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The wheel and axle is one of six simple machines identified by Renaissance scientists drawing from Greek texts on technology.^{[1]} The wheel and axle is generally considered to be a wheel attached to an axle so that these two parts rotate together in which a force is transferred from one to the other. In this configuration a hinge, or bearing, supports the rotation of the axle.
Hero of Alexandria identified the wheel and axle as one of the five simple machines used to lift weights.^{[2]} This is interpreted to be the windlass which consists of 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 a well.^{[3]} This is thought to have been in the form of the windlass which consists of 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 a well.^{[4]}
This system is a version of the lever with loads applied tangentially to the perimeters of the wheel and axle, respectively, that are balanced around the hinge, which is the fulcrum. The mechanical advantage of the wheel and axle is the ratio of the distances from the fulcrum to the applied loads, or what is the same thing the ratio of the radial dimensions of the wheel and axle.^{[5]}
The earliest welldated depiction of a wheeled vehicle (a wagon—four wheels, two axles) is on the Bronocice pot, a ca. 3635–3370 BC ceramic vase, excavated in a Funnelbeaker culture settlement in southern Poland.^{[6]}
The oldest known 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 is 120 cm long and made of oak.^{[7]}
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. Forces applied to the edges of the two disks, or cylinders, provide mechanical advantage. When used as the wheel of a cart 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.^{[8]}^{[9]}
Assuming the wheel and axle does not dissipate or store energy, the power generated by forces applied to the wheel must equal the power out 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.^{[10]}
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_{A} is applied to the edge of the wheel A and the force F_{B} 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
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.^{[5]} 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.^{[11]}
The ideal mechanical advantage of a wheel and axle is calculated with the following formula:
The actual mechanical advantage of a wheel and axle is calculated with the following formula:
where
Basic Machines and How They Work, United States. Bureau of Naval Personnel, Courier Dover Publications 1965, pp. 3–1 and following preview online
