Consider a physical pendulum of mass M and moment of inertia I. Its center of mass is a distance & from the point of support Q, but with attached to the bottom of a vertical spring of force constant k and constrained to move vertically. The top of the spring is fixed to a rigid support. (a) Construct the lagrangian for this system in terms of the generalized coordinates n (the downward displacement of Q) and (the angle of the pendulum away from the vertical down direction). Derive Lagrange's equations. (b) Obtain the same equations from the general principles of rigid-body dynamics concerning the motion of the center of mass and rotations about it (don't forget the force of constraint). (c) For small oscillations, show that the motion uncouples into that of a spring and a pendulum.

Consider a physical pendulum of mass M and moment of inertia I. Its center of mass is a distance & from the point of support Q, but with attached to the bottom of a vertical spring of force constant k and constrained to move vertically. The top of the spring is fixed to a rigid support. (a) Construct the lagrangian for this system in terms of the generalized coordinates n (the downward displacement of Q) and (the angle of the pendulum away from the vertical down direction). Derive Lagrange's equations. (b) Obtain the same equations from the general principles of rigid-body dynamics concerning the motion of the center of mass and rotations about it (don't forget the force of constraint). (c) For small oscillations, show that the motion uncouples into that of a spring and a pendulum.

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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