Tuesday, May 27, 2014

EJSS Circular Motion Ferris Wheel Model

EJSS Circular Motion Ferris Wheel Model
http://weelookang.blogspot.sg/2014/05/ejss-circular-motion-ferris-wheel-model.html
https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_model_ferrisWheelJSwee/ferrisWheelJSwee_Simulation.xhtml
source: https://dl.dropboxusercontent.com/u/44365627/lookangEJSworkspace/export/ejss_src_ferrisWheelJSwee.zip
authors: Mario Belloni, this version by lookang
author of EJS 5: Paco.

EJSS Circular Motion Ferris Wheel Model

this model uses the constant angular velocity to evolve the simulation

Ordinary Differential Equation

$ \frac{d \theta}{dt} = \omega $ 


Position of Ferris Wheel Mass

the coordinates of the mass are spaced out by using

$ x[i] = R sin(1+i) d \theta $

$ y[i] = - R cos(1+i) d \theta $

$ rot[i] = (1+i) d \theta $

where

$ d \theta  = \frac{2 \pi}{N_{1}} $

because there are $ N_{1} =12 carriages 


Newtonian Forces

the following equations are used to represent the forces such as

$ Weight = mg $


Where $ m = 8 kg $

By using the Newton's second law on the carriage in circular motion resolving to a general angle of $ \theta $ where  $ \theta $ is angle made with the vertically downward plumb-line.


 $ F = mg $

typical resolving in the parallel and perpendicular to circular path,

$ N - mg= m \frac{v^2}{R} $

$ mgsin \theta = m \frac{d^{2} \theta}{ d t^{2}}    $

where $ \frac{d^{2} \theta}{ d t^{2}} = 0 $ constant angular velocity motion

resolving into x and y direction

 $ N_{x}       = m \frac{v^2}{R}sin \theta $

$ N_{y} - mg = m \frac{v^2}{R}cos \theta $

The resultant is also resolve as

 $ Resultant_{x} = - m \frac{v^2}{R}sin \theta $

$ Resultant_{y} = - m \frac{v^2}{R}cos \theta $

$ Resultant =  \sqrt{Resultant_{x}^{2}+Resultant_{y}^{2}} $

Conversion of radian to degree


The simple relationship is used to convert radian (used by computer) to degree ( used by human and thus the sliders and input fields)

$ \theta_{deg} = \frac{\theta 180 }{\pi}$

General Description


This program simulates the effect of being on a Ferris Wheel. The simulation shows a wheel that can be varied in radius from 40 m (Ferris' original wheels) to 100 m, or about 10 meters larger than the current world record. In addition, the rotational speed of the wheel can be varied from -20 m/s to 20 m/s. By selecting the check-box, the free-body diagram can be shown.

This simulation is part of a collection of simulations related to amusement park physics. Additional simulations can be found on the OSP ComPADRE site.

Controls
Radius of the wheel (40 - 100 meters).
Speed: sets the speed of the Ferris wheel (-20 to 20 m/s).

Mario Belloni (mabelloni@davidson.edu) and now weelookang@gmail.com


Changes made by lookang include


  1. added velocity array to be display via check-box
  2. added theta to show
  3. added title
  4. made free body diagram to show weight and contact force while resultant force is a dotted line to illustrate the difference, actually FBD only has weight and contact force
  5. added values to aid numerical calculations
  6. made R = the outside hub to aid association of radius of circular motion made by carriage and man and woman
  7. made time step to allow exactly 0 , 90 , 180 , 270, 360 degrees, previously was dt =0.02