Editing Pulley (section)

==Rope and pulley systems==
[[File:Pulley in Oil Well.jpg|thumb|upright|Pulley in oil derrick]]
[[Image:Crane pulley 4x.jpg|thumb|upright|A hoist using the compound pulley system yielding an advantage of 4. The single fixed pulley is installed on the [[hoist (device)|hoist]]. The two movable pulleys (joined) are attached to the [[Lifting hook|hook]]. One end of the rope is attached to the crane frame, another to the winch.]]
A rope and pulley system—that is, a [[block and tackle]]—is characterised by the use of a single continuous rope to transmit a tension force around one or more pulleys to lift or move a load—the rope may be a light line or a strong cable. This system is included in the list of [[simple machine]]s identified by Renaissance scientists.<ref>{{cite book
  | last = Avery
  | first = Elroy
  | title = Elementary physics
  | year = 1878
  | publisher = Sheldon and company
  | url = https://archive.org/details/elementaryphysi01avergoog
| page = [https://archive.org/details/elementaryphysi01avergoog/page/n471 459]
  | quote = wheel and axle.
  }}</ref><ref name="Bowser">{{cite book
  | last = Bowser
  | first = Edward
  | title = An elementary treatise on analytic mechanics: With numerous examples
  | publisher = D. Van Nostrand company
  | url=https://books.google.com/books?id=UdM3AAAAMAAJ&q=simple+machine&pg=PR9
  | edition = 5
  | date = 1890
  | pages = 180}}</ref>

If the rope and pulley system does not dissipate or store energy, then its [[mechanical advantage]] is the number of parts of the rope that act on the load. This can be shown as follows.

Consider the set of pulleys that form the moving block and the parts of the rope that support this block. If there are ''p'' of these parts of the rope supporting the load ''W,'' then a force balance on the moving block shows that the tension in each of the parts of the rope must be ''W/p.'' This means the input force on the rope is ''T''=''W/p.'' Thus, the block and tackle reduces the input force by the factor ''p.''
<gallery>
Image:Polispasto2B.jpg|A gun tackle has a single pulley in both the fixed and moving blocks with two rope parts supporting the load ''W.''
Image:Pulley1a.svg|Separation of the pulleys in the gun tackle show the force balance that results in a rope tension of ''W/2.'' 
</gallery>
<gallery>
Image:Polispasto4.jpg|A double tackle has two pulleys in both the fixed and moving blocks with four rope parts supporting the load ''W.''
Image:Pulley3a.svg|Separation of the pulleys in the double tackle show the force balance that results in a rope tension of ''W/4.''
</gallery>

===Method of operation===
The simplest theory of operation for a pulley system assumes that the pulleys and lines are weightless and that there is no energy loss due to friction. It is also assumed that the lines do not stretch.

In equilibrium, the forces on the moving block must sum to zero. In addition the tension in the rope must be the same for each of its parts. This means that the two parts of the rope supporting the moving block must each support half the load.
<gallery>
Image:Polea-simple-fija.jpg|Fixed pulley
Image:Pulley0.svg|Diagram 1: The load ''F'' on the moving pulley is balanced by the tension in two parts of the rope supporting the pulley.
Image:Polea-simple-movil2.jpg|Movable pulley
Image:Pulley1.svg|Diagram 2: A movable pulley lifting the load ''W'' is supported by two rope parts with tension ''W/2.''
</gallery>

These are different types of pulley systems:
* '''Fixed:''' A ''fixed'' pulley has an axle mounted in bearings attached to a supporting structure. A fixed pulley changes the direction of the force on a rope or belt that moves along its circumference. Mechanical advantage is gained by combining a fixed pulley with a movable pulley or another fixed pulley of a different diameter. 
* '''Movable:''' A ''movable'' pulley has an axle in a movable block. A single movable pulley is supported by two parts of the same rope and has a mechanical advantage of two.
* '''Compound:''' A combination of fixed and movable pulleys forms a [[block and tackle]]. A ''block and tackle'' can have several pulleys mounted on the fixed and moving axles, further increasing the mechanical advantage.

<gallery>
Image:Pulley2.svg|Diagram 3: The gun tackle "rove to advantage" has the rope attached to the moving pulley. The tension in the rope is ''W/3'' yielding an advantage of three.
Image:Pulley2a.svg|Diagram 3a: The Luff tackle adds a fixed pulley "rove to disadvantage." The tension in the rope remains ''W/3'' yielding an advantage of three.
</gallery>

The mechanical advantage of the gun tackle can be increased by interchanging the fixed and moving blocks so the rope is attached to the moving block and the rope is pulled in the direction of the lifted load. In this case the [[block and tackle]] is said to be "rove to advantage."<ref>{{cite web
 | title = Seamanship Reference, Chapter 5, General Rigging
 | publisher = sccheadquarters.com
 | url = http://www.sccheadquarters.com/Data/Sites/1/media/Training/TrainingInstructions/seamanship/Chapter%205%20-%20General%20Rigging.pdf
 }}</ref> Diagram 3 shows that now three rope parts support the load ''W'' which means the tension in the rope is ''W/3''. Thus, the mechanical advantage is three.

By adding a pulley to the fixed block of a gun tackle the direction of the pulling force is reversed though the mechanical advantage remains the same, Diagram 3a. This is an example of the Luff tackle.

===Free body diagrams===
The [[mechanical advantage]] of a pulley system can be analysed using [[free body diagram]]s which balance the [[tension (physics)|tension force]] in the rope with the [[force of gravity]] on the load. In an ideal system, the massless and frictionless pulleys do not dissipate energy and allow for a change of direction of a rope that does not stretch or wear. In this case, a force balance on a free body that includes the load, ''W'', and ''n'' supporting sections of a rope with tension ''T'', yields:

:<math>n T -W = 0.</math>

The ratio of the load to the input tension force is the mechanical advantage ''MA'' of the pulley system,<ref name="Tiner">{{Cite book |last=Tiner |first=John Hudson |url=https://www.worldcat.org/oclc/892430550 |title=Exploring the world of physics : from simple machines to nuclear energy |date=2006-05-01 |publisher=[[New Leaf Publishing Group]] |isbn=978-1-61458-156-7 |location=Green Forest, AR |pages=68 |language=en |lccn=2005936557 |oclc=892430550}}</ref>

:<math>MA = \frac{W}{T} = n.</math>

Thus, the mechanical advantage of the system is equal to the number of sections of rope supporting the load.