Simple machine
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; the term refers to the six classical simple machines which were defined by Renaissance scientists: Lever Wheel and axle Pulley Inclined plane Wedge ScrewA simple machine uses a single applied force to do work against a single load force. Ignoring friction losses, the work done on the load is equal to the work done by the applied force; the machine can increase the amount of the output force, at the cost of a proportional decrease in the distance moved by the load. The ratio of the output to the applied force is called the mechanical advantage. Simple machines can be regarded as the elementary "building blocks" of which all more complicated machines are composed. For example, wheels and pulleys are all used in the mechanism of a bicycle; the mechanical advantage of a compound machine is just the product of the mechanical advantages of the simple machines of which it is composed.
Although they continue to be of great importance in mechanics and applied science, modern mechanics has moved beyond the view of the simple machines as the ultimate building blocks of which all machines are composed, which arose in the Renaissance as a neoclassical amplification of ancient Greek texts. The great variety and sophistication of modern machine linkages, which arose during the Industrial Revolution, is inadequately described by these six simple categories. Various post-Renaissance authors have compiled expanded lists of "simple machines" using terms like basic machines, compound machines, or machine elements to distinguish them from the classical simple machines above. By the late 1800s, Franz Reuleaux had identified hundreds of machine elements, calling them simple machines. Modern machine theory analyzes machines as kinematic chains composed of elementary linkages called kinematic pairs; the idea of a simple machine originated with the Greek philosopher Archimedes around the 3rd century BC, who studied the Archimedean simple machines: lever and screw.
He discovered the principle of mechanical advantage in the lever. Archimedes' famous remark with regard to the lever: "Give me a place to stand on, I will move the Earth." Expresses his realization that there was no limit to the amount of force amplification that could be achieved by using mechanical advantage. Greek philosophers defined the classic five simple machines and were able to calculate their mechanical advantage. For example, Heron of Alexandria in his work Mechanics lists five mechanisms that can "set a load in motion"; however the Greeks' understanding was limited to the statics of simple machines, did not include dynamics, the tradeoff between force and distance, or the concept of work. During the Renaissance the dynamics of the Mechanical Powers, as the simple machines were called, began to be studied from the standpoint of how far they could lift a load, in addition to the force they could apply, leading to the new concept of mechanical work. In 1586 Flemish engineer Simon Stevin derived the mechanical advantage of the inclined plane, it was included with the other simple machines.
The complete dynamic theory of simple machines was worked out by Italian scientist Galileo Galilei in 1600 in Le Meccaniche, in which he showed the underlying mathematical similarity of the machines as force amplifiers. He was the first to explain; the classic rules of sliding friction in machines were discovered by Leonardo da Vinci, but were unpublished and documented in his notebooks, were based on pre-Newtonian science such as believing friction was an ethereal fluid. They were rediscovered by Guillaume Amontons and were further developed by Charles-Augustin de Coulomb. If a simple machine does not dissipate energy through friction, wear or deformation energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, the mechanical advantage can be calculated from its geometric dimensions. Although each machine works differently mechanically, the way they function is similar mathematically. In each machine, a force F in is applied to the device at one point, it does work moving a load, F out at another point.
Although some machines only change the direction of the force, such as a stationary pulley, most machines multiply the magnitude of the force by a factor, the mechanical advantage M A = F out / F in that can be calculated from the machine's geometry and friction. Simple machines do not contain a source of energy, so they cannot do more work than they receive from the input force. A simple machine with no friction or elasticity is called an ideal machine. Due to conservation of energy, in an ideal simple machine, the power output at any time P out is equal to the power input P in P out = P in The power o
Belt (mechanical)
A belt is a loop of flexible material used to link two or more rotating shafts mechanically, most parallel. Belts may be used as a source of motion, to transmit power efficiently or to track relative movement. Belts are looped over pulleys and may have a twist between the pulleys, the shafts need not be parallel. In a two pulley system, the belt can either drive the pulleys in one direction, or the belt may be crossed, so that the direction of the driven shaft is reversed; as a source of motion, a conveyor belt is one application where the belt is adapted to carry a load continuously between two points. The mechanical belt drive, using a pulley machine, was first mentioned in the text the Dictionary of Local Expressions by the Han Dynasty philosopher and politician Yang Xiong in 15 BC, used for a quilling machine that wound silk fibers on to bobbins for weavers' shuttles; the belt drive is an essential component to the invention of the spinning wheel. The belt drive was not only used in textile technologies, it was applied to hydraulic-powered bellows dated from the 1st century AD.
Belts are the cheapest utility for power transmission between shafts that may not be axially aligned. Power transmission pulleys; the demands on a belt-drive transmission system are huge, this has led to many variations on the theme. They run smoothly and with little noise, cushion motor and bearings against load changes, albeit with less strength than gears or chains. However, improvements in belt engineering allow use of belts in systems that only allowed chains or gears. Power transmitted between a belt and a pulley is expressed as the product of difference of tension and belt velocity: P = v, where, T1 and T2 are tensions in the tight side and slack side of the belt respectively, they are related as T 1 T 2 = e μ α, where, μ is the coefficient of friction, α is the angle subtended by contact surface at the centre of the pulley. Belt drives are simple, do not require axially aligned shafts, they help protect machinery from overload and jam, damp and isolate noise and vibration. Load fluctuations are shock-absorbed.
They need no minimal maintenance. They have high efficiency, high tolerance for misalignment, are of low cost if the shafts are far apart. Clutch action is activated by releasing belt tension. Different speeds can be obtained by stepped or tapered pulleys; the angular-velocity ratio may not be constant or equal to that of the pulley diameters, due to slip and stretch. However, this problem has been solved by the use of toothed belts. Working temperatures range from −31 °F to 185 °F. Adjustment of center distance or addition of an idler pulley is crucial to compensate for wear and stretch. Flat belts were used in the 19th and early 20th centuries in line shafting to transmit power in factories, they were used in countless farming and logging applications, such as bucksaws, threshers, silo blowers, conveyors for filling corn cribs or haylofts, water pumps, electrical generators. Flat belts are still used today; the flat belt is a simple system of power transmission, well suited for its day. It can deliver high power in cases of wide belts and large pulleys.
But these wide-belt-large-pulley drives are bulky, consuming lots of space while requiring high tension, leading to high loads, are poorly suited to close-centers applications, so V-belts have replaced flat belts for short-distance power transmission. For example, factory machines now tend to have individual electric motors; because flat belts tend to climb towards the higher side of the pulley, pulleys were made with a convex or "crowned" surface to allow the belt to self-center as it runs. Flat belts tend to slip on the pulley face when heavy loads are applied, many proprietary belt dressings were available that could be applied to the belts to increase friction, so power transmission. Flat belts were traditionally made of fabric. Today most are made of synthetic polymers. Grip of leather belts is better if they are assembled with the hair side of the leather against the pulley, although some belts are instead given a half-twist before joining the ends, so that wear can be evenly distributed on both sides of the belt.
Belts ends are joined by lacing the ends together with leather thonging, steel comb fasteners and/or lacing, or by gluing or welding. Flat belts were traditionally jointed, still are, but they can be made with endless construction. In the mid 19th century, British millwrights discovered that multi-grooved pulleys connected by ropes outperformed flat pulleys connected by leather belts. Wire ropes were used, but cotton, manila hemp and flax rope saw the widest use; the rope connecting two pulleys with multiple V-grooves was spliced into a single loop that t
Egypt (Roman province)
The Roman province of Egypt was established in 30 BC after Octavian defeated his rival Mark Antony, deposed Pharaoh Cleopatra, annexed the Ptolemaic Kingdom to the Roman Empire. The province encompassed most of modern-day Egypt except for the Sinai Peninsula. Aegyptus was bordered by the provinces of Crete and Cyrenaica to Judea to the East; the province came to serve as a major producer of grain for the empire and had a developed urban economy. Aegyptus was by far the wealthiest Eastern Roman province, by far the wealthiest Roman province outside of Italia. In Alexandria, its capital, it possessed the largest port, the second largest city of the Roman Empire; as a key province, but the'crown domain' where the emperors succeeded the divine Pharaohs, Egypt was ruled by a uniquely styled Praefectus augustalis, instead of the traditional senatorial governor of other Roman provinces. The prefect was appointed by the Emperor; the first prefect of Aegyptus, Gaius Cornelius Gallus, brought Upper Egypt under Roman control by force of arms, established a protectorate over the southern frontier district, abandoned by the Ptolemies.
The second prefect, Aelius Gallus, made an unsuccessful expedition to conquer Arabia Petraea and Arabia Felix. The Red Sea coast of Aegyptus was not brought under Roman control until the reign of Claudius; the third prefect, Gaius Petronius, cleared the neglected canals for irrigation, stimulating a revival of agriculture. Petronius led a campaign into present-day central Sudan against the Kingdom of Kush at Meroe, whose queen Imanarenat had attacked Roman Egypt. Failing to acquire permanent gains, in 22 BC he razed the city of Napata to the ground and retreated to the north. From the reign of Nero onward, Aegyptus enjoyed an era of prosperity. Much trouble was caused by religious conflicts between the Greeks and the Jews in Alexandria, which after the destruction of Jerusalem in 70 became the world centre of Jewish religion and culture. Under Trajan a Jewish revolt occurred, resulting in the suppression of the Jews of Alexandria and the loss of all their privileges, although they soon returned.
Hadrian, who twice visited Aegyptus, founded Antinopolis in memory of his drowned lover Antinous. From his reign onward buildings in the Greco-Roman style were erected throughout the country Under Antoninus Pius oppressive taxation led to a revolt in 139, of the native Egyptians, suppressed only after several years of fighting; this Bucolic War, led by one Isidorus, caused great damage to the economy and marked the beginning of Egypt's economic decline. Avidius Cassius, who led the Roman forces in the war, declared himself emperor in 175, was acknowledged by the armies of Syria and Aegyptus. On the approach of Marcus Aurelius, Cassius was deposed and killed and the clemency of the emperor restored peace. A similar revolt broke out in 193, when Pescennius Niger was proclaimed emperor on the death of Pertinax; the Emperor Septimius Severus gave a constitution to Alexandria and the provincial capitals in 202. Caracalla granted Roman citizenship to all Egyptians, in common with the other provincials, but this was to extort more taxes, which grew onerous as the needs of the emperors for more revenue grew more desperate.
There was a series of revolts, both civilian, through the 3rd century. Under Decius, in 250, the Christians again suffered from persecution, but their religion continued to spread; the prefect of Aegyptus in 260, Mussius Aemilianus, first supported the Macriani, usurpers during the rule of Gallienus, in 261, became a usurper himself, but was defeated by Gallienus. Zenobia, queen of Palmyra, took the country away from the Romans when she conquered Aegyptus in 269, declaring herself the Queen of Egypt also; this warrior queen claimed that Egypt was an ancestral home of hers through a familial tie to Cleopatra VII. She was well educated and familiar with the culture of Egypt, its religion, its language, she lost it when the Roman emperor, severed amicable relations between the two countries and retook Egypt in 274. Two generals based in Aegyptus and Domitius Domitianus, led successful revolts and made themselves emperors. Diocletian reorganised the whole province, his edict of 303 against the Christians began a new era of persecution.
This was the last serious attempt to stem the steady growth of Christianity in Egypt, however. As Rome overtook the Ptolemaic system in place for areas of Egypt, they made many changes; the effect of the Roman conquest was at first to strengthen the position of the Greeks and of Hellenism against Egyptian influences. Some of the previous offices and names of offices under the Hellenistic Ptolemaic rule were kept, some were changed, some names would have remained but the function and administration would have changed; the Romans introduced important changes in the administrative system, aimed at achieving a high level of efficiency and maximizing revenue. The duties of the prefect of Aegyptus combined responsibility for military security through command of the legions and cohorts, for the organization of finance and taxation, for the administration of justice; the reforms of the early 4th century had established the basis for another 250 years of comparative prosperity in Aegyptus, at a cost of greater rigidity and more oppressive state control.
Aegyptus was subdivided for administrative purposes into a number of smaller provinces, s
Vacuum cleaner
A vacuum cleaner known as a sweeper or hoover, is a device that uses an air pump, to create a partial vacuum to suck up dust and dirt from floors and from other surfaces such as upholstery and draperies. The dirt is collected by either a dustbag or a cyclone for disposal. Vacuum cleaners, which are used in homes as well as in industry, exist in a variety of sizes and models—small battery-powered hand-held devices, wheeled canister models for home use, domestic central vacuum cleaners, huge stationary industrial appliances that can handle several hundred litres of dust before being emptied, self-propelled vacuum trucks for recovery of large spills or removal of contaminated soil. Specialized shop vacuums can be used to suck up liquids. Although vacuum cleaner and the short form vacuum are neutral names, in some countries hoover is used instead as a genericized trademark, as a verb; the name comes from the Hoover Company, one of the first and more influential companies in the development of the device.
The device is sometimes called a sweeper although the same term refers to a carpet sweeper, a similar invention. The vacuum cleaner evolved from the carpet sweeper via manual vacuum cleaners; the first manual models, using bellows, were developed in the 1860s, the first motorized designs appeared at the turn of the 20th century, with the first decade being the boom decade. In 1860 a manual vacuum cleaner was invented by Daniel Hess of Iowa. Called a'carpet sweeper', It gathered dust with a rotating brush and had a bellows for generating suction. Another early model was the "Whirlwind", invented in Chicago in 1868 by Ives W. McGaffey; the bulky device worked with a belt driven fan cranked by hand that made it awkward to operate, although it was commercially marketed with mixed success. A similar model was constructed by Melville R. Bissell of Grand Rapids, Michigan in 1876, who manufactured carpet sweepers; the company added portable vacuum cleaners to its line of cleaning tools. The end of the 19th century saw the introduction of powered cleaners, although early types used some variation of blowing air to clean instead of suction.
One appeared in 1898 when John S. Thurman of St. Louis, Missouri submitted a patent for a "pneumatic carpet renovator" which blew dust into a receptacle. Thurman's system, powered by an internal combustion engine, traveled to the customers residence on a horse-drawn wagon as part of a door to door cleaning service. Corrine Dufour of Savannah, Georgia received two patents in 1899 and 1900 for another blown air system that seems to have featured the first use of an electric motor. In 1901 powered vacuum cleaners using suction were invented independently by British engineer Hubert Cecil Booth and American inventor David T. Kenney. Booth may have coined the word "vacuum cleaner". Booth's horse drawn combustion engine powered "Puffing Billy", maybe derived from Thurman's blown air design," relied upon just suction with air pumped through a cloth filter and was offered as part of his cleaning services. Kenney's was a stationary 4,000 lb. steam engine powered system with pipes and hoses reaching into all parts of the building.
The first vacuum-cleaning device to be portable and marketed at the domestic market was built in 1905 by Walter Griffiths, a manufacturer in Birmingham, England. His Griffith's Improved Vacuum Apparatus for Removing Dust from Carpets resembled modern-day cleaners. In 1906 James B. Kirby developed his first of many vacuums called the "Domestic Cyclone", it used water for dirt separation. Revisions came to be known as the Kirby Vacuum Cleaner. In 1907 department store janitor James Murray Spangler of Canton, Ohio invented the first portable electric vacuum cleaner, obtaining a patent for the Electric Suction Sweeper on June 2, 1908. Crucially, in addition to suction from an electric fan that blew the dirt and dust into a soap box and one of his wife's pillow cases, Spangler's design utilized a rotating brush to loosen debris. Unable to produce the design himself due to lack of funding, he sold the patent in 1908 to local leather goods manufacturer William Henry Hoover, who had Spangler's machine redesigned with a steel casing and attachments, founding the company that in 1922 was renamed the Hoover Company.
Their first vacuum was the 1908 Model O, which sold for $60. Subsequent innovations included the beater bar in 1919, disposal filter bags in the 1920s, an upright vacuum cleaner in 1926. In Continental Europe, the Fisker and Nielsen company in Denmark was the first to sell vacuum cleaners in 1910; the design could be operated by a single person. The Swedish company Electrolux launched their Model V in 1921 with the innovation of being able to lie on the floor on two thin metal runners. In the 1930s the Germany company Vorwerk started marketing vacuum cleaners of their own design which they sold through direct sales. For many years after their introduction, vacuum cleaners remained a luxury item, but after the Second World War, they became common among the middle classes. Vacuums tend to be more common in Western countries because in most other parts of the world, wall-to-wall carpeting is uncommon and homes have tile or hardwood floors, which are swept, wiped or mopped manually without power assist.
The last decades of the 20th century saw the more widespread use of technologie
Gear
A gear or cogwheel is a rotating machine part having cut teeth, or in the case of a cogwheel, inserted teeth, which mesh with another toothed part to transmit torque. Geared devices can change the speed and direction of a power source. Gears always produce a change in torque, creating a mechanical advantage, through their gear ratio, thus may be considered a simple machine; the teeth on the two meshing gears all have the same shape. Two or more meshing gears, working in a sequence, are called a transmission. A gear can mesh with a linear toothed part, called a rack, producing translation instead of rotation; the gears in a transmission are analogous to the wheels in belt pulley system. An advantage of gears is; when two gears mesh, if one gear is bigger than the other, a mechanical advantage is produced, with the rotational speeds, the torques, of the two gears differing in proportion to their diameters. In transmissions with multiple gear ratios—such as bicycles and cars—the term "gear" as in "first gear" refers to a gear ratio rather than an actual physical gear.
The term describes similar devices when the gear ratio is continuous rather than discrete, or when the device does not contain gears, as in a continuously variable transmission. Early examples of gears date from the 4th century BC in China, which have been preserved at the Luoyang Museum of Henan Province, China; the earliest preserved gears in Europe were found in the Antikythera mechanism, an example of a early and intricate geared device, designed to calculate astronomical positions. Its time of construction is now estimated between 150 and 100 BC. Gears appear in works connected to Hero of Alexandria, in Roman Egypt circa AD 50, but can be traced back to the mechanics of the Alexandrian school in 3rd-century BC Ptolemaic Egypt, were developed by the Greek polymath Archimedes; the segmental gear, which receives/communicates reciprocating motion from/to a cogwheel, consisting of a sector of a circular gear/ring having cogs on the periphery, was invented by Arab engineer Al-Jazari in 1206.
The worm gear was invented in the Indian subcontinent, for use in roller cotton gins, some time during the 13th–14th centuries. Differential gears may have been used in some of the Chinese south-pointing chariots, but the first verifiable use of differential gears was by the British clock maker Joseph Williamson in 1720. Examples of early gear applications include: The Antikythera mechanism Ma Jun used gears as part of a south-pointing chariot; the first geared mechanical clocks were built in China in 725. Al-Jazari invented the segmental gear as part of a water-lifting device; the worm gear was invented as part of a roller cotton gin in the Indian subcontinent. The 1386 Salisbury cathedral clock may be the world's oldest still working geared mechanical clock; the definite ratio that teeth give gears provides an advantage over other drives in precision machines such as watches that depend upon an exact velocity ratio. In cases where driver and follower are proximal, gears have an advantage over other drives in the reduced number of parts required.
The downside is that gears are more expensive to manufacture and their lubrication requirements may impose a higher operating cost per hour. An external gear is one with the teeth formed on the outer surface of a cone. Conversely, an internal gear is one with the teeth formed on the inner surface of a cylinder or cone. For bevel gears, an internal gear is one with the pitch angle exceeding 90 degrees. Internal gears do not cause output shaft direction reversal. Spur gears or straight-cut gears are the simplest type of gear, they consist of a disk with teeth projecting radially. Though the teeth are not straight-sided, the edge of each tooth is straight and aligned parallel to the axis of rotation; these gears mesh together only if fitted to parallel shafts. No axial thrust is created by the tooth loads. Spur gears tend to be noisy at high speeds. Helical or "dry fixed" gears offer a refinement over spur gears; the leading edges of the teeth are set at an angle. Since the gear is curved, this angling makes.
Helical gears can be meshed in crossed orientations. The former refers to. In the latter, the shafts are non-parallel, in this configuration the gears are sometimes known as "skew gears"; the angled teeth engage more than do spur gear teeth, causing them to run more smoothly and quietly. With parallel helical gears, each pair of teeth first make contact at a single point at one side of the gear wheel. In spur gears, teeth meet at a line contact across their entire width, causing stress and noise. Spur gears make a characteristic whine at high speeds. For this reason spur gears are used in low-speed applications and in situations where noise control is not a problem, helical gears are used in high-speed applications, large power transmission, or where noise abatement is important; the speed is considered high. A disadvantage of helical gears is a resultant thrust along the axis of the gear, which must
Wheel
In its primitive form, a wheel is a circular block of a hard and durable material at whose center has been bored a circular hole through, placed an axle bearing about which the wheel rotates when a moment is applied by gravity or torque to the wheel about its axis, thereby making together one of the six simple machines. When placed vertically under a load-bearing platform or case, the wheel turning on the horizontal axle makes it possible to transport heavy loads; the English word wheel comes from the Old English word hweol, from Proto-Germanic *hwehwlan, *hwegwlan, from Proto-Indo-European *kwekwlo-, an extended form of the root *kwel- "to revolve, move around". Cognates within Indo-European include Icelandic hjól "wheel, tyre", Greek κύκλος kúklos, Sanskrit chakra, the latter two both meaning "circle" or "wheel"; the invention of the wheel falls into the late Neolithic, may be seen in conjunction with other technological advances that gave rise to the early Bronze Age. This implies the passage of several wheel-less millennia after the invention of agriculture and of pottery, during the Aceramic Neolithic.
4500–3300 BCE: Copper Age, invention of the potter's wheel. Precursors of wheels, known as "tournettes" or "slow wheels", were known in the Middle East by the 5th millennium 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 were in use in Mesopotamia by 3500 BCE and as early as 4000 BCE, the oldest surviving example, found in Ur, dates to 3100 BCE; the first evidence of wheeled vehicles appears in the second half of the 4th millennium BCE, near-simultaneously in Mesopotamia, the Northern and South Caucasus, Eastern Europe, so the question of which culture invented the wheeled vehicle is still unresolved. The earliest well-dated depiction of a wheeled vehicle is on the 3500–3350 BCE Bronocice clay pot excavated in a Funnelbeaker culture settlement in southern Poland. In nearby Olszanica 5000 BCE 2.2 m wide door were constructed for wagon entry. This barn was 40 m long with 3 doors; the oldest securely dated real wheel-axle combination, that from Stare Gmajne near Ljubljana in Slovenia is now dated within two standard deviations to 3340–3030 BCE, the axle to 3360–3045 BCE.
Two types of early Neolithic European wheel and axle are known. They both are dated to c. 3200–3000 BCE. In China, the wheel was present with the adoption of the chariot in c. 1200 BCE, although Barbieri-Low argues for earlier Chinese wheeled vehicles, c. 2000 BCE. In Britain, a large wooden wheel, measuring about 1 m in diameter, was uncovered at the Must Farm site in East Anglia in 2016; the specimen, dating from 1,100–800 BCE, represents the most complete and earliest of its type found in Britain. The wheel's hub is present. A horse's spine found; the wheel was found in a settlement built on stilts over wetland, indicating that the settlement had some sort of link to dry land. Although large-scale use of wheels did not occur in the Americas prior to European contact, numerous small wheeled artifacts, identified as children's toys, have been found in Mexican archeological sites, some dating to about 1500 BCE, it is thought that the primary obstacle to large-scale development of the wheel in the Americas was the absence of domesticated large animals which could be used to pull wheeled carriages.
The closest relative of cattle present in Americas in pre-Columbian times, the American Bison, is difficult to domesticate and was never domesticated by Native Americans. The only large animal, domesticated in the Western hemisphere, the llama, a pack animal but not physically suited to use as a draft animal to pull wheeled vehicles, did not spread far beyond the Andes by the time of the arrival of Columbus. Nubians from after about 400 BCE used wheels as water wheels, it is thought. It is known that Nubians used horse-drawn chariots imported from Egypt; the wheel was used, with the exception of the Horn of Africa, in Sub-Saharan Africa well into the 19th century but this changed with the arrival of the Europeans. Early wheels were simple wooden disks with a hole for the axle; some of the earliest wheels were made from horizontal slices of tree trunks
Power (physics)
In physics, power is the rate of doing work or of transferring heat, i.e. the amount of energy transferred or converted per unit time. Having no direction, it is a scalar quantity. In the International System of Units, the unit of power is the joule per second, known as the watt in honour of James Watt, the eighteenth-century developer of the condenser steam engine. Another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written: power = work time As a physical concept, power requires both a change in the physical system and a specified time in which the change occurs; this is distinct from the concept of work, only measured in terms of a net change in the state of the physical system. The same amount of work is done when carrying a load up a flight of stairs whether the person carrying it walks or runs, but more power is needed for running because the work is done in a shorter amount of time; the output power of an electric motor is the product of the torque that the motor generates and the angular velocity of its output shaft.
The power involved in moving a vehicle is the product of the traction force of the wheels and the velocity of the vehicle. The rate at which a light bulb converts electrical energy into light and heat is measured in watts—the higher the wattage, the more power, or equivalently the more electrical energy is used per unit time; the dimension of power is energy divided by time. The SI unit of power is the watt, equal to one joule per second. Other units of power include ergs per second, metric horsepower, foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the power required to lift 550 pounds by one foot in one second, is equivalent to about 746 watts. Other units include a logarithmic measure relative to a reference of 1 milliwatt. Power, as a function of time, is the rate at which work is done, so can be expressed by this equation: P = d W d t where P is power, W is work, t is time; because work is a force F applied over a distance x, W = F ⋅ x for a constant force, power can be rewritten as: P = d W d t = d d t = F ⋅ d x d t = F ⋅ v In fact, this is valid for any force, as a consequence of applying the fundamental theorem of calculus.
As a simple example, burning one kilogram of coal releases much more energy than does detonating a kilogram of TNT, but because the TNT reaction releases energy much more it delivers far more power than the coal. If ΔW is the amount of work performed during a period of time of duration Δt, the average power Pavg over that period is given by the formula P a v g = Δ W Δ t, it is the average amount of energy converted per unit of time. The average power is simply called "power" when the context makes it clear; the instantaneous power is the limiting value of the average power as the time interval Δt approaches zero. P = lim Δ t → 0 P a v g = lim Δ t → 0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration t is given by: W = P t. In the context of energy conversion, it is more customary to use the symbol E rather than W. Power in mechanical systems is the combination of forces and movement. In particular, power is the product of a force on an object and the object's velocity, or the product of a torque on a shaft and the shaft's angular velocity.
Mechanical power is described as the time derivative of work. In mechanics, the work done by a force F on an object that travels along a curve C is given by the line integral: W C = ∫ C F ⋅ v d t = ∫ C F ⋅ d x, where x defines the path C and v is the velocity along this path. If the force F is derivable from a potential applying the gradi
