Consider the pendulum of figure 7.4, suspended inside a railroad car, but suppose that the can is oscillating back and forth, so that the point of suspension has position x=Acos(wt), y=0. Use the angle Φ as the generalized coordinate and write down the equations that give the Cartesian coordinates of the bob in terms of Φ and vice versa.
See attached© BrainMass Inc. brainmass.com October 24, 2018, 10:49 pm ad1c9bdddf
Hello and thank you for posting your question to Brainmass!
The solution ...
Detailed calculations in attached documents are examined. The expert writes down the equations that give the Cartesian coordinates of the bob in terms of Φ and vice versa.
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.