Wheel and axle
The wheel and axle is one of six simple machines identified by Renaissance scientists drawing from Greek texts on technology. The wheel and axle consists 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. A hinge or bearing supports the axle, allowing rotation. It can amplify force; a small force applied to the periphery of the large wheel can move a larger load attached to the axle.
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. 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 diameter of the wheel and axle. A major application is in wheeled vehicles, in which the wheel and axle are used to reduce friction of the moving vehicle with the ground. Other examples of devices which use the wheel and axle are capstans, belt drives and gears.
Precursors of wheels, 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 effort to turn. True (freely-spinning) wheels were apparently in use in Mesopotamia by 3500 BCE and possibly as early as 4000 BCE, and the oldest surviving example, which was found in Ur (modern day Iraq), dates to approximately 3100 BCE.
Evidence of wheeled vehicles appears in the second half of the 4th millennium BCE, near-simultaneously in Mesopotamia (Sumerian civilization), the Northern Caucasus (Maykop culture) and Central Europe (Cucuteni-Trypillian culture).
An early well-dated depiction of a wheeled vehicle (a wagon—four wheels, two axles) is on the Bronocice pot, a ca. 3635–3370 BCE ceramic vase, excavated in a Funnelbeaker culture settlement in southern Poland.
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.
In Roman Egypt, Hero of Alexandria identified the wheel and axle as one of the simple machines used to lift weights. 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.
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.
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.
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
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. 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.
Ideal mechanical advantage
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:
- is the efficiency of the wheel, the ratio of power output to power input
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