# Dynamics of a Spring-Pendulum System

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A string of fixed length is free to move over two frictionless rollers fixed at the top. A spring is at one end of the string and a bob is at the other end, forming a pendulum of variable length.

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Contributed by: Erik Mahieu (October 2014)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The spring and pendulum system has two degrees of freedom: , the angular displacement of the pendulum bob, and , the vertical displacement of the spring.

Lagrangian mechanics can be used to simulate this system.

The potential energy of the system at time is:

.

The kinetic energy of the system at time is:

.

The Lagrangian is given by

.

The algebraic constraint expresses the constant length of the string:

.

The resulting equations of motion are

,

.

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