Equation of Motion for pendulum suspended from a spring

• stigg
In summary, the conversation discusses the derivation of Newton's and Lagrange's equations of motion for a pendulum system with a massless rod of length L, a mass m, and a spring of stiffness k attached to the ceiling. The differences between the two equations are also discussed, and it is shown how Newton's equations can be reduced to Lagrange's equations. The system is assumed to have arbitrarily large θ and motion is constrained to the y direction. The conversation also mentions the attempt at finding the Lagrange EOM and the question of how many degrees of freedom and equations of motion should be expected.
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'

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.

1. What is the equation of motion for a pendulum suspended from a spring?

The equation of motion for a pendulum suspended from a spring is a second-order differential equation that relates the position of the pendulum to time. It is given by d2x/dt2 + (k/m)x = 0, where x is the displacement of the pendulum from its equilibrium position, t is time, k is the spring constant, and m is the mass of the pendulum.

2. How does the equation of motion for a pendulum suspended from a spring differ from a simple pendulum?

A simple pendulum is a mass suspended from a string, while a pendulum suspended from a spring has an additional force acting on it due to the spring. As a result, the equation of motion for a pendulum suspended from a spring includes the spring constant and mass, while the equation of motion for a simple pendulum does not.

3. How does the spring constant affect the motion of a pendulum suspended from a spring?

The spring constant, k, determines the strength of the spring and how much it will resist the displacement of the pendulum. A higher spring constant will result in a shorter period of oscillation and a stiffer spring, while a lower spring constant will result in a longer period of oscillation and a more flexible spring.

4. Can the equation of motion for a pendulum suspended from a spring be solved analytically?

Yes, the equation of motion for a pendulum suspended from a spring can be solved analytically using techniques such as the method of undetermined coefficients or the method of variation of parameters. However, for more complex systems, numerical methods may be necessary to obtain a solution.

5. What are some real-life applications of the equation of motion for a pendulum suspended from a spring?

The equation of motion for a pendulum suspended from a spring can be used to model the motion of a variety of systems, including simple harmonic oscillators, musical instruments, and pendulum clocks. It is also used in seismology to study the motion of the Earth's crust during earthquakes.

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