6. A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particleof mass m moves without friction on its inner surface while being subject to the gravitationalforce -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 equationof 0o in terms of m, g, a and a constant of motion.

Lagrangian is nothing but the difference between kinetic energy and the potential energy of an…

Answered: A hemispherical bowl of radius a is…

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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Similar questions

A locomotive of mass M moves on a frictionless rail. Inside the locomotive is a simple pendulum oflength l and mass m. Let the acceleration of gravity be constant and its value g. Also assume that the angle is very small (θ << 1). In this case, the expansion will be made in the Lagrangian expression,including the terms of 2.So sin (θ) ≈ θ can be written as cos (θ) ≈ 1- θ2 / 2.Let X be the displacement in the horizontal direction. a) Write the Lagrangian of the system in terms of generalized coordinates X and θ.b) Write the equations of motion of the system.
Using Lagrange's equation, obtain the equation of motion for a particle in plane polar coordinates.
Determine the polar equation for orbit r(0) of a particle in a potential energy field given by B V(r) = where a > 0 and β ≥ 0.
2. Consider a particle of mass m moving along a circle of radius R and subject to a periodic potential field V (9 + 27) = V (8) for 0 to using the path integral representation.
Consider a system spin-1/2 system, denoted by A, interacting with another system spin-1/2 system, denoted by B, such that the state of the combined system is AB) a++ B|-+). Find (a) the density matrix PA for system A corresponding to this state and (b) obtain the formulas for (()).
5. Consider the two state system with basis |+) which diagonalizes the Pauli matrix 03. Generally the state of the system at time t can be written as |W(t)) = c+(t)|+) + c_(t)|-). (i) For the Hamiltonian of the system, first take H = functions c+(t) given the initial condition that at time t = 0 Eo03. Solve for the coefficient |W(0)) = |-).
Please obtain the same result as in the book.
The Hamiltonian matrix has been constructed using an orthonormal basis. (1 1 0V (1 0 1) A = (2 1 0 )+(0 2 2 \2 1 4 where H = Hº + V and cis a constant. 1 2 0/ b) Use time-independent perturbation theory to determine the eigenvalues with corrections up to second order.
Consider a 3-body system again. This time, let the particles be the Sun, the Earth, and the Moon. (a) Write the relative vector equations of motion for the Moon relative to the Earth. (b) At the very instance where the following right angle triangle geometry is achieved, compute the dominant and perturbing accelerations. Compare the magnitudes of both accelerations, is it reasonable to model the Moon's motion by considering a two-body problem with the Earth? Explain your answer.

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