Survey of the Elementary Principles — Herbert Goldstein, Charles Poole, John Safko
Physics — Mechanics · spring-pendulum-lagrangian
Problem
A spring of rest length La (no tension) is connected to a support at one end and has a mass M attached at the other. Neglect the mass of the spring, the dimension of the mass M, and assume that the motion is confined to a vertical plane. Also, assume that the spring only stretches without bending but it can swing in the plane.
((a)) Using the angular displacement of the mass from the vertical and the length that the string has stretched from its rest length (hanging with the mass m), find Lagrange's equations.
((b)) Solve these equations for small stretching and angular displacements.
((c)) Solve the equations in part (a) to the next order in both stretching and angular displacement. This part is amenable to hand calculations. Using some reasonable assumptions about the spring constant, the mass, and the rest length, discuss the motion. Is a resonance likely under the assumptions stated in the problem?
((d)) (For analytic computer programs.) Consider the spring to have a total mass m≪M. Neglecting the bending of the spring, set up Lagrange's equations correctly to first order in m and the angular and linear displacements.
((e)) (For numerical computer analysis.) Make sets of reasonable assumptions of the constants in part (a) and make a single plot of the two coordinates as functions of time.