EOM for Pendulum hanging from spring

[stigg]

Homework Statement

Derive Newton's and Lagrange's equation of motion for the system. Discuss differences and show how Newton's equations can be reduced to lagrange's equations. Assume arbitrarily large θ.

The system is a pendulum consisting of a massless rod of length L with a mass m attached to the end. The point of rotation is attached to a spring of stiffness k which is then attached to the ceiling and constrained to move in the y direction.

I have acquired what i believe to be the solution for the Lagrange EOM but am hung up on the Newtonian solution.

Homework Equations

Newtonian mechanics

The Attempt at a Solution

summing forces in the y direction i get my''-ky+mg=0 and summing toques about the rotation point i get mL²θ''+mgLsin(θ)=0

i defined positive y as going upward and positive moments as counterclockwise

I feel like this is incomplete and I am missing something.

For reference the lagrange EOM i got is 0=ML²θ'' + mLsin(θ)y'' + mLcos(θ)y'θ' - mL²θ'-mLsin(θ)y'

 

[stigg]

The set up looks like this diagram, with the spring's motion being confined to the y direction: