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A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of mass m moves without friction on its inner surface while being subject to the gravitational force -mg2 (see figure). (a) Find the Lagrangian of the system in terms of spherical polar coordinates. (b) Derive the corresponding equations of motion using the Euler-Lagrange formulation. (c) Show that it is possible to have circular orbits with 0 00, a constant. Find an equation of Oo in terms of m, g, a and a constant of motion.

A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of mass m moves without friction on its inner surface while being subject to the gravitational force -mg2 (see figure). (a) Find the Lagrangian of the system in terms of spherical polar coordinates. (b) Derive the corresponding equations of motion using the Euler-Lagrange formulation. (c) Show that it is possible to have circular orbits with 0 00, a constant. Find an equation of Oo in terms of m, g, a and a constant of motion.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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6. A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of mass m moves without friction on its inner surface while being subject to the gravitational force -mg2 (see figure). (a) Find the Lagrangian of the system in terms of spherical polar coordinates. (b) Derive the corresponding equations of motion using the Euler-Lagrange formulation. (c) Show that it is possible to have circular orbits with 0 00, a constant. Find an equation of 0o in terms of m, g, a and a constant of motion.
Transcribed Image Text:6. A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of mass m moves without friction on its inner surface while being subject to the gravitational force -mg2 (see figure). (a) Find the Lagrangian of the system in terms of spherical polar coordinates. (b) Derive the corresponding equations of motion using the Euler-Lagrange formulation. (c) Show that it is possible to have circular orbits with 0 00, a constant. Find an equation of 0o in terms of m, g, a and a constant of motion.
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