# Wheel and axle

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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, respectively, that are balanced around the hinge which is the 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. [1]

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 BCE. One of the earliest examples was discovered at Tepe Pardis, Iran, and dated to 5200–4700 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 BCE. [2] The oldest surviving example, which was found in Ur (modern day Iraq), dates to approximately 3100 BCE. [3]

Evidence of wheeled vehicles appeared by the late 4th millennium BCE. Depictions of wheeled wagons found on clay tablet pictographs at the Eanna district of Uruk, in the Sumerian civilization of Mesopotamia, are dated between 3700–3500 BCE. [4] In the second half of the 4th millennium 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 BCE in the Bronocice clay pot excavated in a Funnelbeaker culture settlement in southern Poland. [5] In nearby Olszanica, a 2.2 m wide door was constructed (2.2 wide doors were constructed) for wagon entry; this barn was 40 m long and had 3 doors. [6] 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 BCE, the axle to 3360–3045 BCE. [7] Two types of early Neolithic European wheel and axle are known; a 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 BCE. [8] Historians believe that there was a diffusion of the wheeled vehicle from the Near East to Europe around the mid-4th millennium BCE. [9]

An early example of a wooden wheel and its axle was found in 2002 at the Ljubljana Marshes some 20 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 ash and oak and had a radius of 70 cm and the axle was 120 cm long and made of oak. [10]

In Roman Egypt, Hero of Alexandria identified the wheel and axle as one of the simple machines used to lift weights. [11] 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. [12]

The wheel and axle was identified as one of six simple machines by Renaissance scientists, drawing from Greek texts on technology. [13]

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. [14] [15]

Assuming the wheel and axle does not dissipate or store energy, that is it has no friction or elasticity, the 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. [16]

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 FA is applied to the edge of the wheel A and the force FB 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

${\displaystyle MA={\frac {F_{B}}{F_{A}}}={\frac {a}{b}}.}$

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. [17] 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, gears) are used. As a wheel and axle is a type of lever, a system of wheels and axles is like a compound lever. [18]

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 is 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:

${\displaystyle \mathrm {IMA} ={F_{\text{out}} \over F_{\text{in}}}={\mathrm {Radius} _{\text{wheel}} \over \mathrm {Radius} _{\text{axle}}}}$

### 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:

${\displaystyle \mathrm {AMA} ={F_{\text{out}} \over F_{\text{in}}}=\eta \cdot {\mathrm {Radius} _{\text{wheel}} \over \mathrm {Radius} _{\text{axle}}}}$

where

${\displaystyle \eta ={P_{\text{out}} \over P_{\text{in}}}}$ is the efficiency of the wheel, the ratio of power output to power input

## Related Research Articles

Mechanical advantage is a measure of the force amplification achieved by using a tool, mechanical device or machine system. The device trades off input forces against movement to obtain a desired amplification in the output force. The model for this is the law of the lever. Machine components designed to manage forces and movement in this way are called mechanisms. An ideal mechanism transmits power without adding to or subtracting from it. This means the ideal mechanism does not include a power source, is frictionless, and is constructed from rigid bodies that do not deflect or wear. The performance of a real system relative to this ideal is expressed in terms of efficiency factors that take into account departures from the ideal.

A pulley is a wheel on an axle or shaft that is designed to support movement and change of direction of a taut cable or belt, or transfer of power between the shaft and cable or belt. In the case of a pulley supported by a frame or shell that does not transfer power to a shaft, but is used to guide the cable or exert a force, the supporting shell is called a block, and the pulley may be called a sheave.

In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.

A simple machine is a mechanical device that changes the direction or magnitude of a force. In general, they can be defined as the simplest mechanisms that use mechanical advantage to multiply force. Usually the term refers to the six classical simple machines that were defined by Renaissance scientists:

A wheel is a circular component that is intended to rotate on an axle bearing. The wheel is one of the key components of the wheel and axle which is one of the six simple machines. Wheels, in conjunction with axles, allow heavy objects to be moved easily facilitating movement or transportation while supporting a load, or performing labor in machines. Wheels are also used for other purposes, such as a ship's wheel, steering wheel, potter's wheel and flywheel.

A lever is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. A lever is a rigid body capable of rotating on a point on itself. On the basis of the locations of fulcrum, load and effort, the lever is divided into three types. Also, leverage is mechanical advantage gained in a system. It is one of the six simple machines identified by Renaissance scientists. A lever amplifies an input force to provide a greater output force, which is said to provide leverage. The ratio of the output force to the input force is the mechanical advantage of the lever. As such, the lever is a mechanical advantage device, trading off force against movement.The formula for mechanical advantage of a lever is

An inclined plane, also known as a ramp, is a flat supporting surface tilted at an angle, with one end higher than the other, used as an aid for raising or lowering a load. The inclined plane is one of the six classical simple machines defined by Renaissance scientists. Inclined planes are widely used to move heavy loads over vertical obstacles; examples vary from a ramp used to load goods into a truck, to a person walking up a pedestrian ramp, to an automobile or railroad train climbing a grade.

A machine is any physical system with ordered structural and functional properties. It may represent human-made or naturally occurring device molecular machine that uses power to apply forces and control movement to perform an action. Machines can be driven by animals and people, by natural forces such as wind and water, and by chemical, thermal, or electrical power, and include a system of mechanisms that shape the actuator input to achieve a specific application of output forces and movement. They can also include computers and sensors that monitor performance and plan movement, often called mechanical systems.

A differential is a gear train with three shafts that has the property that the rotational speed of one shaft is the average of the speeds of the others, or a fixed multiple of that average.

As used in mechanical engineering, the term tractive force can either refer to the total traction a vehicle exerts on a surface, or the amount of the total traction that is parallel to the direction of motion.

A limited-slip differential (LSD) is a type of differential that allows its two output shafts to rotate at different speeds but limits the maximum difference between the two shafts. Limited-slip differentials are often known by the generic trademark Positraction, a brand name owned by General Motors.

A locking differential is a mechanical component, commonly used in vehicles, designed to overcome the chief limitation of a standard open differential by essentially "locking" both wheels on an axle together as if on a common shaft. This forces both wheels to turn in unison, regardless of the traction available to either wheel individually.

Rolling is a type of motion that combines rotation and translation of that object with respect to a surface, such that, if ideal conditions exist, the two are in contact with each other without sliding.

Rolling resistance, sometimes called rolling friction or rolling drag, is the force resisting the motion when a body rolls on a surface. It is mainly caused by non-elastic effects; that is, not all the energy needed for deformation of the wheel, roadbed, etc., is recovered when the pressure is removed. Two forms of this are hysteresis losses, and permanent (plastic) deformation of the object or the surface. Note that the slippage between the wheel and the surface also results in energy dissipation. Although some researchers have included this term in rolling resistance, some suggest that this dissipation term should be treated separately from rolling resistance because it is due to the applied torque to the wheel and the resultant slip between the wheel and ground, which is called slip loss or slip resistance. In addition, only the so-called slip resistance involves friction, therefore the name "rolling friction" is to an extent a misnomer.

A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular (normal) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width. Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle.

A gear train is a mechanical system formed by mounting gears on a frame so the teeth of the gears engage.

An adhesion railway relies on adhesion traction to move the train. Adhesion traction is the friction between the drive wheels and the steel rail. The term "adhesion railway" is used only when it is necessary to distinguish adhesion railways from railways moved by other means, such as by a stationary engine pulling on a cable attached to the cars or by railways that are moved by a pinion meshing with a rack.

A screw is a mechanism that converts rotational motion to linear motion, and a torque to a linear force. It is one of the six classical simple machines. The most common form consists of a cylindrical shaft with helical grooves or ridges called threads around the outside. The screw passes through a hole in another object or medium, with threads on the inside of the hole that mesh with the screw's threads. When the shaft of the screw is rotated relative to the stationary threads, the screw moves along its axis relative to the medium surrounding it; for example rotating a wood screw forces it into wood. In screw mechanisms, either the screw shaft can rotate through a threaded hole in a stationary object, or a threaded collar such as a nut can rotate around a stationary screw shaft. Geometrically, a screw can be viewed as a narrow inclined plane wrapped around a cylinder.

A simple machine that exhibits mechanical advantage is called a mechanical advantage device - e.g.:

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