 13:44

Length of differential element along a cone.

1 Answers1