Particle sliding down an arbitrary concave curve in the Lagrangian formalism

Balart, Leonardo; Belmar-Herrera, Sebastián

doi:10.1119/10.0000037

Cited by 7 publications

(6 citation statements)

References 7 publications

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“…We wish to compare the motion of an incompressible fluid with the motion of a particle under a time-independent holonomic constraint. For this reason, we will repeat the familiar application of d'Alembert's principle of virtual work in the case of the motion of a particle on a time-independent surface (for the Lagrangian perspective, see [10]). Let a single particle with mass m moves on the surface defined by f x y z , , 0 ( )= , as shown in figure 2.…”

Section: Motion Of a Particle On A Surfacementioning

confidence: 99%

Pressure gradient in an incompressible fluid as a reaction force and the preservation of the principle of ‘cause and effect’

Simeonov

2023

Eur. J. Phys.

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When considering the motion of an incompressible fluid, it is common practice to take the curl on both sides of the Navier-Stokes (or Euler) equations and cancel the pressure force. The governing equations are sufficient to derive the velocity field of the fluid \textit{without} any knowledge of the pressure. In fact, the pressure is only calculated \textit{after} obtaining the velocity field. This raises a number of conceptual problems. For instance, why is the pressure unnecessary for obtaining the velocity field? Traditionally, forces have been considered as the ``causes`` of motion, and the resulting acceleration as the ``effect``. However, the acceleration (the effect) and the resulting velocity field can be obtained without any recourse to the pressure (the cause), seemingly violating the principle of ``cause`` and ``effect``. We address these questions by deriving the pressure force of an incompressible fluid, starting from d'Alembert's principle of virtual work, as a ``reaction force`` that maintains the incompressibility condition. Next, we show that taking the curl on both sides of the Navier-Stokes (or Euler) equations is \textit{equivalent} to using d'Alembert's principle of virtual work, which cancels out the virtual work of the pressure gradient. This shows that abstract procedures, such as taking the curl on both sides of an equation, can actually be tacit applications of rich physical principles, without one realizing it. This can be quite instructive in a classroom of undergraduate students.

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Section: Motion Of a Particle On A Surfacementioning

confidence: 99%

Pressure gradient in an incompressible fluid as a reaction force and the preservation of the principle of ‘cause and effect’

Simeonov

2023

Eur. J. Phys.

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“…The inertia tensor for arbitrary masses and length parameters can be computed by making use of (6). In this section, we compute that tensor.…”

Section: When CM Is Identical To the Incentermentioning

confidence: 99%

“…Because A , B , and C are coplanar, those linear combinations are not unique. We employ Lagrange's method of undetermined multipliers to find the inverse transformation [2][3][4][5][6] . The complicated dependence on the multipliers eventually cancels when we compute a physically measurable quantities like the scalar products of any two vectors.…”

Section: Introductionmentioning

confidence: 99%

Three-Body Inertia Tensor

Ee,

Jung,

Kim

et al. 2020

Preprint

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We derive a general formula for the inertia tensor of a three-body system. By employing three independent Lagrange undetermined multipliers to express the vectors corresponding to the sides in terms of the position vectors of the vertices, we present the general covariant expression for the inertia tensor of the three particles of different masses. If m a /a = m b /b = m c /c = ρ, then the center of mass coincides with the incenter of the triangle and the moment of inertia about the normal axis passing the center of mass is I = ρabc, where m a , m b , and m c are the masses of the particles at A, B, and C, respectively, and a, b, and c are the lengths of the line segments BC, CA, and AB, respectively. The derivation and the corresponding results are closely related to the famous Heron's formula for the area of a triangle.

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“…However, sometimes there are variations on this class of problems by allowing the inclined plane to be affected by the reaction force between the object and the inclined plane (see problem 15-98 of [3]). In the literature, similar problems have been addressed with different approaches like the use of Lagrangian mechanics [4], including friction [5][6][7], or the experimental demonstration [8].…”

Section: Introductionmentioning

confidence: 99%

Sliding down over a horizontally moving semi-sphere

Lineros¹

2022

Eur. J. Phys.

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We studied the dynamics of an object sliding down on a semi-sphere with radius $R$. We consider the physical setup where the semi-sphere is free to move over a flat surface. For simplicity, we assume that all surfaces are friction-less. We analyze the values for the last contact angle $\theta^\star$, corresponding to the angle when the object and the semi-sphere detach one of each other. We consider all possible scenarios with different combination of mass values: $m_A$ and $m_B$. We found that the last contact angle only depends on the ratio between the masses, and it is independent of the acceleration of gravity and semi-sphere's radius. In addition, we found that the largest possible value of $\theta^\star$ is $48.19^{\circ}$ that coincides with the case of a fixed semi-sphere. On the opposite case, the minimum value of $\theta^\star$ is $0^\circ$ and it occurs then the object on the semi-sphere is extremely heavy, occurring the detachment as soon as the sliding body touches the semi-sphere.

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Particle sliding down an arbitrary concave curve in the Lagrangian formalism

Abstract: We apply the method of Lagrange multipliers to the problem of a particle sliding on an arbitrary concave downward surface under the action of gravity to obtain the point where it leaves the surface.

Cited by 7 publications

References 7 publications

Pressure gradient in an incompressible fluid as a reaction force and the preservation of the principle of ‘cause and effect’

Pressure gradient in an incompressible fluid as a reaction force and the preservation of the principle of ‘cause and effect’

Three-Body Inertia Tensor

Sliding down over a horizontally moving semi-sphere

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