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The belt problem

The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with radius r₁ and r₂ whose centers are separated by a distance P. 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 $\varphi$, the length of the belt is

$CO+DO+EO+FO+{\text{arc}}CD+{\text{arc}}EF\,\!$

$=2r_{1}\tan(\varphi )+2r_{2}\tan(\varphi )+(2\pi -2\varphi )r_{1}+(2\pi -2\varphi )r_{2}\,\!$

$=2(r_{1}+r_{2})(\tan(\varphi )+\pi -\varphi )\,\!$

This uses the fact that the length of an arc = the radius × the measure of the angle facing the arc in radians.

To find $\varphi$ we see from the similarity of triangles ACO and BEO that

${\frac {AO}{BO}}={\frac {AC}{BE}}\,\!$

$\Rightarrow {\frac {P-x}{x}}={\frac {r_{1}}{r_{2}}}\,\!$

$\Rightarrow {\frac {P}{x}}={\frac {r_{1}+r_{2}}{r_{2}}}\,\!$

$\Rightarrow {x}={\frac {Pr_{2}}{r_{1}+r_{2}}}\,\!$

$\cos(\varphi )={\frac {r_{2}}{x}}={\frac {r_{2}}{\left({\dfrac {Pr_{2}}{r_{1}+r_{2}}}\right)}}={\frac {r_{1}+r_{2}}{P}}\,\!$

$\Rightarrow \varphi =\cos ^{-1}\left({\frac {r_{1}+r_{2}}{P}}\right)\,\!$

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

The pulley problem

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

$2P\sin \left({\frac {\theta }{2}}\right)+r_{1}(2\pi -\theta )+r_{2}{\theta }\,,$

where r₁ represents the radius of the larger pulley, r₂ represents the radius of the smaller one, and:

$\theta =2\cos ^{-1}\left({\frac {r_{1}-r_{2}}{P}}\right)\,.$

Applications

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

See also

• Tangent lines to circles

References

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