Pulley Problem
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The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
''r''1 and ''r''2 whose centers are separated by a distance ''P''. The solution of the belt problem requires
trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...
and the concepts of the bitangent line, the vertical angle, and
congruent angles In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be t ...
.


Solution

Clearly
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
s ACO and ADO are congruent
right angled triangle A right triangle (American English) or right-angled triangle ( British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right ...
s, as are
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
s BEO and BFO. In addition,
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
s ACO and BEO are similar. Therefore
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s CAO, DAO, EBO and FBO are all equal. Denoting this
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
by \varphi (denominated in
radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s), the length of the belt is :CO + DO + EO + FO + \text CD + \text 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 exploits the convenience of denominating angles in radians that the length of an arc = the
radius In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
× the measure of the
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
facing the arc. To find \varphi we see from the
similarity Similarity may refer to: In mathematics and computing * Similarity (geometry), the property of sharing the same shape * Matrix similarity, a relation between matrices * Similarity measure, a function that quantifies the similarity of two objects * ...
of
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...
s ACO and BEO that :\frac = \frac \,\! :\Rightarrow \frac = \frac \,\! :\Rightarrow \frac = \frac \,\! :\Rightarrow = \frac \,\! : \cos(\varphi) = \frac = \frac = \frac \,\! :\Rightarrow \varphi=\cos^\left(\frac\right) \,\! For fixed ''P'' the length of the belt depends only on the sum of the radius values ''r''1 + ''r''2, and not on their individual values.


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 :2 P \sin\left(\frac\right)+r_1(2\pi-\theta)+r_2\, , where ''r''1 represents the radius of the larger pulley, ''r''2 represents the radius of the smaller one, and: :\theta=2\cos^\left(\frac\right)\, .


Applications

The belt problem is used in the design of
aeroplane An airplane or aeroplane (informally plane) is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, propeller, or rocket engine. Airplanes come in a variety of sizes, shapes, and wing configurations. The broad spec ...
s,
bicycle gearing Bicycle gearing is the aspect of a bicycle drivetrain that determines the relation between the cadence, the rate at which the rider pedals, and the rate at which the drive wheel turns. On some bicycles there is only one gear and, therefore, ...
,
car A car or automobile is a motor vehicle with wheels. Most definitions of ''cars'' say that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people instead of goods. The year 1886 is regarded as t ...
s, and other items with pulleys or belts that cross each other. The pulley problem is also used in the design of conveyor belts found in
airport An airport is an aerodrome with extended facilities, mostly for commercial air transport. Airports usually consists of a landing area, which comprises an aerially accessible open space including at least one operationally active surfa ...
luggage Baggage or luggage consists of bags, cases, and containers which hold a traveler's personal articles while the traveler is in transit. A modern traveler can be expected to have packages containing clothing, toiletries, small possessions, trip ...
belts and
automated Automation describes a wide range of technologies that reduce human intervention in processes, namely by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machines ...
factory A factory, manufacturing plant or a production plant is an industrial facility, often a complex consisting of several buildings filled with machinery, where workers manufacture Manufacturing is the creation or production of goods with t ...
lines.Trigonometry used in conveyor belts


See also

*
Tangent lines to circles In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...


References

{{reflist Trigonometry