EOM for Pendulum hanging from spring

In summary, the conversation discussed deriving Newton's and Lagrange's equations of motion for a pendulum system consisting of a massless rod with a mass attached to the end, attached to a spring constrained to move in the y direction. The differences between the two equations were also discussed, and it was shown how Newton's equations can be reduced to Lagrange's equations. The conversation also assumed arbitrarily large θ. The attempted solution involved summing forces and torques in the y direction, resulting in equations for y'' and θ''. The conversation also referenced a diagram of the system, with the spring's motion confined to the y direction.
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stigg
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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'
 
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The set up looks like this diagram, with the spring's motion being confined to the y direction:

upload_2015-3-17_20-2-28.png
 

1. What is the equation of motion (EOM) for a pendulum hanging from a spring?

The EOM for a pendulum hanging from a spring is given by:
mLθ'' + kθ = 0
where m is the mass of the pendulum, L is the length of the pendulum, k is the spring constant, and θ is the angle of displacement from the equilibrium position.

2. How is the EOM for a pendulum hanging from a spring derived?

The EOM for a pendulum hanging from a spring can be derived using Newton's second law of motion and Hooke's law. The forces acting on the pendulum are the tension in the spring and the force of gravity. By equating these forces to the mass times acceleration, the EOM can be obtained.

3. Can the EOM for a pendulum hanging from a spring be simplified for small angles of displacement?

Yes, for small angles of displacement (θ < 10°), the sine function in the EOM can be approximated by the angle itself. This simplifies the equation to:
mLθ'' + kθ = 0
which is known as the simple harmonic motion equation.

4. How does the mass of the pendulum affect the EOM for a pendulum hanging from a spring?

The mass of the pendulum affects the EOM by changing the inertia term (mL) in the equation. A larger mass will result in a larger inertia, which will require more force to move the pendulum and therefore affect the motion of the pendulum.

5. Can the EOM for a pendulum hanging from a spring be used to calculate the period of oscillation?

Yes, the period of oscillation for a pendulum hanging from a spring can be calculated using the EOM and the equation for the period of a simple harmonic motion:
T = 2π√(m/k)
where T is the period, m is the mass of the pendulum, and k is the spring constant.

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