Fundamental principles governing the movement of the mysterious double cone

Luz-Burgoa, K.; Nogales, José A.C.

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Revista Brasileira de Ensino de Física

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Artigos Gerais • Rev. Bras. Ensino Fís. 46 • 2024 • https://doi.org/10.1590/1806-9126-RBEF-2023-0265 copy

Fundamental principles governing the movement of the mysterious double cone

Authorship SCIMAGO INSTITUTIONS RANKINGS

The movement of a double cone along inclined tracks arranged in a V shape is famous in both formal and informal educational environments. When the inclination of the tracks is less than a certain geometric relationship between the double cone and the opening of the tracks, the double cone appears to defy gravity and roll “spontaneously” up the inclined tracks, sparking intense debates between scientific knowledge and common sense. However, basic physics textbooks do not address this phenomenon, and scientific articles encapsulate the mystery behind the motion. This work develops the dynamics of the double cone by applying Newton’s laws to rigid bodies, following the explanations and resources found in basic physics textbooks. The results reveal that the physical interpretation of the geometric condition involves the angle between the normal force and the vertical when the tracks are not inclined. This angle, implicit in the geometry of the double cone on the tracks, surprisingly organizes the equations. As a result, the study delves deeper into the vectorial understanding of the phenomenon and confirms the theorems of work-kinetic energy and parallel axes in the context of double cone motion.

Keywords:
Dynamics of the double cone; Physics education; Scientific methodology; Low-cost experiments; Scientific education

$\vec{r} = r (\sin θ \hat{i} - \cos θ \hat{j})$	$\vec{q} = \frac{R - r}{η} \hat{a}$	$\vec{d} = \frac{r}{\cos θ_{0}} [\sin (θ - θ_{0}) \hat{i} - \cos (θ - θ_{0}) \hat{j}]$
${\vec{r}}_{c p} = \frac{(R - r)}{η \cos β} (\cos β \hat{a} + \sin β \hat{k})$	${\vec{r}}_{c m} = \frac{R}{η} \hat{a} - \frac{r}{\sin θ_{0}} \hat{b}$	$\hat{a} \cdot \hat{b} = \cos θ_{0}$
${\vec{v}}_{c p} = \frac{- \dot{r}}{η \cos β} (\cos β \hat{a} + \sin β \hat{k})$	${\vec{v}}_{c m} = - \frac{\dot{r}}{\sin θ_{0}} \hat{b}$	${\vec{v}}_{t} = - \dot{\vec{q}} = \vec{ω} \times \vec{r} = \frac{\dot{r}}{η} \hat{a}$
$\vec{ω} = - ω \hat{k} = \frac{\dot{r}}{r η} \hat{k}$	$v_{c m} = v_{c p} \frac{\cos β}{\cos θ_{0}}$	$v_{c m} = \frac{v_{t}}{\cos θ_{0}}$
${\vec{a}}_{c p} = \frac{- \ddot{r}}{η \cos β} (\cos β \hat{a} + \sin β \hat{k})$	${\vec{a}}_{c m} = - \frac{\ddot{r}}{\sin θ_{0}} \hat{b}$	${\vec{a}}_{t} = \vec{α} \times \vec{r} + \vec{ω} \times \dot{\vec{r}}$
$\vec{α} = - α \hat{k}$	$\dot{\vec{ω}} = \frac{(\ddot{r} - {\dot{r}}^{2} / r)}{r η} \hat{k}$	$\vec{α} = \dot{\vec{ω}}$
$\vec{P} = - M g \hat{j}$	${\vec{f}}_{s} = f_{s b} \hat{b}$	$\vec{N} = N [- \sin (θ - θ_{0}) \hat{i} + \cos (θ - θ_{0}) \hat{j}]$

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