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Ball Bearings

A ball bearing consists of a central cylinder of radius surrounded by tangent balls of radius . All the balls are circumscribed by a hollow cylinder with radius , for simplicity.
In the cross section, the cylinder is shown as an orange disk, the balls are shown as purple disks, and the hollow cylinder is shown as a circle.
The radius of the central cylinder is .
The radius of any purple ball is .

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Joining the center of the orange disk O to the centers P and Q of two successive purple disks shows that the center angle is , where is the number of purple disks. The line segment connecting O to the tangent point of two consecutive purple disks bisects this angle, so each smaller angle measures . The line segment connecting P and Q is perpendicular to the line segment between the center of the orange disk and the tangent point of the two blue balls. Therefore:
.
Simplifying, .
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