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Lagrange of a simple pendulum

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A pendulum consists of a mass m suspended by a massless spring with unextended length of b and spring constant k. Find Lagrange's equation of motion. Assume that the pendulum is constrained to swing in a single plane.

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Let the generalized coordinates be (r, θ).

Kinetic Energy T= 1/2 m ( )
Potential Energy V = 1/2 k (r-b)2 - mg ( r cos θ)

Lagrange, L = T - V = 1/2 m ( ) - 1/2 k (r-b)2 + mg ( r cos θ)

L = 1/2 m ( ) - 1/2 k (r-b)2 + mg ...

Solution Summary

Equation of motion for a simple pendulum was developed starting from the generalized coordinates. Two cases were discussed. First the case of a simple pendulum was analyzed. Then the case of a pendulum oscillating vertically was analyzed. Very detailed solution is provided in a 3-page word document. Please download this question/answer pair if you are interested in understanding how to solve classical mechanics problems using Lagrange method.

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Lagrangian for a Simple Pendulum

A) Write the Lagrangian for a simple pendulum consisting of a mass m suspended at the end of a massless string of length l. Derive the equation of motion from the Euler-Lagrange equation, and solve for the motion in the small angle approximation.
B) Assume the massless string can stretch with a restoring force F=-k(r-ro) where ro is the unstretched length. Write the new Lagrangian and find the equations of motion.

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