Mechanics (Greekμηχανική) is that area of science concerned with the behaviour of physical bodies when subjected to forces or displacements, and the subsequent effects of the bodies on their environment.The scientific discipline has its origins in Ancient Greece with the writings of Aristotle and Archimedes[1][2][3] (see History of classical mechanics and Timeline of classical mechanics). During the early modern period, scientists such as Galileo, Kepler, and Newton laid the foundation for what is now known as classical mechanics.It is a branch of classical physics that deals with particles that are either at rest or are moving with velocities significantly less than the speed of light. It can also be defined as a branch of science which deals with the motion of and forces on objects. The field is yet less widely understood in terms of quantum theory.
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Historically, classical mechanics came first and quantum mechanics is a comparatively recent development. Classical mechanics originated with Isaac Newton's laws of motion in Philosophiæ Naturalis Principia Mathematica; Quantum Mechanics was developed in the early 20th century. Both are commonly held to constitute the most certain knowledge that exists about physical nature.
Classical mechanics has especially often been viewed as a model for other so-called exact sciences. Essential in this respect is the extensive use of mathematics in theories, as well as the decisive role played by experiment in generating and testing them.
Quantum mechanics is of a bigger scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence principle, there is no contradiction or conflict between the two subjects, each simply pertains to specific situations. The correspondence principle states that the behavior of systems described by quantum theories reproduces classical physics in the limit of large quantum numbers. Quantum mechanics has superseded classical mechanics at the foundation level and is indispensable for the explanation and prediction of processes at the molecular, atomic, and sub-atomic level. However, for macroscopic processes classical mechanics is able to solve problems which are unmanageably difficult in quantum mechanics and hence remains useful and well used.Modern descriptions of such behavior begin with a careful definition of such quantities as displacement (distance moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however, motion was explained from a very different point of view. For example, following the ideas of Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down because its natural position is in the Earth; the sun, the moon, and the stars travel in circles around the earth because it is the nature of heavenly objects to travel in perfect circles.
Often cited as father to modern science, Galileo brought together the ideas of other great thinkers of his time and began to calculate motion in terms of distance traveled from some starting position and the time that it took. He showed that the speed of falling objects increases steadily during the time of their fall. This acceleration is the same for heavy objects as for light ones, provided air friction (air resistance) is discounted. The English mathematician and physicist Isaac Newton improved this analysis by defining force and mass and relating these to acceleration. For objects traveling at speeds close to the speed of light, Newton's laws were superseded by Albert Einstein’s theory of relativity. [A sentence illustrating the computational complication of Einstein's theory of relativity.] For atomic and subatomic particles, Newton's laws were superseded by quantum theory. For everyday phenomena, however, Newton's three laws of motion remain the cornerstone of dynamics, which is the study of what causes motion.
In analogy to the distinction between quantum and classical mechanics, Einstein's general and special theories of relativity have expanded the scope of Newton and Galileo's formulation of mechanics. The differences between relativistic and Newtonian mechanics become significant and even dominant as the velocity of a massive body approaches the speed of light. For instance, in Newtonian mechanics, Newton's laws of motion specify that F = ma, whereas in relativistic mechanics and Lorentz transformations, which were first discovered by Hendrik Lorentz, F = γma (where γ is the Lorentz factor, which is almost equal to 1 for low speeds).
Relativistic corrections are also needed for quantum mechanics, although general relativity has not been integrated. The two theories remain incompatible, a hurdle which must be overcome in developing a theory of everything.
The main theory of mechanics in antiquity was Aristotelian mechanics.[4] A later developer in this tradition is Hipparchus.[5]
In the Middle Ages, Aristotle's theories were criticized and modified by a number of figures, beginning with John Philoponus in the 6th century. A central problem was that of projectile motion, which was discussed by Hipparchus and Philoponus.
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Persian Islamic polymath Ibn Sīnā published his theory of motion in The Book of Healing (1020). He said that an impetus is imparted to a projectile by the thrower, and viewed it as persistent, requiring external forces such as air resistance to dissipate it.[6][7][8] Ibn Sina made distinction between 'force' and 'inclination' (called 'mayl'), and argued that an object gained mayl when the object is in opposition to its natural motion. So he concluded that continuation of motion is attributed to the inclination that is transferred to the object, and that object will be in motion until the mayl is spent. He also claimed that projectile in a vacuum would not stop unless it is acted upon. This conception of motion is consistent with Newton's first law of motion, inertia. Which states that an object in motion will stay in motion unless it is acted on by an external force.[9] This idea which dissented from the Aristotelian view was later described as 'impetus' by John Buridan, who was influenced by Ibn Sina's Book of Healing.[10]
On the question of a body subject to a constant (uniform) force, the 12th-century Jewish-Arab scholar Hibat Allah Abu'l-Barakat al-Baghdaadi (born Nathanel, Iraqi, of Baghdad) stated that constant force imparts constant acceleration. According to Shlomo Pines, al-Baghdaadi's theory of motion was 'the oldest negation of Aristotle's fundamental dynamic law [namely, that a constant force produces a uniform motion], [and is thus an] anticipation in a vague fashion of the fundamental law of classical mechanics [namely, that a force applied continuously produces acceleration].'[11] The same century, Ibn Bajjah proposed that for every force there is always a reaction force. While he did not specify that these forces be equal, it is still an early version of the third law of motion which states that for every action there is an equal and opposite reaction.[12]
Influenced by earlier writers such as Ibn Sina[10] and al-Baghdaadi,[13] the 14th-century French priest Jean Buridan developed the theory of impetus, which later developed into the modern theories of inertia, velocity, acceleration and momentum. This work and others was developed in 14th-century England by the Oxford Calculators such as Thomas Bradwardine, who studied and formulated various laws regarding falling bodies. The concept that the main properties of a body are uniformly accelerated motion (as of falling bodies) was worked out by the 14th-century Oxford Calculators.
Two central figures in the early modern age are Galileo Galilei and Isaac Newton. Galileo's final statement of his mechanics, particularly of falling bodies, is his Two New Sciences (1638). Newton's 1687 Philosophiæ Naturalis Principia Mathematica provided a detailed mathematical account of mechanics, using the newly developed mathematics of calculus and providing the basis of Newtonian mechanics.[5]
There is some dispute over priority of various ideas: Newton's Principia is certainly the seminal work and has been tremendously influential, and the systematic mathematics therein did not and could not have been stated earlier because calculus had not been developed. However, many of the ideas, particularly as pertain to inertia (impetus) and falling bodies had been developed and stated by earlier researchers, both the then-recent Galileo and the less-known medieval predecessors. Precise credit is at times difficult or contentious because scientific language and standards of proof changed, so whether medieval statements are equivalent to modern statements or sufficient proof, or instead similar to modern statements and hypotheses is often debatable.
Two main modern developments in mechanics are general relativity of Einstein, and quantum mechanics, both developed in the 20th century based in part on earlier 19th-century ideas. The development in the modern continuum mechanics, particularly in the areas of elasticity, plasticity, fluid dynamics, electrodynamics and thermodynamics of deformable media, started in the second half of the 20th century.
The often-used term body needs to stand for a wide assortment of objects, including particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids (gases and liquids), etc.
Other distinctions between the various sub-disciplines of mechanics, concern the nature of the bodies being described. Particles are bodies with little (known) internal structure, treated as mathematical points in classical mechanics. Rigid bodies have size and shape, but retain a simplicity close to that of the particle, adding just a few so-called degrees of freedom, such as orientation in space.
Otherwise, bodies may be semi-rigid, i.e. elastic, or non-rigid, i.e. fluid. These subjects have both classical and quantum divisions of study.
For instance, the motion of a spacecraft, regarding its orbit and attitude (rotation), is described by the relativistic theory of classical mechanics, while the analogous movements of an atomic nucleus are described by quantum mechanics.
The following are two lists of various subjects that are studied in mechanics.
Note that there is also the 'theory of fields' which constitutes a separate discipline in physics, formally treated as distinct from mechanics, whether classical fields or quantum fields. But in actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for instance, forces that act on particles are frequently derived from fields (electromagnetic or gravitational), and particles generate fields by acting as sources. In fact, in quantum mechanics, particles themselves are fields, as described theoretically by the wave function.
The following are described as forming classical mechanics:
The following are categorized as being part of quantum mechanics:
| Look up mechanics in Wiktionary, the free dictionary. |
A simple machine is a mechanical device that changes the direction or magnitude of a force.[2] In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force.[3] Usually the term refers to the six classical simple machines which were defined by Renaissance scientists:[4]
A 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 (sometimes called 'compound machines'[5][6]) are composed.[3][7] For example, wheels, levers, and pulleys are all used in the mechanism of a bicycle.[8][9] 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', often using terms like basic machines,[8]compound machines,[5] or machine elements to distinguish them from the classical simple machines above. By the late 1800s, Franz Reuleaux[10] had identified hundreds of machine elements, calling them simple machines.[11] 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, pulley, and screw.[3][12] He discovered the principle of mechanical advantage in the lever.[13] Archimedes' famous remark with regard to the lever: 'Give me a place to stand on, and I will move the Earth.' (Greek: δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω)[14] expresses his realization that there was no limit to the amount of force amplification that could be achieved by using mechanical advantage. Later Greek philosophers defined the classic five simple machines (excluding the inclined plane) and were able to calculate their (ideal) mechanical advantage.[6] For example, Heron of Alexandria (c. 10–75 AD) in his work Mechanics lists five mechanisms that can 'set a load in motion'; lever, windlass, pulley, wedge, and screw,[12] and describes their fabrication and uses.[15] However the Greeks' understanding was limited to the statics of simple machines (the balance of forces), and 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 eventually to the new concept of mechanical work. In 1586 Flemish engineer Simon Stevin derived the mechanical advantage of the inclined plane, and 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 (On Mechanics), in which he showed the underlying mathematical similarity of the machines as force amplifiers.[16][17] He was the first to explain that simple machines do not create energy, only transform it.[16]
The classic rules of sliding friction in machines were discovered by Leonardo da Vinci (1452–1519), but were unpublished and merely documented in his notebooks, and were based on pre-Newtonian science such as believing friction was an ethereal fluid. They were rediscovered by Guillaume Amontons (1699) and were further developed by Charles-Augustin de Coulomb (1785).[18]
If a simple machine does not dissipate energy through friction, wear or deformation, then energy is conserved and it is called an ideal simple machine. In this case, the power into the machine equals the power out, and the mechanical advantage can be calculated from its geometric dimensions.
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Although each machine works differently mechanically, the way they function is similar mathematically.[19] In each machine, a force is applied to the device at one point, and it does work moving a load, at another point.[20] 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
that can be calculated from the machine's geometry and friction.
Simple machines do not contain a source of energy,[21] so they cannot do more work than they receive from the input force.[20] A simple machine with no friction or elasticity is called an ideal machine.[22][23][24] Due to conservation of energy, in an ideal simple machine, the power output (rate of energy output) at any time is equal to the power input
The power output equals the velocity of the load multiplied by the load force . Similarly the power input from the applied force is equal to the velocity of the input point multiplied by the applied force .Therefore,
So the mechanical advantage of an ideal machine is equal to the velocity ratio, the ratio of input velocity to output velocity
The velocity ratio is also equal to the ratio of the distances covered in any given period of time[25][26][27]
Therefore the mechanical advantage of an ideal machine is also equal to the distance ratio, the ratio of input distance moved to output distance moved
This can be calculated from the geometry of the machine. For example, the mechanical advantage and distance ratio of the lever is equal to the ratio of its lever arms.
The mechanical advantage can be greater or less than one:
In the screw, which uses rotational motion, the input force should be replaced by the torque, and the velocity by the angular velocity the shaft is turned.
All real machines have friction, which causes some of the input power to be dissipated as heat. If is the power lost to friction, from conservation of energy
The mechanical efficiency of a machine (where ) is defined as the ratio of power out to the power in, and is a measure of the frictional energy losses
As above, the power is equal to the product of force and velocity, so
Therefore,
So in non-ideal machines, the mechanical advantage is always less than the velocity ratio by the product with the efficiency η. So a machine that includes friction will not be able to move as large a load as a corresponding ideal machine using the same input force.
A compound machine is a machine formed from a set of simple machines connected in series with the output force of one providing the input force to the next. For example, a bench vise consists of a lever (the vise's handle) in series with a screw, and a simple gear train consists of a number of gears (wheels and axles) connected in series.
The mechanical advantage of a compound machine is the ratio of the output force exerted by the last machine in the series divided by the input force applied to the first machine, that is
Because the output force of each machine is the input of the next, , this mechanical advantage is also given by
Thus, the mechanical advantage of the compound machine is equal to the product of the mechanical advantages of the series of simple machines that form it
Similarly, the efficiency of a compound machine is also the product of the efficiencies of the series of simple machines that form it
In many simple machines, if the load force Fout on the machine is high enough in relation to the input force Fin, the machine will move backwards, with the load force doing work on the input force.[28] So these machines can be used in either direction, with the driving force applied to either input point. For example, if the load force on a lever is high enough, the lever will move backwards, moving the input arm backwards against the input force. These are called 'reversible', 'non-locking' or 'overhauling' machines, and the backward motion is called 'overhauling'. However, in some machines, if the frictional forces are high enough, no amount of load force can move it backwards, even if the input force is zero. This is called a 'self-locking', 'nonreversible', or 'non-overhauling' machine.[28] These machines can only be set in motion by a force at the input, and when the input force is removed will remain motionless, 'locked' by friction at whatever position they were left.
Self-locking occurs mainly in those machines with large areas of sliding contact between moving parts: the screw, inclined plane, and wedge:
A machine will be self-locking if and only if its efficiency η is below 50%:[28]
Whether a machine is self-locking depends on both the friction forces (coefficient of static friction) between its parts, and the distance ratio din/dout (ideal mechanical advantage). If both the friction and ideal mechanical advantage are high enough, it will self-lock.
When a machine moves in the forward direction from point 1 to point 2, with the input force doing work on a load force, from conservation of energy[29][30] the input work is equal to the sum of the work done on the load force and the work lost to friction
(Eq. 1) |
If the efficiency is below 50%
From Eq. 1
When the machine moves backward from point 2 to point 1 with the load force doing work on the input force, the work lost to friction is the same
So the output work is
Thus the machine self-locks, because the work dissipated in friction is greater than the work done by the load force moving it backwards even with no input force
Simple machines are elementary examples of kinematic chains that are used to model mechanical systems ranging from the steam engine to robot manipulators. The bearings that form the fulcrum of a lever and that allow the wheel and axle and pulleys to rotate are examples of a kinematic pair called a hinged joint. Similarly, the flat surface of an inclined plane and wedge are examples of the kinematic pair called a sliding joint. The screw is usually identified as its own kinematic pair called a helical joint.
Two levers, or cranks, are combined into a planar four-bar linkage by attaching a link that connects the output of one crank to the input of another. Additional links can be attached to form a six-bar linkage or in series to form a robot.[23]
The identification of simple machines arises from a desire for a systematic method to invent new machines. Therefore, an important concern is how simple machines are combined to make more complex machines. One approach is to attach simple machines in series to obtain compound machines.
However, a more successful strategy was identified by Franz Reuleaux, who collected and studied over 800 elementary machines. He realized that a lever, pulley, and wheel and axle are in essence the same device: a body rotating about a hinge. Similarly, an inclined plane, wedge, and screw are a block sliding on a flat surface.[31]
This realization shows that it is the joints, or the connections that provide movement, that are the primary elements of a machine. Starting with four types of joints, the revolute joint, sliding joint, cam joint and gear joint, and related connections such as cables and belts, it is possible to understand a machine as an assembly of solid parts that connect these joints.[23]