This tutorial demonstrates how to model the motion of a bead constrained to slide without friction on a helical wire. This example will be worked in the Cartesian coordinates x, y, and z. Cartesian coordinates are chosen in order to illustrate a 3D example. Furthermore the plots generated in 3D are more intuitive. For a more thorough derivation, consult the Webnucleo technical report, 2006-05-27-2. If you have not already done so, launch the Newton Tool.
Acceleration Equations
The acceleration equations for this example can be derived using Lagrangian mechanics and Lagrange multipliers. The resulting equations become:
Where r is the radius of the helix, and β is related to the number of turns of a given length of helix. Setting β and r to 1, enter the input on the acceleration panel as shown below.
Initial Conditions
Enter the initial conditions as shown below
The initial conditions chosen mean the helix is centered around the origin and the bead is dropped from z=0.
Plots
With three variables and their associated velocities and accelerations, there are many different combinations we can plot in the plot tool. The three most obvious choices are x vs. t, y vs. t, and z vs. t. These three plots are shown below.
Movie
Though many other two dimensional plots can be generated with the data from the newton tool, you can also create three dimensional plots or even three dimensional movies. The movie provided here was generated using IDL. It shows the motion of the bead with time as well as that motion projected onto the xy and yz planes. Try using the data generated from the newton tool in many different programs to best visualize the motion of the system.
Challenge
Can you add a term to the acceleration equations that will represent a drag force caused by friction or air resistance?
