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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Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.

Question

Question 2.

than 5-6 lines, plus drawings.)
2. 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.
3. A Thumbtack on an inclined plane
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A rigid body in the shape of a thumbtack formed from a thin
disk of mass M and radius a and a massless stem is placed on
an inclined plane that makes an angle a with the horizontal.
The head of the tack rolls along a circle of radius b. Introduce
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