Geometry request

[Occasional Sage]

My reference books are in a box someplace. Could somebody problem solve for me?

I have three spheres of diameter 1. I want them to sit in a channel of width 3/8 (depth is irrelevant, large enough that spheres do not hit the bottom). What is the minimum channel length?

[warty goblin]

I make 2 and 3/8, assuming that the ends of the channel are shaped to exactly fit the spheres, by the following logic.


This is most easily done by counting parts. Divide each sphere into hemispheres. Each hemisphere that is in contact with a hemisphere of another sphere requires 1/2 lengths, since the spheres cannot get closer than touching and have radius 1/2. This gets us length 2. Now to account for the remaining two hemispheres. Since the channel is of width 3/8, the diameter of the circular profile of the sphere defined by the contact points with the channel is 3/8. Each edge hemisphere then requires 3/16 length channel to accommodate, and there are two of them, hence the remaining 3/8.

[Occasional Sage]

Thanks, I appreciate the help! I came up with another way to logic through this and came to the same number, which is reassuring.