Pulley Problem
   HOME

TheInfoList



Click Here for Items Related To - Pulley Problem
OR:
Pulley Problem on:   [Wikipedia]   [Google]   [Amazon]

The belt problem is a
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
problem which requires finding the length of a crossed belt that connects two circular
pulley Sheave without a rope A pulley is a wheel on an axle or shaft enabling a taut cable or belt passing over the wheel to move and change direction, or transfer power between itself and a shaft. A pulley may have a groove or grooves between flan ...
s with
radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
''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 concerned with relationships between angles and side lengths of triangles. In particular, the trigonometric functions relate the angles of a right triangle with ratios of its side lengths. The fiel ...
and the concepts of the bitangent line, the
vertical angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
, 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 corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s ACO and ADO are
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modu ...
right angled triangle A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two Edge (geometry), sides are perpendicular, forming a right angle ( turn (unit), turn or 90 degree (angle), degree ...
s, as are
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s BEO and BFO. In addition,
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s ACO and BEO are similar. Therefore
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s CAO, DAO, EBO and FBO are all equal. Denoting this
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
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. It is defined such that one radian is the angle subtended at ...
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 or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
× the measure of the
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
facing the arc. To find \varphi we see from the similarity of
triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...
s ACO and BEO that :\frac = \frac \,\! :\Rightarrow \frac = \frac \,\! :\Rightarrow \frac = \frac \,\! :\Rightarrow = \frac \,\! : \cos(\varphi) = \frac = \frac = \frac \,\! :\Rightarrow \varphi=\arccos\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 Sheave without a rope A pulley is a wheel on an axle or shaft enabling a taut cable or belt passing over the wheel to move and change direction, or transfer power between itself and a shaft. A pulley may have a groove or grooves between flan ...
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\arccos\left(\frac\right)\, .


Applications

The belt problem is used in the design of
aeroplane An airplane (American English), or aeroplane (Commonwealth English), informally plane, is a fixed-wing aircraft that is propelled forward by thrust from a jet engine, Propeller (aircraft), propeller, or rocket engine. Airplanes come in a vari ...
s,
bicycle gearing Bicycle gearing is the aspect of a Bicycle drivetrain systems, bicycle drivetrain that determines the relation between the cadence (cycling), cadence, the rate at which the rider pedals, and the rate at which the drive Bicycle wheel, wheel tur ...
,
car A car, or an automobile, is a motor vehicle with wheels. Most definitions of cars state that they run primarily on roads, seat one to eight people, have four wheels, and mainly transport people rather than cargo. There are around one billio ...
s, and other items with
pulley Sheave without a rope A pulley is a wheel on an axle or shaft enabling a taut cable or belt passing over the wheel to move and change direction, or transfer power between itself and a shaft. A pulley may have a groove or grooves between flan ...
s or belts that cross each other. The pulley problem is also used in the design of
conveyor belt A conveyor belt is the carrying medium of a belt conveyor system (often shortened to a belt conveyor). A belt conveyor system consists of two or more pulleys (sometimes referred to as drums), with a closed loop of carrying medium—the conveyor b ...
s found in
airport An airport is an aerodrome with extended facilities, mostly for commercial Aviation, air transport. They usually consist of a landing area, which comprises an aerially accessible open space including at least one operationally active surf ...
luggage Baggage, or luggage, consists of bags, cases, and containers which hold a traveler's personal articles while the traveler is in transport, transit. A modern tourist, traveler can be expected to have packages containing clothing, toiletries, sma ...
belts and
automated Automation describes a wide range of technologies that reduce human intervention in processes, mainly by predetermining decision criteria, subprocess relationships, and related actions, as well as embodying those predeterminations in machine ...
factory A factory, manufacturing plant or production plant is an industrial facility, often a complex consisting of several buildings filled with machinery, where workers manufacture items or operate machines which process each item into another. Th ...
lines.Trigonometry used in conveyor belts


See also

*
Tangent lines to circles In Euclidean geometry, Euclidean plane geometry, a tangent line to a circle is a Line (geometry), line that touches the circle at exactly one Point (geometry), point, never entering the circle's interior. Tangent lines to circles form the subject ...


References

{{reflist Trigonometry