There is a perfect sphere of diameter 100 cms. resting up against a perfectly straight wall and a perfectly straight floor.
What is the diameter of the largest sphere that can pass through the gap between the wall, floor and the sphere ?
The situation at the point at which the smaller sphere is most constricted is that at which the centers of the two spheres and their points of contact with the wall and floor all lie on one plane.
At that moment, the given plane contains the largest cross section of each sphere, with circles equal in diameter to those of the spheres.
Construct a square on that plane with corners at the center of the larger circle (and sphere), the corner where the wall meets the floor and the points of tangency of the larger sphere with the wall and with the floor. The edge of this square is 50 cm. Its diagonal is therefore 50 √(2).
Construct the same type of square with the smaller circle, making one corner the center of that sphere (and circle). Let the smaller circle's radius be called r, which is also the length of one side of the new square.
The smaller square's diagonal lies along the larger square's diagonal, and we can find the length of the the smaller one by subtracting lengths that we know from that of the larger.
From the center of the larger circle (the beginning of the larger diagonal) to the point of tangency with the smaller circle is the 50-cm radius of the larger circle. From there to the center of the smaller circle (where the smaller diagonal begins) is r cm.
The ratio of the side of the smaller square to its diagonal is still 1/√(2), so
r/(50(√2) - 50 - r) = 1/√(2)
r√(2) = 50(√2) - 50 - r
r((√2)+1) = 50 ((√2) - 1)
r = 50((√2)-1)/((√2)+1)
so the diameter is
100((√2)-1)/((√2)+1) cm or about 17.157 cm.
Posted by Charlie
on 2003-09-13 13:31:48