Equation of Motion for pendulum suspended from a 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.

spring motion is constrained to the y direction

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'

 

[Dr.D]

How many degrees of freedom do you think this system has? How many equations of motion should you expect to find?

For the Newtonian approach, first work through the kinematics, in terms of the same generalized variable you used for the Lagrange approach. Then write F = m a. That's all there is to it.