A thin hoop of radius R and mass M oscillates in its own vertical plane with one point of the hoop fixed. Attached to the hoop is a point mass M constrained to move without friction along the hoop. (a) Write the transformation equations for the center of the hoop and the position of the bead, and their time derivatives. Find the kinetic energy, the potential energy, and the full lagrangian for the system. (b) Expand the lagrangian to 2nd order in small angular deviations from equilibrium. Find the matrices T and V. (c) Find the normal mode frequencies (d) Find the normal mode eigenvectors. Sketch the motion corresponding to each one of them. You don't need to find the normalization constants for the eigenvectors.

A thin hoop of radius R and mass M oscillates in its own vertical plane with one point of the hoop fixed. Attached to the hoop is a point mass M constrained to move without friction along the hoop. (a) Write the transformation equations for the center of the hoop and the position of the bead, and their time derivatives. Find the kinetic energy, the potential energy, and the full lagrangian for the system. (b) Expand the lagrangian to 2nd order in small angular deviations from equilibrium. Find the matrices T and V. (c) Find the normal mode frequencies (d) Find the normal mode eigenvectors. Sketch the motion corresponding to each one of them. You don't need to find the normalization constants for the eigenvectors.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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