A wheel with a rod, called an axle, through its center. both parts move together

The wheel and axle is a type of simple machine used to make tasks easier in terms of manipulating force by applying the concept of mechanical advantage. The wheel and axle consists of a round disk, known as a wheel, with a rod through the centre of it, known as the axle. This system uses angular momentum and torque to do work on objects, typically against the force of gravity. The wheel and axle simple machine is closely related to gears.

Like all other simple machines the wheel and axle system changes the force by changing the distance over which the force must be applied; if the input force is reduced to [math]\frac{1}{5}[/math] the output force, then the force must be applied over five times the distance. The work done is always force times distance, and this must always be the same because of the conservation of energy.

The wheel and axle both rotate at the same rate. What this means is that both the axle and the wheel will complete one full rotation in the same amount of time (as opposed to how gears work). Due to the size difference in the radius of the wheel and axle, this means that the distance the two parts rotate through in the same amount of time is different. This is due to the difference in the circumferences of the wheel itself and the axle that supports it. This supplies the conditions for mechanical advantage.

Mechanical advantage

where [math]R[/math] is the radius of the wheel, and [math]r[/math] is the radius of the axle, shown in Figure 2.

A wheel and axle system need a force in order to lift the load, but that force can be less than the weight of the object. Although the force a person needs to apply may not be very large compared to the force it does on the object, the distance they need to rotate the wheel is much larger than the distance the axle rotates through.

Ideally there are no losses during the energy transfer, but in reality there is no such thing as a system with 100% efficiency. Energy will be lost due to non-conservative forces such as friction, but wheel and axle systems often have very high efficiencies.

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simple machine, any of several devices with few or no moving parts that are used to modify motion and the magnitude of a force in order to perform work. They are the simplest mechanisms known that can use leverage (or mechanical advantage) to increase force. The simple machines are the inclined plane, lever, wedge, wheel and axle, pulley, and screw.

An inclined plane consists of a sloping surface; it is used for raising heavy bodies. The plane offers a mechanical advantage in that the force required to move an object up the incline is less than the weight being raised (discounting friction). The steeper the slope, or incline, the more nearly the required force approaches the actual weight. Expressed mathematically, the force F required to move a block D up an inclined plane without friction is equal to its weight W times the sine of the angle the inclined plane makes with the horizontal (θ). The equation is F = W sin θ.

inclined plane

In this representation of an inclined plane, D represents a block to be moved up the plane, F represents the force required to move the block, and W represents the weight of the block. Expressed mathematically, and assuming the plane to be without friction, F = W sin θ.

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The principle of the inclined plane is used widely—for example, in ramps and switchback roads, where a small force acting for a distance along a slope can do a large amount of work.

A lever is a bar or board that rests on a support called a fulcrum. A downward force exerted on one end of the lever can be transferred and increased in an upward direction at the other end, allowing a small force to lift a heavy weight.

levers

Two examples of levers(Left) A crowbar, supported and turning freely on a fulcrum f, multiplies a downward force F applied at point a such that it can overcome the load P exerted by the mass of the rock at point b. If, for example, the length af is five times bf, the force F will be multiplied five times. (Right) A nutcracker is essentially two levers connected by a pin joint at a fulcrum f. If af is three times bf, the force F exerted by hand at point a will be multiplied three times at b, easily overcoming the compressive strength P of the nutshell.

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All early people used the lever in some form, for example, for moving heavy stones or as digging sticks for land cultivation. The principle of the lever was used in the swape, or shadoof, a long lever pivoted near one end with a platform or water container hanging from the short arm and counterweights attached to the long arm. A man could lift several times his own weight by pulling down on the long arm. This device is said to have been used in Egypt and India for raising water and lifting soldiers over battlements as early as 1500 bce.

shadoof

Shadoof, central Anatolia, Turkey.

Noumenon

A wedge is an object that tapers to a thin edge. Pushing the wedge in one direction creates a force in a sideways direction. It is usually made of metal or wood and is used for splitting, lifting, or tightening, as in securing a hammer head onto its handle.

wedge

Wedge used for splitting wood.

Shakespeare

The wedge was used in prehistoric times to split logs and rocks; an ax is also a wedge, as are the teeth on a saw. In terms of its mechanical function, the screw may be thought of as a wedge wrapped around a cylinder.

A wheel and axle is made up of a circular frame (the wheel) that revolves on a shaft or rod (the axle). In its earliest form it was probably used for raising weights or water buckets from wells.

Its principle of operation is best explained by way of a device with a large gear and a small gear attached to the same shaft. The tendency of a force, F, applied at the radius R on the large gear to turn the shaft is sufficient to overcome the larger force W at the radius r on the small gear. The force amplification, or mechanical advantage, is equal to the ratio of the two forces (W:F) and also equal to the ratio of the radii of the two gears (R:r).

wheel and axle arrangements

Two wheel and axle arrangements(A) With a large gear and a small gear attached to the same shaft, or axle, a force F applied at the radius R on the large gear is sufficient to overcome the larger force W at the radius r on the small gear, turning the axle. (B) In a drum and rope arrangement capable of raising weights, a large drum of radius R can be used to turn a small drum. An increase in mechanical advantage can be obtained by using the large drum to turn a small drum with two radii as well as a pulley block. When a force F is applied to the rope wrapped around the large drum, the rope wrapped around the small two-radius drum winds off of d (radius r1) and onto D (radius r2). The force W on the radius of the pulley block P is easily overcome, and the attached weight is lifted.

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If the large and small gears are replaced with large- and small-diameter drums that are wrapped with ropes, the wheel and axle becomes capable of raising weights. The weight being lifted is attached to the rope on the small drum, and the operator pulls the rope on the large drum. In this arrangement the mechanical advantage is the radius of the large drum divided by the radius of the small drum. An increase in the mechanical advantage can be obtained by using a small drum with two radii, r1 and r2, and a pulley block. When a force is applied to the large drum, the rope on the small drum winds onto D and off of d.

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A measure of the force amplification available with the pulley-and-rope system is the velocity ratio, or the ratio of the velocity at which the force is applied to the rope (VF) to the velocity at which the weight is raised (VW). This ratio is equal to twice the radius of the large drum divided by the difference in the radii of the smaller drums D and d. Expressed mathematically, the equation is VF/VW = 2R/(r2 - r1). The actual mechanical advantage W/F is less than this velocity ratio, depending on friction. A very large mechanical advantage may be obtained with this arrangement by making the two smaller drums D and d of nearly equal radius.