EJSS simple harmonic motion pendulum model with t vs $ \theta $ graph

based on models and ideas by

- lookang http://weelookang.blogspot.sg/2014/02/ejss-shm-model-with-vs-x-and-v-vs-x.html
- lookang http://weelookang.blogspot.sg/2010/06/ejs-open-source-simple-harmonic-motion.html?q=SHM
- lookang http://weelookang.blogspot.sg/2013/02/ejs-open-source-vertical-spring-mass.html?q=vertical+spring
- lookang http://weelookang.blogspot.sg/2010/06/physical-quantities-and-units.html?q=pendulum
- Wolfgang Christian and Francisco Esquembre http://www.opensourcephysics.org/items/detail.cfm?ID=13103

commons terms associated with pendulum http://en.wikipedia.org/wiki/File:Simple_gravity_pendulum.svg#file |

## Assumption of this simple pendulum model:

- The rod or cord on which the bob swings is massless, inextensible and always remains taut;
- The bob is a point mass;
- Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
- The motion does not lose energy to friction or air resistance.

## Assumptions of SHM comparable to pendulum:

- Motion approximates SHM when the angle $ \theta$ of oscillation is small, where $ sin \theta \approx \theta $

## The equations that model the motion of the pendulum system are:

$ \frac{\delta \theta}{\delta t} = \omega $

$ \frac{\delta \omega}{\delta t} = -\frac{g}{L}( sin \theta) $

where the terms

$ L $ represents the fixed length of the pendulum

$ g $ represents the gravity force component as a result of Earth's pull.

## If the motion starts to the positive amplitude position:

$ \theta = 30 degree $ |

$ \theta = 40 degree $ |

$ \theta = 50 degree $ |

$ \theta = 60 degree $ |

$ \theta = 70 degree $ |

$ \theta = 80 degree $ |

$ \theta = 90 degree $ |

## General Rule of Thumb

Thus, in general, the pendulum's swing motion is approximately a simple harmonic motion only when the Motion's angle $ \theta$ of oscillation is small, where $ sin \theta \approx \theta $, rule of thumb is about 5 degree!