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{{Short description|Simple machine consisting of a wheel attached to a tiny axel}}
[[Image:Wheelaxle quackenbos.gif|thumb|The [[windlass]] is a well-known application of the wheel and axle.]]

The '''wheel and axle''' is a [[simple machine]], consisting of a [[wheel]] attached to a smaller [[axle]] so that these two parts rotate together, in which a force is transferred from one to the other. The wheel and axle can be viewed as a version of the [[Lever]], with a drive force applied tangentially to the perimeter of the wheel, and a load force applied to the axle supported in a [[Bearing (mechanical)|bearing]], which serves as a fulcrum.

==History==

The [[Halaf]] culture of 6500–5100 BCE has been credited with the earliest depiction of a wheeled vehicle, but this is doubtful as there is no evidence of Halafians using either wheeled vehicles or even pottery wheels.<ref>{{cite book|title=New Light on the Most Ancient East|url=https://archive.org/details/in.ernet.dli.2015.185120|author=V. Gordon Childe|year=1928|page=110}}</ref>

One of the first applications of the wheel to appear was the [[potter's wheel]], used by prehistoric cultures to fabricate clay pots. The earliest type, known as "tournettes" or "slow wheels", were known in the [[Middle East]] by the 5th millennium&nbsp;BCE. One of the earliest examples was discovered at Tepe Pardis, [[Iran]], and dated to 5200–4700&nbsp;BCE. These were made of stone or clay and secured to the ground with a peg in the center, but required significant effort to turn. True potter's wheels, which are freely-spinning and have a wheel and axle mechanism, were developed in [[Mesopotamia]] ([[Iraq]]) by 4200–4000&nbsp;BCE.<ref>{{cite book|title=A Companion to the Archaeology of the Ancient Near East|author=D.T. Potts|year=2012|page=285}}</ref> The oldest surviving example, which was found in [[Ur]] (modern day [[Iraq]]), dates to approximately 3100&nbsp;BCE.<ref name=Moorey1994>{{cite book|last=Moorey|first=Peter Roger Stuart|date=1999|orig-year=1994|title=Ancient Mesopotamian Materials and Industries: The Archaeological Evidence|url=https://books.google.com/books?id=P_Ixuott4doC|location=Winona Lake, IN|publisher=Eisenbrauns|isbn=978-1-57506-042-2|page=146}}</ref>

Evidence of wheeled vehicles appeared by the late [[4th millennium&nbsp;BCE]]. Depictions of wheeled [[wagon]]s found on [[clay tablet]] [[pictographs]] at the [[Uruk#Eanna District|Eanna district]] of [[Uruk]], in the [[Sumer]]ian civilization of Mesopotamia, are dated between 3700–3500 BCE.<ref>{{cite journal |last1=Attema |first1=P. A. J. |last2=Los-Weijns |first2=Ma |last3=Pers |first3=N. D. Maring-Van der |title=Bronocice, Flintbek, Uruk, JEbel Aruda and Arslantepe: The Earliest Evidence Of Wheeled Vehicles In Europe And The Near East |journal=Palaeohistoria |date=December 2006 |volume=47/48 |publisher=[[University of Groningen]] |pages=10–28 (11) |isbn=9789077922187 |url=https://books.google.com/books?id=qqEqjtKJQ3YC&pg=PA11}}</ref> In the second half of the 4th millennium&nbsp;BCE, evidence of wheeled vehicles appeared near-simultaneously in the [[Northern Caucasus]] ([[Maykop culture]]) and [[Eastern Europe]] ([[Cucuteni–Trypillian culture]]). Depictions of a wheeled vehicle appeared between 3500 and 3350&nbsp;BCE in the [[Bronocice pot|Bronocice clay pot]] excavated in a [[Funnelbeaker culture]] settlement in southern [[Poland]].<ref>{{cite book |author=Anthony, David A. |title=The horse, the wheel, and language: how Bronze Age riders from the Eurasian steppes shaped the modern world |url=https://archive.org/details/horsewheellangua00davi |url-access=limited |publisher=Princeton University Press |location=Princeton, NJ |year=2007 |page=[https://archive.org/details/horsewheellangua00davi/page/n39 67] |isbn=978-0-691-05887-0 }}</ref> In nearby [[Zwierzyniec (Kraków)|Olszanica]], a 2.2 m wide door was constructed for wagon entry; this barn was 40 m long and had 3 doors.<ref>{{Cite web | url=http://structuralarchaeology.blogspot.com/2009/10/35-olszanica-longhouse-6-why-has-it-got.html | title=35. Olszanica Longhouse 6: Why has it got wide doors?| date=2018-10-26}}</ref> Surviving evidence of a wheel–axle combination, from Stare Gmajne near Ljubljana in Slovenia ([[Ljubljana Marshes Wooden Wheel]]), is dated within two [[standard deviations]] to 3340–3030&nbsp;BCE, the axle to 3360–3045&nbsp;BCE.<ref>Velušček, A.; Čufar, K. and Zupančič, M. (2009) "Prazgodovinsko leseno kolo z osjo s kolišča Stare gmajne na Ljubljanskem barju", pp. 197–222 in A. Velušček (ed.). ''Koliščarska naselbina Stare gmajne in njen as. Ljubljansko barje v 2. polovici 4''. tisočletja pr. Kr. Opera Instituti Archaeologici Sloveniae 16. Ljubljana.</ref> Two types of early Neolithic European wheel and axle are known; a [[Prehistoric pile dwellings around the Alps|circumalpine]] type of wagon construction (the wheel and axle rotate together, as in Ljubljana Marshes Wheel), and that of the [[Baden culture]] in Hungary (axle does not rotate). They both are dated to c. 3200–3000&nbsp;BCE.<ref>Fowler, Chris; Harding, Jan and Hofmann, Daniela (eds.) (2015). [https://books.google.com/books?id=2PAkBwAAQBAJ&pg=PA109 ''The Oxford Handbook of Neolithic Europe.''] OUP Oxford. {{ISBN|0-19-166688-2}}. p. 109.</ref> Historians believe that there was a diffusion of the wheeled vehicle from the [[Near East]] to Europe around the mid-4th millennium&nbsp;BCE.<ref>{{cite journal |last1=Attema |first1=P. A. J. |last2=Los-Weijns |first2=Ma |last3=Pers |first3=N. D. Maring-Van der |title=Bronocice, Flintbek, Uruk, JEbel Aruda and Arslantepe: The Earliest Evidence Of Wheeled Vehicles In Europe And The Near East |journal=Palaeohistoria |date=December 2006 |volume=47/48 |publisher=[[University of Groningen]] |pages=10–28 (19–20) |isbn=9789077922187 |url=https://books.google.com/books?id=qqEqjtKJQ3YC&pg=PA19}}</ref>

An early example of a wooden wheel and its axle was found in 2002 at the Ljubljana Marshes some 20&nbsp;km south of [[Ljubljana]], the capital of Slovenia. According to [[radiocarbon dating]], it is between 5,100 and 5,350 years old. The wheel was made of [[Fraxinus excelsior|ash]] and [[oak]] and had a [[radius]] of 70&nbsp;cm and the axle was 120&nbsp;cm long and made of oak.<ref name="Slovenia">{{cite web |title=World's Oldest Wheel Found in Slovenia |url=http://www.ukom.gov.si/en/media_room/background_information/culture/worlds_oldest_wheel_found_in_slovenia/ |date=March 2003 |publisher=Government Communication Office of the Republic of Slovenia |author=Aleksander Gasser |access-date=19 August 2010}}</ref>

In [[China]], the earliest evidence of spoked wheels comes from [[Qinghai]] in the form of two wheel hubs from a site dated between 2000 and 1500 BCE.<ref>{{Cite web|title=Chinese Bronze Age Wheeled Vehicles|url=http://www.sino-platonic.org/abstracts/spp099_wheeled_vehicles.html|access-date=2022-01-28|website=www.sino-platonic.org}}</ref>

In [[Roman Egypt]], [[Hero of Alexandria]] identified the wheel and axle as one of the [[simple machine]]s used to lift weights.<ref name="Usher">{{cite book 
  | last = Usher
  | first = Abbott Payson 
  | title = A History of Mechanical Inventions
  | publisher = Courier Dover Publications
  | year = 1988
  | location = US
  | pages = 98
  | url = https://books.google.com/books?id=xuDDqqa8FlwC&q=wedge+and+screw&pg=PA196
  | isbn = 048625593X}}</ref> This is thought to have been in the form of the [[windlass]] which consists of a crank or [[pulley]] connected to a cylindrical barrel that provides mechanical advantage to wind up a rope and lift a load such as a bucket from the well.<ref>Elroy McKendree Avery, [[iarchive:elementaryphysi01avergoog/page/n470|<!-- pg=459 quote=wheel and axle. --> Elementary Physics]], New York : Sheldon & Company, 1878.</ref>

The wheel and axle was identified as one of six simple machines by Renaissance scientists, drawing from Greek texts on technology.<ref>Wheel and Axle, The World Book Encyclopedia, World Book Inc., 1998, pp. 280–281</ref>

==Mechanical advantage==

The simple machine called a ''wheel and axle'' refers to the assembly formed by two disks, or cylinders, of different diameters mounted so they rotate together around the same axis. The thin rod which needs to be turned is called the axle and the wider object fixed to the axle, on which we apply force is called the wheel. A tangential force applied to the periphery of the large disk can exert a larger force on a load attached to the axle, achieving [[mechanical advantage]]. When used as the [[wheel]] of a [[wheeled vehicle]] the smaller cylinder is the axle of the wheel, but when used in a [[windlass]], [[winch]], and other similar applications (see medieval mining lift to right) the smaller cylinder may be separate from the axle mounted in the bearings. It cannot be used separately.<ref>Prater, Edward L. (1994), [http://www.constructionknowledge.net/public_domain_documents/Div_1_General/Basic_Skills/Basic%20Machines%20NAVEDTRA%2014037%201994.pdf Basic Machines], Naval Education and Training Professional Development and Technology Center, NAVEDTRA 14037</ref><ref>Bureau of Naval Personnel, 1971, [http://www.webpal.org/SAFE/aaarecovery/5_simple_technology/basic_machines.pdf Basic Machines and How They Work], Dover Publications.</ref>

Assuming the wheel and axle does not dissipate or store energy, that is it has no [[friction]] or [[elasticity (physics)|elasticity]], the [[power (physics)|power]] input by the force applied to the wheel must equal the power output at the axle. As the wheel and axle system rotates around its bearings, points on the circumference, or edge, of the wheel move faster than points on the circumference, or edge, of the axle. Therefore, a force applied to the edge of the wheel must be less than the force applied to the edge of the axle, because power is the product of force and velocity.<ref>J. J. Uicker, G. R. Pennock, and J. E. Shigley, 2003, '''Theory of Machines and Mechanisms,''' Oxford University Press, New York.</ref>

Let ''a'' and ''b'' be the distances from the center of the bearing to the edges of the wheel ''A'' and the axle ''B.'' If the input force ''F<sub>A</sub>'' is applied to the edge of the wheel ''A'' and the force ''F<sub>B</sub>'' at the edge of the axle ''B'' is the output, then the ratio of the velocities of points ''A'' and ''B'' is given by ''a/b'', so the ratio of the output force to the input force, or [[mechanical advantage]], is given by
:<math>MA = \frac{F_B}{F_A} = \frac{a}{b}.</math>

The mechanical advantage of a simple machine like the wheel and axle is computed as the ratio of the resistance to the effort. The larger the ratio the greater the multiplication of force (torque) created or distance achieved. By varying the radii of the axle and/or wheel, any amount of mechanical advantage may be gained.<ref name="Bowser">Bowser, Edward Albert, 1890, [https://books.google.com/books?id=UdM3AAAAMAAJ&dq=wheel+and+axle&pg=PR9 An elementary treatise on analytic mechanics: with numerous examples.] (Originally from the University of Michigan) D. Van Nostrand Company, pp. 190</ref> In this manner, the size of the wheel may be increased to an inconvenient extent. In this case a system or combination of wheels (often toothed, that is, [[gear]]s) are used. As a wheel and axle is a type of lever, a system of wheels and axles is like a compound lever.<ref>Baker, C.E. The Principles and Practice of Statics and Dynamics{{nbsp}}... for the Use of Schools and Private Students. London: John Weale, 59, High Holborn. 1851 pp. 26–29 [https://archive.org/details/principlesandpr01bakegoog/page/n35 <!-- pg=26 quote="wheel and axle". --> read online or download full text]</ref>

On a powered wheeled vehicle the transmission exerts a force on the axle which has a smaller radius than the wheel. The mechanical advantage is therefore much less than 1. The wheel and axle of a car are therefore not representative of a simple machine (whose purpose is to increase the force). The friction between wheel and road is actually quite low, so even a small force exerted on the axle is sufficient. The actual advantage lies in the large rotational speed at which the axle is rotating thanks to the transmission.

===Ideal mechanical advantage===

The mechanical advantage of a wheel and axle with no [[friction]] is called the [[ideal mechanical advantage]] (IMA). It is calculated with the following formula:

:<math>\mathrm{IMA} = {F_\text{out} \over F_\text{in}} = { \mathrm{Radius}_\text{wheel} \over \mathrm{Radius}_\text{axle}} </math>

===Actual mechanical advantage===

All actual wheels have friction, which dissipates some of the power as heat. The [[actual mechanical advantage]] (AMA) of a wheel and axle is calculated with the following formula:

:<math>\mathrm{AMA} = {F_\text{out} \over F_\text{in}} = \eta \cdot { \mathrm{Radius}_\text{wheel} \over \mathrm{Radius}_\text{axle}} </math>

where 
:<math>\eta = {P_\text{out} \over P_\text{in} }</math> is the efficiency of the wheel, the ratio of power output to power input

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==References==

{{Reflist}}

==Additional resources==

Basic Machines and How They Work, United States. Bureau of Naval Personnel, Courier Dover Publications 1965, pp.&nbsp;3–1 and following [https://books.google.com/books?id=yDKzy4rKEg0C&dq=%22wheel+and+axle%22&pg=SA3-PA1 preview online]

{{Simple machines}}
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[[Category:Historic machinery]]
[[Category:Simple machines]]
[[Category:Vehicle technology]]