Math Problem ..Help Genius.

[CHADS]

Hi i'm not hot with math .. I wonder if some of you guys can help find the right math to analyse Something....

 

I have three Equal Spheres clumped togeather making a triangle shape..

 

Now I see that each sphere contacts two other spheres and these points of contact act as the tracks to allow one sphere to traval a complete 360 around an axis of the other two balls point of contact.

I need to find the math that will describe What would happen completly when All spheres want to move along these tracks .. and in addition what will happen with varying raddii of the 3 Spheres .... Also what Angular paths the varying tracks will mapout and where to find these tracks on the surface of each Sphere at any time .... plus what math would describe the tracks of the Spheres if the Spheres have 3 components of rotation about there individual centres giving the tracks more mapping potentially over all of the Spheres.

 

I dont know whether putting this evolving system inside a box helps with the Analysis ...IM Not A Mathemagician ... Can any body help please ?!!!!!!!!!!!

[modest]

If any one sphere must always touch the other two then consider a point at the center of each sphere. The three points will form an equilateral triangle with sides equal to the diameter of any given sphere (assuming they’re all the same size). All angles for the triangle will be 60 degrees. The length of this triangle’s sides and value of its angles are the only limitations to the spheres' movement in three dimensions.

 

Essentially, it's no different than rotating an equilateral triangle.

 

-modest

[CHADS]

Thanks modest ....

Ermm .. with the tracks the spheres traverse how would i map the possibilities with respect to their varying rotations?

 

Im assuming that every possible point of contact is on the 4 pi r^2 of each sphere taking into consideration :

 

relative velocities , changing Radii , Individual rotations in 3d would give the paths taken.

 

Of course 3 identical spheres arranged in this way with the same velocities wouldnt move but with 3 different velocities the system will evolve .

 

I also thought what would happen to 1 spheres motion if the other 2 spheres where near infinite size ..... I only picture a small sphere wedged between two straight lines ... how would this evolve in such a system .. probably move in a circle ...How could this deduce the size of the other two ?

[modest]

umm...

 

You could add the equation for a sphere on to the coordinates of your triangle. More than that, I'm not exactly sure what you're talking about.

 

-modest

[CraigD]

Assuming no external force is applied to the object consisting of the 3 rigidly joined spheres (or any collection of rigidly joined objects), and that the entire object is at rest relative to the coordinates used to measure it (that is, the point defined by its center of mass has zero velocity), every point in or on the surface of every sphere will follow a circular path defined by the body’s axis of rotation and a single angular velocity. Every point will occupy the same position at time [math]t=np[/math], where [math]n[/math] is an integer, and [math]p[/math] is the body’s period of rotation.

 

Mathematically, a simple way to describe every point in the object is with 3 coordinates: the “length” on its axis of rotation where a line perpendicular to the axis and passing through the point touches the axis, the “radius” distance along this line from the axis to the point, and the angle of the line relative to an arbitrary reference point. This is roughly equivalent to describing a point above, on, or below the surface of the Earth by its latitude, longitude, and altitude.

 

From this, you can calculate the position of any point at any time. Selecting the axis of rotation to coincide with the z axis in a Cartesian coordinate system, the position of a point [math](z_n, r_n, \theta_n)[/math] at time [math]t[/math] is [math](r_n \cos (tw + \theta_n), r_n \sin (tw + \theta_n), z_n)[/math], where [math]w[/math] is the body’s angular speed.

 

Though setting up all of the detailed parameters for this calculation can be laborious, the system is about as simple as any rotating three-dimensional object can be. It doesn’t “evolve” in any interesting or complicated way.

 

Working such a calculation out in detail is a good exercise, especially if you allow the axis or rotation to be oriented other than along the z axis. Why don’t you (and any other interested person), give it a try, CHADS, and post your work here?

[modest]

Mathematically, a simple way to describe every point in the object is with 3 coordinates: the “length” on its axis of rotation where a line perpendicular to the axis and passing through the point touches the axis, the “radius” distance along this line from the axis to the point, and the angle of the line relative to an arbitrary reference point. This is roughly equivalent to describing a point above, on, or below the surface of the Earth by its latitude, longitude, and altitude.

 

Well said. If I may also reference wikipedia's: Spherical coordinate system and a java applet for graphing them:

 

Mathlets: Spherical Coordinates (3-D Graphing)

 

-modest

[CHADS]

Do you know what guys .... Thats terrific help thank you very much . I will set about understanding the math staright away and post my dubious exercise results .

 

Thanks again.!!!:shade: