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Three Balls in a Bowl (Posted on 2005-09-07) Difficulty: 5 of 5
Here is a problem I have been developing. Maybe somebody can tell me if it can be solved or if more information is needed.

Three solid balls of radii a, b, and c are placed in a bowl whose inner surface is a hemisphere of radius d. The following information is known:

1) a < b < c < d,

2) d is large enough so that each ball touches a point on the inner surface of the bowl,

3) a is large enough so that each ball touches the other two balls,

4) the balls are made of the same material so that their weights are proportional to their volumes,

5) the forces that the balls exert on each other and the bowl are directed along the lines determined by their centers.

After the balls come to rest, what is the angle between the plane determined by the centers of the balls and the horizontal in terms of a, b, c, and d ?

No Solution Yet Submitted by Bractals    
Rating: 3.6667 (9 votes)

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Hugo's Challenge from CB | Comment 5 of 18 |

I do not have the mathematics to fully see this through, but I believe that this path will allow one better equipped to come to an appropriate solution. Hopefully my memories of Physics still stand by me, reasonably.

 

The Volume of a sphere is (4*Pi*R^3)/3.

We have 3 spheres which are of identical composition, so with respect to each other, we can merely consider their respective volumes as the cubes of their radii (a^3, b^3 and c^3).

 

SHM (simple harmonic motion) allows us to describe properties of a body exhibiting vertical circular motion.

 

In this instance the property is Potential Energy. 

The Potential Energy of each sphere tracks a path which is circular (or spherical, depending upon one's viewpoint); each circle has a respective radius of d-a, d-b and d-c.

 

When all spheres are at equal PE, the respective "height" of each centre above the 'base' of the hemisphere (radius d) can then be calculated.  From this the plane through the three centres can then be 'described'.

 

After that, calculate the angle that Bractal requires.

 

 

 

Edited on September 18, 2005, 12:33 pm
  Posted by brianjn on 2005-09-18 12:28:23

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