Belt problem

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The belt problem is the problem of finding the length of a crossed belt that connects two circular pulleys with radius r1 and r2 whose centres are separated by a distance P (see diagram below). The solution of the belt problem requires trigonometry and the concepts of the bitangent line, the vertical angle, and congruent angles.

The belt problem.
The belt problem.

[edit] Solution

Clearly triangles ACO and ADO are congruent right angled triangles, as are triangles BEO and BFO. In addition, triangles ACO and BEO are similar. Therefore angles CAO, DAO, EBO and FBO are all equal. Denoting this angle by φ, the length of the belt is

CO + DO + EO + FO + arc CD + arc EF
= 2r1tan(φ) + 2r2tan(φ) + (2π − 2φ)r1 + (2π − 2φ)r2
= 2(r1 + r2)(tan(φ) + π − φ).

To find φ we see from the similarity of triangles ACO and BEO that

\frac{AO}{BO}=\frac{AC}{BE}
\Rightarrow \frac{P-x}{x} = \frac{r_1}{r_2}
\Rightarrow \frac{P}{x} = \frac{r_1+r_2}{r_2}
\Rightarrow cos(\phi) = \frac{r_2}{x} = \frac{r_1+r_2}{P}
\Rightarrow \phi=cos^{-1}(\frac{r_1+r_2}{P}).

For fixed P the length of the belt depends only on the sum of the radius values r1+r2, and not on their individual values.

[edit] Pulley problem

The pulley problem.
The pulley problem.

There are other types of problems similar to the belt problem. The pulley problem, as shown above, is similar to the belt problem, except that the belt does not cross itself. In the pulley problem the length of the belt is

2Psin(φ) + 2(r1 + r2)(π − φ)

where

\phi=cos^{-1}(\frac{r_1-r_2}{P}).

[edit] Applications

The belt problem is used in real life in the design of aeroplanes, bicycle gears, cars, and other items with pulleys or belts that cross each other like the belt problem does. The pulley problem is used in the design of airport luggage belts and automated factory lines.