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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
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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3. Consider a physical pendulum of mass \( M \) and moment of inertia \( I \). Its center of mass is a distance \( \ell \) from the point of support \( Q \), but with \( Q \) 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 \( \eta \) (the downward displacement of \( Q \)) and \( \phi \) (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.
Transcribed Image Text:3. Consider a physical pendulum of mass \( M \) and moment of inertia \( I \). Its center of mass is a distance \( \ell \) from the point of support \( Q \), but with \( Q \) 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 \( \eta \) (the downward displacement of \( Q \)) and \( \phi \) (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.
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