Lagrangian Mechanics Question
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Consider a bead of mass m sliding without friction on a straight wire that passes through the origin and is inclined from the vertical by a constant angle alpha. The wire is being spun with constant angular velocity w about its vertical axis as shown in the figure. The bead is attached to the origin with a spring of force constant k as shown, with the unstretched length of the spring assumed to be zero. A characteristic frequency of the problem is thus w = sqrt(k/m).
Choose for generalized coordinate the distance r of the mass from the origin, and solve for its motion by the Lagrangian method. Your solution should clearly exhibit the following steps:
i) Write down expressions for the kinetic energy T and the potential energy U.
ii) Write down the Lagrangian.
iii) Write down the Lagrange equation of motion for r.
iv) Solve the Lagrange equation of motion.
v) Discusses the conditions under which there is a position of stable equilibrium, and the frequency of small oscillations about this position.
vi) Discuss the motion when there is not a position of stable equilibrium.
vii) Discuss relevant limiting cases for the parameters of the problem.
PDF attached answers 7 questions on a bead sliding on a wire, including the Lagrangian equation of motion and conditions for a stable equilibrium.