How are the number of ball bearings in a roller bearing determined?

[sketchtrack]

Six circles fit tightly around one all of equal length, 12 spheres fit tightly around one sphere all of equal diameter.
How many circles can fit around one which is 6 times the other circles diameter?
How many spheres can fit around one sphere which is 6 times the diameter of the others?

 

[neu]

My solution:

Spoiler

well I'd make a quick guess that as the circumference of a circle increases as 2\pi r and the surface area of a sphere as 4\pi r^2 then I'd say you'd get 6*12 = 72 circles around the new circle, and 36*12 = 432 spheres around the new sphere

 

[Jimmy Snyder]

Spoiler

Let r be the radius of the smaller circles, so 6r is the radius of the larger central circle. Consider the triangle formed by the center of the larger circle and two adjacent smaller circles. The distance between the centers of the two smaller circles is 2r and the distance from the center of either smaller circle and the center of the larger circle is 7r.
The formula for the sides of a triangle is c^2 = a^2 + b^2 - 2ab \cdot cos(\theta) where \theta is the central angle. So the central angle is roughly 16.42 degrees and 21 smaller circles will fit around the circumference of the larger circle with almost enough room for a 22nd one. I.e. they do not fit tightly. In the OP it mentions that the 6 circles fit tightly around the same size central circle, but does not say tightly for the larger central circle.

I was busy dipping the tips of Amy Krumplemeyer's pigtails in the inkwell in my desk on the day we did solid geometry, so I have not figured out the second part yet.

 

[sketchtrack]

What ratios of inner circle diameter to outer circle diameters result in tight fits?

 

[Jimmy Snyder]

What ratios of inner circle diameter to outer circle diameters result in tight fits?

I don't know of any but 1:1. However the following come close.

112 225 338 451 564 677 790 903

I wonder how they determine the number of ball bearings to put into a roller bearing.