A Belt Around Two Pulleys

  1. Given two pulleys of radius (X) and (Y), and the distance (D) between their centers, find the circumference (Z) of the belt which goes around both pulleys. Assume the belt touches at least part of each pulley, and the pulleys can overlap (D can be less than X+Y).
  2. With the definitions as above, if you're given X, Y, and Z, find D.
Source: Reader David Quinn, though I wouldn't be surprised to hear of other sources.
Solutions were received from Nick McGrath, Allan Christensen, Sudipta Das, Philippe Fondanaiche, Jozef Hanenberg, and Paul Botham. All solvers found a solution for part 1 and pointed out that part 2 was either impossible or had to be solved by approximations. Thanks to Sudipta Das for mentioning a similar problem at the SMSU problem corner.

Paul Botham's solutions are quite succinct, yet not difficult to understand their derivation:

1. Z in terms of X, Y, D

Assume Y > X (or Y = X)

Let A be the angle each straight section of belt makes to horizontal


Z = 2 * D * COS[A] + PI * (X+Y) + 2 * (Y-X) * A

where SIN[A] = (Y-X) / D
PI = 3.141...

2. D in terms of X, Y, Z

[Z - PI * (X+Y)] / [2 * (Y-X) ] = COT[A] + A

solve numerically for A

then  D = (Y-X) / SIN[A]

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