belt problem
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.
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
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.
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
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.
This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors (see full disclaimer)
