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University Physics

University Physics, informally known as the Sears & Zemansky, is the name of a two-volume physics textbook written by Hugh Young and Roger Freedman. The first edition of University Physics was published by Mark Zemansky and Francis Sears in 1949. Hugh Young became a coauthor with Sears and Zemansky in 1973. Now in its 15th edition, University Physics is among the most widely used introductory textbooks in the world. University Physics by Pearson is not to be confused with a free textbook by the same name, available from OpenStax. Contents Volume 1. Classic mechanics, Waves/acoustics, and Thermodynamics Mechanics Units, Physical Quantities, and Vectors Motion Along a Straight Line Motion in Two or Three Dimensions Newton's Laws of Motion Applying Newton’s Laws Work and Kinetic Energy Potential Energy and Energy Conservation Momentum, Impulse, and Collisions Rotation of Rigid Bodies Dynamics of Rotational Motion Equilibrium and Elasticity Fluid Mechanics Gravitation Periodic Motion Waves/Acoustics Mechanical Waves Sound and Hearing Thermodynamics Temperature and Heat Thermal Properties of Matter The First Law of Thermodynamics The Second Law of Thermodynamics Volume 2. Electromagnetism, optics, and modern physics Electromagnetism Electric Charge and Electric Field Gauss’s Law Electric Potential Capacitance and Dielectrics Current, Resistance, and Electromotive Force Direct-Current Circuits Magnetic Field and Magnetic Forces Sources of Magnetic Field Electromagnetic Induction Inductance Alternating Current Electromagnetic Waves Optics The Nature and Propagation of Light Geometric Optics Interference Diffraction Modern Physics Relativity Photons: Light Waves Behaving as Particles Particles Behaving as Waves Quantum Mechanics Atomic Structure Molecules and Condensed Matter Nuclear Physics Particle Physics and Cosmology References Physics textbooks

Energy transformation

Energy transformation, also known as energy conversion, is the process of changing energy from one form to another. In physics, energy is a quantity that provides the capacity to perform work or moving (e.g. lifting an object) or provides heat. In addition to being converted, according to the law of conservation of energy, energy is transferable to a different location or object, but it cannot be created or destroyed. The energy in many of its forms may be used in natural processes, or to provide some service to society such as heating, refrigeration, lighting or performing mechanical work to operate machines. For example, to heat a home, the furnace burns fuel, whose chemical potential energy is converted into thermal energy, which is then transferred to the home's air to raise its temperature. Limitations in the conversion of thermal energy Conversions to thermal energy from other forms of energy may occur with 100% efficiency. Conversion among non-thermal forms of energy may occur with fairly high efficiency, though there is always some energy dissipated thermally due to friction and similar processes. Sometimes the efficiency is close to 100%, such as when potential energy is converted to kinetic energy as an object falls in a vacuum. This also applies to the opposite case; for example, an object in an elliptical orbit around another body converts its kinetic energy (speed) into gravitational potential energy (distance from the other object) as it moves away from its parent body. When it reaches the furthest point, it will reverse the process, accelerating and converting potential energy into kinetic. Since space is a near-vacuum, this process has close to 100% efficiency. Thermal energy is unique because it in most cases (willow) cannot be converted to other forms of energy. Only a difference in the density of thermal/heat energy (temperature) can be used to perform work, and the efficiency of this conversion will be (much) less than 100%. This is because thermal energy represents a particularly disordered form of energy; it is spread out randomly among many available states of a collection of microscopic particles constituting the system (these combinations of position and momentum for each of the particles are said to form a phase space). The measure of this disorder or randomness is entropy, and its defining feature is that the entropy of an isolated system never decreases. One cannot take a high-entropy system (like a hot substance, with a certain amount of thermal energy) and convert it into a low entropy state (like a low-temperature substance, with correspondingly lower energy), without that entropy going somewhere else (like the surrounding air). In other words, there is no way to concentrate energy without spreading out energy somewhere else. Thermal energy in equilibrium at a given temperature already represents the maximal evening-out of energy between all possible states because it is not entirely convertible to a "useful" form, i.e. one that can do more than just affect temperature. The second law of thermodynamics states that the entropy of a closed system can never decrease. For this reason, thermal energy in a system may be converted to other kinds of energy with efficiencies approaching 100% only if the entropy of the universe is increased by other means, to compensate for the decrease in entropy associated with the disappearance of the thermal energy and its entropy content. Otherwise, only a part of that thermal energy may be converted to other kinds of energy (and thus useful work). This is because the remainder of the heat must be reserved to be transferred to a thermal reservoir at a lower temperature. The increase in entropy for this process is greater than the decrease in entropy associated with the transformation of the rest of the heat into other types of energy. In order to make energy transformation more efficient, it is desirable to avoid thermal conversion. For example, the efficiency of nuclear reactors, where the kinetic energy of the nuclei is first converted to thermal energy and then to electrical energy, lies at around 35%. By direct conversion of kinetic energy to electric energy, effected by eliminating the intermediate thermal energy transformation, the efficiency of the energy transformation process can be dramatically improved. History of energy transformation Energy transformations in the universe over time are usually characterized by various kinds of energy, which have been available since the Big Bang, later being "released" (that is, transformed to more active types of energy such as kinetic or radiant energy) by a triggering mechanism. Release of energy from gravitational potential A direct transformation of energy occurs when hydrogen produced in the Big Bang collects into structures such as planets, in a process during which part of the gravitational potential is to be converted directly into heat. In Jupiter, Saturn, and Neptune, for example, such heat from the continued collapse of the planets' large gas atmospheres continue to drive most of the planets' weather systems. These systems, consisting of atmospheric bands, winds, and powerful storms, are only partly powered by sunlight. However, on Uranus, little of this process occurs. On Earth, a significant portion of the heat output from the interior of the planet, estimated at a third to half of the total, is caused by the slow collapse of planetary materials to a smaller size, generating heat. Release of energy from radioactive potential Familiar examples of other such processes transforming energy from the Big Bang include nuclear decay, which releases energy that was originally "stored" in heavy isotopes, such as uranium and thorium. This energy was stored at the time of the nucleosynthesis of these elements. This process uses the gravitational potential energy released from the collapse of Type II supernovae to create these heavy elements before they are incorporated into star systems such as the Solar System and the Earth. The energy locked into uranium is released spontaneously during most types of radioactive decay, and can be suddenly released in nuclear fission bombs. In both cases, a portion of the energy binding the atomic nuclei together is released as heat. Release of energy from hydrogen fusion potential In a similar chain of transformations beginning at the dawn of the universe, nuclear fusion of hydrogen in the Sun releases another store of potential energy which was created at the time of the Big Bang. At that time, according to one theory, space expanded and the universe cooled too rapidly for hydrogen to completely fuse into heavier elements. This resulted in hydrogen representing a store of potential energy which can be released by nuclear fusion. Such a fusion process is triggered by heat and pressure generated from the gravitational collapse of hydrogen clouds when they produce stars, and some of the fusion energy is then transformed into starlight. Considering the solar system, starlight, overwhelmingly from the Sun, may again be stored as gravitational potential energy after it strikes the Earth. This occurs in the case of avalanches, or when water evaporates from oceans and is deposited as precipitation high above sea level (where, after being released at a hydroelectric dam, it can be used to drive turbine/generators to produce electricity). Sunlight also drives many weather phenomena on Earth. One example is a hurricane, which occurs when large unstable areas of warm ocean, heated over months, give up some of their thermal energy suddenly to power a few days of violent air movement. Sunlight is also captured by plants as a chemical potential energy via photosynthesis, when carbon dioxide and water are converted into a combustible combination of carbohydrates, lipids, and oxygen. The release of this energy as heat and light may be triggered suddenly by a spark, in a forest fire; or it may be available more slowly for animal or human metabolism when these molecules are ingested, and catabolism is triggered by enzyme action. Through all of these transformation chains, the potential energy stored at the time of the Big Bang is later released by intermediate events, sometimes being stored in several different ways for long periods between releases, as more active energy. All of these events involve the conversion of one kind of energy into others, including heat. Examples Examples of sets of energy conversions in machines A coal-fired power plant involves these energy transformations: Chemical energy in the coal is converted into thermal energy in the exhaust gases of combustion Thermal energy of the exhaust gases converted into thermal energy of steam through heat exchange Kinetic energy of steam converted to mechanical energy in the turbine Mechanical energy of the turbine is converted to electrical energy by the generator, which is the ultimate output In such a system, the first and fourth steps are highly efficient, but the second and third steps are less efficient. The most efficient gas-fired electrical power stations can achieve 50% conversion efficiency. Oil- and coal-fired stations are less efficient. In a conventional automobile, the following energy transformations occur: Chemical energy in the fuel is converted into kinetic energy of expanding gas via combustion Kinetic energy of expanding gas converted to the linear piston movement Linear piston movement converted to rotary crankshaft movement Rotary crankshaft movement passed into transmission assembly Rotary movement passed out of transmission assembly Rotary movement passed through a differential Rotary movement passed out of differential to drive wheels Rotary movement of drive wheels converted to linear motion of the vehicle Other energy conversions There are many different machines and transducers that convert one energy form into another. A short list of examples follows: ATP hydrolysis (chemical energy in adenosine triphosphate → mechanical energy) Battery (electricity) (chemical energy → electrical energy) Electric generator (kinetic energy or mechanical work → electrical energy) Electric heater (electric energy → heat) Fire (chemical energy → heat and light) Friction (kinetic energy → heat) Fuel cell (chemical energy → electrical energy) Geothermal power (heat→ electrical energy) Heat engines, such as the internal combustion engine used in cars, or the steam engine (heat → mechanical energy) Hydroelectric dam (gravitational potential energy → electrical energy) Electric lamp (electrical energy → heat and light) Microphone (sound → electrical energy) Ocean thermal power (heat → electrical energy) Photosynthesis (electromagnetic radiation → chemical energy) Piezoelectrics (strain → electrical energy) Thermoelectric (heat → electrical energy) Wave power (mechanical energy → electrical energy) Windmill (wind energy → electrical energy or mechanical energy) See also Chaos theory Conservation law Conservation of energy Conservation of mass Energy accounting Energy quality Groundwater energy balance Laws of thermodynamics Noether's theorem Ocean thermal energy conversion Thermodynamic equilibrium Thermoeconomics Uncertainty principle References Further reading Energy Transfer and Transformation | Core knowledge science Energy (physics)

Bernoulli's principle

Bernoulli's principle is a key concept in fluid dynamics that relates pressure, speed and height. Bernoulli's principle states that an increase in the speed of a parcel of fluid occurs simultaneously with a decrease in either the pressure or the height above a datum. The principle is named after the Swiss mathematician and physicist Daniel Bernoulli, who published it in his book Hydrodynamica in 1738. Although Bernoulli deduced that pressure decreases when the flow speed increases, it was Leonhard Euler in 1752 who derived Bernoulli's equation in its usual form. Bernoulli's principle can be derived from the principle of conservation of energy. This states that, in a steady flow, the sum of all forms of energy in a fluid is the same at all points that are free of viscous forces. This requires that the sum of kinetic energy, potential energy and internal energy remains constant. Thus an increase in the speed of the fluid—implying an increase in its kinetic energy—occurs with a simultaneous decrease in (the sum of) its potential energy (including the static pressure) and internal energy. If the fluid is flowing out of a reservoir, the sum of all forms of energy is the same because in a reservoir the energy per unit volume (the sum of pressure and gravitational potential ) is the same everywhere. Bernoulli's principle can also be derived directly from Isaac Newton's second Law of Motion. If a small volume of fluid is flowing horizontally from a region of high pressure to a region of low pressure, then there is more pressure behind than in front. This gives a net force on the volume, accelerating it along the streamline. Fluid particles are subject only to pressure and their own weight. If a fluid is flowing horizontally and along a section of a streamline, where the speed increases it can only be because the fluid on that section has moved from a region of higher pressure to a region of lower pressure; and if its speed decreases, it can only be because it has moved from a region of lower pressure to a region of higher pressure. Consequently, within a fluid flowing horizontally, the highest speed occurs where the pressure is lowest, and the lowest speed occurs where the pressure is highest. Bernoulli's principle is only applicable for isentropic flows: when the effects of irreversible processes (like turbulence) and non-adiabatic processes (e.g. thermal radiation) are small and can be neglected. However, the principle can be applied to various types of flow within these bounds, resulting in various forms of Bernoulli's equation. The simple form of Bernoulli's equation is valid for incompressible flows (e.g. most liquid flows and gases moving at low Mach number). More advanced forms may be applied to compressible flows at higher Mach numbers. Incompressible flow equation In most flows of liquids, and of gases at low Mach number, the density of a fluid parcel can be considered to be constant, regardless of pressure variations in the flow. Therefore, the fluid can be considered to be incompressible, and these flows are called incompressible flows. Bernoulli performed his experiments on liquids, so his equation in its original form is valid only for incompressible flow. A common form of Bernoulli's equation is: where: is the fluid flow speed at a point, is the acceleration due to gravity, is the elevation of the point above a reference plane, with the positive -direction pointing upward—so in the direction opposite to the gravitational acceleration, is the pressure at the chosen point, and is the density of the fluid at all points in the fluid. Bernoulli's equation and the Bernoulli constant are applicable throughout any region of flow where the energy per unit mass is uniform. Because the energy per unit mass of liquid in a well-mixed reservoir is uniform throughout, Bernoulli's equation can be used to analyze the fluid flow everywhere in that reservoir (including pipes or flow fields that the reservoir feeds) except where viscous forces dominate and erode the energy per unit mass. The following assumptions must be met for this Bernoulli equation to apply: the flow must be steady, that is, the flow parameters (velocity, density, etc.) at any point cannot change with time, the flow must be incompressible—even though pressure varies, the density must remain constant along a streamline; friction by viscous forces must be negligible. For conservative force fields (not limited to the gravitational field), Bernoulli's equation can be generalized as: where is the force potential at the point considered. For example, for the Earth's gravity . By multiplying with the fluid density , equation can be rewritten as: or: where is dynamic pressure, is the piezometric head or hydraulic head (the sum of the elevation and the pressure head) and is the stagnation pressure (the sum of the static pressure and dynamic pressure ). The constant in the Bernoulli equation can be normalized. A common approach is in terms of total head or energy head : The above equations suggest there is a flow speed at which pressure is zero, and at even higher speeds the pressure is negative. Most often, gases and liquids are not capable of negative absolute pressure, or even zero pressure, so clearly Bernoulli's equation ceases to be valid before zero pressure is reached. In liquids—when the pressure becomes too low—cavitation occurs. The above equations use a linear relationship between flow speed squared and pressure. At higher flow speeds in gases, or for sound waves in liquid, the changes in mass density become significant so that the assumption of constant density is invalid. Simplified form In many applications of Bernoulli's equation, the change in the term is so small compared with the other terms that it can be ignored. For example, in the case of aircraft in flight, the change in height is so small the term can be omitted. This allows the above equation to be presented in the following simplified form: where is called total pressure, and is dynamic pressure. Many authors refer to the pressure as static pressure to distinguish it from total pressure and dynamic pressure . In Aerodynamics, L.J. Clancy writes: "To distinguish it from the total and dynamic pressures, the actual pressure of the fluid, which is associated not with its motion but with its state, is often referred to as the static pressure, but where the term pressure alone is used it refers to this static pressure." The simplified form of Bernoulli's equation can be summarized in the following memorable word equation: Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure and dynamic pressure . Their sum is defined to be the total pressure . The significance of Bernoulli's principle can now be summarized as "total pressure is constant in any region free of viscous forces". If the fluid flow is brought to rest at some point, this point is called a stagnation point, and at this point the static pressure is equal to the stagnation pressure. If the fluid flow is irrotational, the total pressure is uniform and Bernoulli's principle can be summarized as "total pressure is constant everywhere in the fluid flow". It is reasonable to assume that irrotational flow exists in any situation where a large body of fluid is flowing past a solid body. Examples are aircraft in flight and ships moving in open bodies of water. However, Bernoulli's principle importantly does not apply in the boundary layer such as in flow through long pipes. Unsteady potential flow The Bernoulli equation for unsteady potential flow is used in the theory of ocean surface waves and acoustics. For an irrotational flow, the flow velocity can be described as the gradient of a velocity potential . In that case, and for a constant density , the momentum equations of the Euler equations can be integrated to: which is a Bernoulli equation valid also for unsteady—or time dependent—flows. Here denotes the partial derivative of the velocity potential with respect to time , and is the flow speed. The function depends only on time and not on position in the fluid. As a result, the Bernoulli equation at some moment applies in the whole fluid domain. This is also true for the special case of a steady irrotational flow, in which case and are constants so equation can be applied in every point of the fluid domain. Further can be made equal to zero by incorporating it into the velocity potential using the transformation: resulting in: Note that the relation of the potential to the flow velocity is unaffected by this transformation: . The Bernoulli equation for unsteady potential flow also appears to play a central role in Luke's variational principle, a variational description of free-surface flows using the Lagrangian mechanics. Compressible flow equation Bernoulli developed his principle from observations on liquids, and Bernoulli's equation is valid for ideal fluids: those that are incompressible, irrotational, inviscid, and subjected to conservative forces. It is sometimes valid for the flow of gases: provided that there is no transfer of kinetic or potential energy from the gas flow to the compression or expansion of the gas. If both the gas pressure and volume change simultaneously, then work will be done on or by the gas. In this case, Bernoulli's equation—in its incompressible flow form—cannot be assumed to be valid. However, if the gas process is entirely isobaric, or isochoric, then no work is done on or by the gas (so the simple energy balance is not upset). According to the gas law, an isobaric or isochoric process is ordinarily the only way to ensure constant density in a gas. Also the gas density will be proportional to the ratio of pressure and absolute temperature; however, this ratio will vary upon compression or expansion, no matter what non-zero quantity of heat is added or removed. The only exception is if the net heat transfer is zero, as in a complete thermodynamic cycle or in an individual isentropic (frictionless adiabatic) process, and even then this reversible process must be reversed, to restore the gas to the original pressure and specific volume, and thus density. Only then is the original, unmodified Bernoulli equation applicable. In this case the equation can be used if the flow speed of the gas is sufficiently below the speed of sound, such that the variation in density of the gas (due to this effect) along each streamline can be ignored. Adiabatic flow at less than Mach 0.3 is generally considered to be slow enough. It is possible to use the fundamental principles of physics to develop similar equations applicable to compressible fluids. There are numerous equations, each tailored for a particular application, but all are analogous to Bernoulli's equation and all rely on nothing more than the fundamental principles of physics such as Newton's laws of motion or the first law of thermodynamics. Compressible flow in fluid dynamics For a compressible fluid, with a barotropic equation of state, and under the action of conservative forces, where: is the pressure is the density and indicates that it is a function of pressure is the flow speed is the potential associated with the conservative force field, often the gravitational potential In engineering situations, elevations are generally small compared to the size of the Earth, and the time scales of fluid flow are small enough to consider the equation of state as adiabatic. In this case, the above equation for an ideal gas becomes: where, in addition to the terms listed above: is the ratio of the specific heats of the fluid is the acceleration due to gravity is the elevation of the point above a reference plane In many applications of compressible flow, changes in elevation are negligible compared to the other terms, so the term can be omitted. A very useful form of the equation is then: where: is the total pressure is the total density Compressible flow in thermodynamics The most general form of the equation, suitable for use in thermodynamics in case of (quasi) steady flow, is: Here is the enthalpy per unit mass (also known as specific enthalpy), which is also often written as (not to be confused with "head" or "height"). Note that where is the thermodynamic energy per unit mass, also known as the specific internal energy. So, for constant internal energy the equation reduces to the incompressible-flow form. The constant on the right-hand side is often called the Bernoulli constant and denoted . For steady inviscid adiabatic flow with no additional sources or sinks of energy, is constant along any given streamline. More generally, when may vary along streamlines, it still proves a useful parameter, related to the "head" of the fluid (see below). When the change in can be ignored, a very useful form of this equation is: where is total enthalpy. For a calorically perfect gas such as an ideal gas, the enthalpy is directly proportional to the temperature, and this leads to the concept of the total (or stagnation) temperature. When shock waves are present, in a reference frame in which the shock is stationary and the flow is steady, many of the parameters in the Bernoulli equation suffer abrupt changes in passing through the shock. The Bernoulli parameter remains unaffected. An exception to this rule is radiative shocks, which violate the assumptions leading to the Bernoulli equation, namely the lack of additional sinks or sources of energy. Unsteady potential flow For a compressible fluid, with a barotropic equation of state, the unsteady momentum conservation equation With the irrotational assumption, namely, the flow velocity can be described as the gradient of a velocity potential . The unsteady momentum conservation equation becomes which leads to In this case, the above equation for isentropic flow becomes: Derivations Applications In modern everyday life there are many observations that can be successfully explained by application of Bernoulli's principle, even though no real fluid is entirely inviscid, and a small viscosity often has a large effect on the flow. Bernoulli's principle can be used to calculate the lift force on an airfoil, if the behaviour of the fluid flow in the vicinity of the foil is known. For example, if the air flowing past the top surface of an aircraft wing is moving faster than the air flowing past the bottom surface, then Bernoulli's principle implies that the pressure on the surfaces of the wing will be lower above than below. This pressure difference results in an upwards lifting force. Whenever the distribution of speed past the top and bottom surfaces of a wing is known, the lift forces can be calculated (to a good approximation) using Bernoulli's equations, which were established by Bernoulli over a century before the first man-made wings were used for the purpose of flight. The carburetor used in many reciprocating engines contains a venturi to create a region of low pressure to draw fuel into the carburetor and mix it thoroughly with the incoming air. The low pressure in the throat of a venturi can be explained by Bernoulli's principle; in the narrow throat, the air is moving at its fastest speed and therefore it is at its lowest pressure. An injector on a steam locomotive or a static boiler. The pitot tube and static port on an aircraft are used to determine the airspeed of the aircraft. These two devices are connected to the airspeed indicator, which determines the dynamic pressure of the airflow past the aircraft. Bernoulli's principle is used to calibrate the airspeed indicator so that it displays the indicated airspeed appropriate to the dynamic pressure. A De Laval nozzle utilizes Bernoulli's principle to create a force by turning pressure energy generated by the combustion of propellants into velocity. This then generates thrust by way of Newton's third law of motion. The flow speed of a fluid can be measured using a device such as a Venturi meter or an orifice plate, which can be placed into a pipeline to reduce the diameter of the flow. For a horizontal device, the continuity equation shows that for an incompressible fluid, the reduction in diameter will cause an increase in the fluid flow speed. Subsequently, Bernoulli's principle then shows that there must be a decrease in the pressure in the reduced diameter region. This phenomenon is known as the Venturi effect. The maximum possible drain rate for a tank with a hole or tap at the base can be calculated directly from Bernoulli's equation and is found to be proportional to the square root of the height of the fluid in the tank. This is Torricelli's law, which is compatible with Bernoulli's principle. Increased viscosity lowers this drain rate; this is reflected in the discharge coefficient, which is a function of the Reynolds number and the shape of the orifice. The Bernoulli grip relies on this principle to create a non-contact adhesive force between a surface and the gripper. During a cricket match, bowlers continually polish one side of the ball. After some time, one side is quite rough and the other is still smooth. Hence, when the ball is bowled and passes through air, the speed on one side of the ball is faster than on the other, and this results in a pressure difference between the sides; this leads to the ball rotating ("swinging") while travelling through the air, giving advantage to the bowlers. Misconceptions Airfoil lift One of the most common erroneous explanations of aerodynamic lift asserts that the air must traverse the upper and lower surfaces of a wing in the same amount of time, implying that since the upper surface presents a longer path the air must be moving over the top of the wing faster than over the bottom. Bernoulli's principle is then cited to conclude that the pressure on top of the wing must be lower than on the bottom. Equal transit time applies to the flow around a body generating no lift, but there is no physical principle that requires equal transit time in cases of bodies generating lift. In fact, theory predicts – and experiments confirm – that the air traverses the top surface of a body experiencing lift in a shorter time than it traverses the bottom surface; the explanation based on equal transit time is false. While the equal-time explanation is false, it is not the Bernoulli principle that is false, because this principle is well established; Bernoulli's equation is used correctly in common mathematical treatments of aerodynamic lift. Common classroom demonstrations There are several common classroom demonstrations that are sometimes incorrectly explained using Bernoulli's principle. One involves holding a piece of paper horizontally so that it droops downward and then blowing over the top of it. As the demonstrator blows over the paper, the paper rises. It is then asserted that this is because "faster moving air has lower pressure". One problem with this explanation can be seen by blowing along the bottom of the paper: if the deflection was caused by faster moving air, then the paper should deflect downward; but the paper deflects upward regardless of whether the faster moving air is on the top or the bottom. Another problem is that when the air leaves the demonstrator's mouth it has the same pressure as the surrounding air; the air does not have lower pressure just because it is moving; in the demonstration, the static pressure of the air leaving the demonstrator's mouth is equal to the pressure of the surrounding air. A third problem is that it is false to make a connection between the flow on the two sides of the paper using Bernoulli's equation since the air above and below are different flow fields and Bernoulli's principle only applies within a flow field. As the wording of the principle can change its implications, stating the principle correctly is important. What Bernoulli's principle actually says is that within a flow of constant energy, when fluid flows through a region of lower pressure it speeds up and vice versa. Thus, Bernoulli's principle concerns itself with changes in speed and changes in pressure within a flow field. It cannot be used to compare different flow fields. A correct explanation of why the paper rises would observe that the plume follows the curve of the paper and that a curved streamline will develop a pressure gradient perpendicular to the direction of flow, with the lower pressure on the inside of the curve. Bernoulli's principle predicts that the decrease in pressure is associated with an increase in speed; in other words, as the air passes over the paper, it speeds up and moves faster than it was moving when it left the demonstrator's mouth. But this is not apparent from the demonstration. Other common classroom demonstrations, such as blowing between two suspended spheres, inflating a large bag, or suspending a ball in an airstream are sometimes explained in a similarly misleading manner by saying "faster moving air has lower pressure". See also Torricelli's law Coandă effect Euler equations – for the flow of an inviscid fluid Hydraulics – applied fluid mechanics for liquids Navier–Stokes equations – for the flow of a viscous fluid Teapot effect Terminology in fluid dynamics Notes References External links The Flow of Dry Water - The Feynman Lectures on Physics Science 101 Q: Is It Really Caused by the Bernoulli Effect? Bernoulli equation calculator Millersville University – Applications of Euler's equation NASA – Beginner's guide to aerodynamics Misinterpretations of Bernoulli's equation – Weltner and Ingelman-Sundberg Fluid dynamics Eponymous laws of physics Equations of fluid dynamics 1738 in science

Kinetic energy

In physics, the kinetic energy of an object is the form of energy that it possesses due to its motion. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is . The kinetic energy of an object is equal to the work, force (F) times displacement (s), needed to achieve its stated velocity. Having gained this energy during its acceleration, the mass maintains this kinetic energy unless its speed changes. The same amount of work is done by the object when decelerating from its current speed to a state of rest. The SI unit of kinetic energy is the joule, while the English unit of kinetic energy is the foot-pound. In relativistic mechanics, is a good approximation of kinetic energy only when v is much less than the speed of light. History and etymology The adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality. The principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva. Willem 's Gravesande of the Netherlands provided experimental evidence of this relationship in 1722. By dropping weights from different heights into a block of clay, Willem 's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet recognized the implications of the experiment and published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–1851. Rankine, who had introduced the term "potential energy" in 1853, and the phrase "actual energy" to complement it, later cites William Thomson and Peter Tait as substituting the word "kinetic" for "actual". Overview Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, and rest energy. These can be categorized in two main classes: potential energy and kinetic energy. Kinetic energy is the movement energy of an object. Kinetic energy can be transferred between objects and transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction. The chemical energy has been converted into kinetic energy, the energy of motion, but the process is not completely efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top. The kinetic energy has now largely been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling. The energy is not destroyed; it has only been converted to another form by friction. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent. The bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity that is a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an entirely circular orbit, this kinetic energy remains constant because there is almost no friction in near-earth space. However, it becomes apparent at re-entry when some of the kinetic energy is converted to heat. If the orbit is elliptical or hyperbolic, then throughout the orbit kinetic and potential energy are exchanged; kinetic energy is greatest and potential energy lowest at closest approach to the earth or other massive body, while potential energy is greatest and kinetic energy the lowest at maximum distance. Disregarding loss or gain however, the sum of the kinetic and potential energy remains constant. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down dramatically, and the ball it hit accelerates as the kinetic energy is passed on to it. Collisions in billiards are effectively elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, sound and binding energy (breaking bound structures). Flywheels have been developed as a method of energy storage. This illustrates that kinetic energy is also stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula mv2 given by classical mechanics is suitable. However, if the speed of the object is comparable to the speed of light, relativistic effects become significant and the relativistic formula is used. If the object is on the atomic or sub-atomic scale, quantum mechanical effects are significant, and a quantum mechanical model must be employed. Kinetic energy for non-relativistic velocity Treatments of kinetic energy depend upon the relative velocity of objects compared to the fixed speed of light. Speeds experienced directly by humans are non-relativisitic; higher speeds require the theory of relativity. Kinetic energy of rigid bodies In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well as its speed. The kinetic energy is equal to 1/2 the product of the mass and the square of the speed. In formula form: where is the mass and is the speed (magnitude of the velocity) of the body. In SI units, mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules. For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as When a person throws a ball, the person does work on it to give it speed as it leaves the hand. The moving ball can then hit something and push it, doing work on what it hits. The kinetic energy of a moving object is equal to the work required to bring it from rest to that speed, or the work the object can do while being brought to rest: net force × displacement = kinetic energy, i.e., Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force. As a consequence of this quadrupling, it takes four times the work to double the speed. The kinetic energy of an object is related to its momentum by the equation: where: is momentum is mass of the body For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass , whose center of mass is moving in a straight line with speed , as seen above is equal to where: is the mass of the body is the speed of the center of mass of the body. The kinetic energy of any entity depends on the reference frame in which it is measured. However, the total energy of an isolated system, i.e. one in which energy can neither enter nor leave, does not change over time in the reference frame in which it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames would however disagree on the value of this conserved energy. The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole. Derivation Without vector calculus The work W done by a force F on an object over a distance s parallel to F equals . Using Newton's Second Law with m the mass and a the acceleration of the object and the distance traveled by the accelerated object in time t, we find with for the velocity v of the object With vector calculus The work done in accelerating a particle with mass m during the infinitesimal time interval dt is given by the dot product of force F and the infinitesimal displacement dx where we have assumed the relationship p = m v and the validity of Newton's Second Law. (However, also see the special relativistic derivation below.) Applying the product rule we see that: Therefore, (assuming constant mass so that dm = 0), we have, Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy: This equation states that the kinetic energy (Ek) is equal to the integral of the dot product of the momentum (p) of a body and the infinitesimal change of the velocity (v) of the body. It is assumed that the body starts with no kinetic energy when it is at rest (motionless). Rotating bodies If a rigid body Q is rotating about any line through the center of mass then it has rotational kinetic energy which is simply the sum of the kinetic energies of its moving parts, and is thus given by: where: ω is the body's angular velocity r is the distance of any mass dm from that line is the body's moment of inertia, equal to . (In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape). Kinetic energy of systems A system of bodies may have internal kinetic energy due to the relative motion of the bodies in the system. For example, in the Solar System the planets and planetoids are orbiting the Sun. In a tank of gas, the molecules are moving in all directions. The kinetic energy of the system is the sum of the kinetic energies of the bodies it contains. A macroscopic body that is stationary (i.e. a reference frame has been chosen to correspond to the body's center of momentum) may have various kinds of internal energy at the molecular or atomic level, which may be regarded as kinetic energy, due to molecular translation, rotation, and vibration, electron translation and spin, and nuclear spin. These all contribute to the body's mass, as provided by the special theory of relativity. When discussing movements of a macroscopic body, the kinetic energy referred to is usually that of the macroscopic movement only. However, all internal energies of all types contribute to a body's mass, inertia, and total energy. Fluid dynamics In fluid dynamics, the kinetic energy per unit volume at each point in an incompressible fluid flow field is called the dynamic pressure at that point. Dividing by V, the unit of volume: where is the dynamic pressure, and ρ is the density of the incompressible fluid. Frame of reference The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary to an observer moving with the same velocity as the bullet, and so has zero kinetic energy. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case, the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame. The total kinetic energy of a system depends on the inertial frame of reference: it is the sum of the total kinetic energy in a center of momentum frame and the kinetic energy the total mass would have if it were concentrated in the center of mass. This may be simply shown: let be the relative velocity of the center of mass frame i in the frame k. Since Then, However, let the kinetic energy in the center of mass frame, would be simply the total momentum that is by definition zero in the center of mass frame, and let the total mass: . Substituting, we get: Thus the kinetic energy of a system is lowest to center of momentum reference frames, i.e., frames of reference in which the center of mass is stationary (either the center of mass frame or any other center of momentum frame). In any different frame of reference, there is additional kinetic energy corresponding to the total mass moving at the speed of the center of mass. The kinetic energy of the system in the center of momentum frame is a quantity that is invariant (all observers see it to be the same). Rotation in systems It sometimes is convenient to split the total kinetic energy of a body into the sum of the body's center-of-mass translational kinetic energy and the energy of rotation around the center of mass (rotational energy): where: Ek is the total kinetic energy Et is the translational kinetic energy Er is the rotational energy or angular kinetic energy in the rest frame Thus the kinetic energy of a tennis ball in flight is the kinetic energy due to its rotation, plus the kinetic energy due to its translation. Relativistic kinetic energy If a body's speed is a significant fraction of the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy. In relativity, the total energy is given by the energy-momentum relation: Here we use the relativistic expression for linear momentum: , where . with being an object's (rest) mass, speed, and c the speed of light in vacuum. Then kinetic energy is the total relativistic energy minus the rest energy: At low speeds, the square root can be expanded and the rest energy drops out, giving the Newtonian kinetic energy. Derivation Start with the expression for linear momentum , where . Integrating by parts yields Since , is a constant of integration for the indefinite integral. Simplifying the expression we obtain is found by observing that when and , giving resulting in the formula This formula shows that the work expended accelerating an object from rest approaches infinity as the velocity approaches the speed of light. Thus it is impossible to accelerate an object across this boundary. The mathematical by-product of this calculation is the mass–energy equivalence formula—the body at rest must have energy content At a low speed (v ≪ c), the relativistic kinetic energy is approximated well by the classical kinetic energy. This is done by binomial approximation or by taking the first two terms of the Taylor expansion for the reciprocal square root: So, the total energy can be partitioned into the rest mass energy plus the non-relativistic kinetic energy at low speeds. When objects move at a speed much slower than light (e.g. in everyday phenomena on Earth), the first two terms of the series predominate. The next term in the Taylor series approximation is small for low speeds. For example, for a speed of the correction to the non-relativistic kinetic energy is 0.0417 J/kg (on a non-relativistic kinetic energy of 50 MJ/kg) and for a speed of 100 km/s it is 417 J/kg (on a non-relativistic kinetic energy of 5 GJ/kg). The relativistic relation between kinetic energy and momentum is given by This can also be expanded as a Taylor series, the first term of which is the simple expression from Newtonian mechanics: This suggests that the formulae for energy and momentum are not special and axiomatic, but concepts emerging from the equivalence of mass and energy and the principles of relativity. General relativity Using the convention that where the four-velocity of a particle is and is the proper time of the particle, there is also an expression for the kinetic energy of the particle in general relativity. If the particle has momentum as it passes by an observer with four-velocity uobs, then the expression for total energy of the particle as observed (measured in a local inertial frame) is and the kinetic energy can be expressed as the total energy minus the rest energy: Consider the case of a metric that is diagonal and spatially isotropic (gtt, gss, gss, gss). Since where vα is the ordinary velocity measured w.r.t. the coordinate system, we get Solving for ut gives Thus for a stationary observer (v = 0) and thus the kinetic energy takes the form Factoring out the rest energy gives: This expression reduces to the special relativistic case for the flat-space metric where In the Newtonian approximation to general relativity where Φ is the Newtonian gravitational potential. This means clocks run slower and measuring rods are shorter near massive bodies. Kinetic energy in quantum mechanics In quantum mechanics, observables like kinetic energy are represented as operators. For one particle of mass m, the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator . The kinetic energy operator in the non-relativistic case can be written as Notice that this can be obtained by replacing by in the classical expression for kinetic energy in terms of momentum, In the Schrödinger picture, takes the form where the derivative is taken with respect to position coordinates and hence The expectation value of the electron kinetic energy, , for a system of N electrons described by the wavefunction is a sum of 1-electron operator expectation values: where is the mass of the electron and is the Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons. The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. Given an electron density , the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as where is known as the von Weizsäcker kinetic energy functional. See also Escape velocity Foot-pound Joule Kinetic energy penetrator Kinetic energy per unit mass of projectiles Kinetic projectile Parallel axis theorem Potential energy Recoil Notes References External links Dynamics (mechanics) Forms of energy

Motion

In physics, motion is when an object changes its position with respect to a reference point in a given time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed, and frame of reference to an observer, measuring the change in position of the body relative to that frame with a change in time. The branch of physics describing the motion of objects without reference to their cause is called kinematics, while the branch studying forces and their effect on motion is called dynamics. If an object is not in motion relative to a given frame of reference, it is said to be at rest, motionless, immobile, stationary, or to have a constant or time-invariant position with reference to its surroundings. Modern physics holds that, as there is no absolute frame of reference, Newton's concept of absolute motion cannot be determined. Everything in the universe can be considered to be in motion. Motion applies to various physical systems: objects, bodies, matter particles, matter fields, radiation, radiation fields, radiation particles, curvature, and space-time. One can also speak of the motion of images, shapes, and boundaries. In general, the term motion signifies a continuous change in the position or configuration of a physical system in space. For example, one can talk about the motion of a wave or the motion of a quantum particle, where the configuration consists of the probabilities of the wave or particle occupying specific positions. Equations of motion Laws of motion In physics, the motion of bodies is described through two related sets of laws of mechanics. Classical mechanics for super atomic (larger than an atom) objects (such as cars, projectiles, planets, cells, and humans) and quantum mechanics for atomic and sub-atomic objects (such as helium, protons, and electrons). Historically, Newton and Euler formulated three laws of classical mechanics: Classical mechanics Classical mechanics is used for describing the motion of macroscopic objects moving at speeds significantly slower than the speed of light, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results within these domains and is one of the oldest and largest scientific descriptions in science, engineering, and technology. Classical mechanics is fundamentally based on Newton's laws of motion. These laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work Philosophiæ Naturalis Principia Mathematica, which was first published on July 5, 1687. Newton's three laws are: A body at rest will remain at rest, and a body in motion will remain in motion unless it is acted upon by an external force. (This is known as the law of inertia.) Force is equal to the change in momentum per change in time. For a constant mass, force equals mass times acceleration. For every action, there is an equal and opposite reaction. (In other words, whenever one body exerts a force onto a second body, (in some cases, which is standing still) the second body exerts the force back onto the first body. and are equal in magnitude and opposite in direction. So, the body that exerts will be pushed backward.) Newton's three laws of motion were the first to accurately provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of celestial bodies and the motion of objects on Earth. Relativistic mechanics Modern kinematics developed with study of electromagnetism and refers all velocities to their ratio to speed of light . Velocity is then interpreted as rapidity, the hyperbolic angle for which the hyperbolic tangent function . Acceleration, the change of velocity over time, then changes rapidity according to Lorentz transformations. This part of mechanics is special relativity. Efforts to incorporate gravity into relativistic mechanics were made by W. K. Clifford and Albert Einstein. The development used differential geometry to describe a curved universe with gravity; the study is called general relativity. Quantum mechanics Quantum mechanics is a set of principles describing physical reality at the atomic level of matter (molecules and atoms) and the subatomic particles (electrons, protons, neutrons, and even smaller elementary particles such as quarks). These descriptions include the simultaneous wave-like and particle-like behavior of both matter and radiation energy as described in the wave–particle duality. In classical mechanics, accurate measurements and predictions of the state of objects can be calculated, such as location and velocity. In quantum mechanics, due to the Heisenberg uncertainty principle, the complete state of a subatomic particle, such as its location and velocity, cannot be simultaneously determined. In addition to describing the motion of atomic level phenomena, quantum mechanics is useful in understanding some large-scale phenomena such as superfluidity, superconductivity, and biological systems, including the function of smell receptors and the structures of protein. Orders of magnitude Humans, like all known things in the universe, are in constant motion; however, aside from obvious movements of the various external body parts and locomotion, humans are in motion in a variety of ways that are more difficult to perceive. Many of these "imperceptible motions" are only perceivable with the help of special tools and careful observation. The larger scales of imperceptible motions are difficult for humans to perceive for two reasons: Newton's laws of motion (particularly the third), which prevents the feeling of motion on a mass to which the observer is connected, and the lack of an obvious frame of reference that would allow individuals to easily see that they are moving. The smaller scales of these motions are too small to be detected conventionally with human senses. Universe Spacetime (the fabric of the universe) is expanding, meaning everything in the universe is stretching, like a rubber band. This motion is the most obscure as it is not physical motion, but rather a change in the very nature of the universe. The primary source of verification of this expansion was provided by Edwin Hubble who demonstrated that all galaxies and distant astronomical objects were moving away from Earth, known as Hubble's law, predicted by a universal expansion. Galaxy The Milky Way Galaxy is moving through space and many astronomers believe the velocity of this motion to be approximately relative to the observed locations of other nearby galaxies. Another reference frame is provided by the Cosmic microwave background. This frame of reference indicates that the Milky Way is moving at around . Sun and Solar System The Milky Way is rotating around its dense Galactic Center, thus the Sun is moving in a circle within the galaxy's gravity. Away from the central bulge, or outer rim, the typical stellar velocity is between . All planets and their moons move with the Sun. Thus, the Solar System is in motion. Earth The Earth is rotating or spinning around its axis. This is evidenced by day and night, at the equator the earth has an eastward velocity of . The Earth is also orbiting around the Sun in an orbital revolution. A complete orbit around the Sun takes one year, or about 365 days; it averages a speed of about . Continents The Theory of Plate tectonics tells us that the continents are drifting on convection currents within the mantle, causing them to move across the surface of the planet at the slow speed of approximately per year. However, the velocities of plates range widely. The fastest-moving plates are the oceanic plates, with the Cocos Plate advancing at a rate of per year and the Pacific Plate moving per year. At the other extreme, the slowest-moving plate is the Eurasian Plate, progressing at a typical rate of about per year. Internal body The human heart is regularly contracting to move blood throughout the body. Through larger veins and arteries in the body, blood has been found to travel at approximately 0.33 m/s. Though considerable variation exists, and peak flows in the venae cavae have been found between . additionally, the smooth muscles of hollow internal organs are moving. The most familiar would be the occurrence of peristalsis, which is where digested food is forced throughout the digestive tract. Though different foods travel through the body at different rates, an average speed through the human small intestine is . The human lymphatic system is also constantly causing movements of excess fluids, lipids, and immune system related products around the body. The lymph fluid has been found to move through a lymph capillary of the skin at approximately 0.0000097 m/s. Cells The cells of the human body have many structures and organelles that move throughout them. Cytoplasmic streaming is a way in which cells move molecular substances throughout the cytoplasm, various motor proteins work as molecular motors within a cell and move along the surface of various cellular substrates such as microtubules, and motor proteins are typically powered by the hydrolysis of adenosine triphosphate (ATP), and convert chemical energy into mechanical work. Vesicles propelled by motor proteins have been found to have a velocity of approximately 0.00000152 m/s. Particles According to the laws of thermodynamics, all particles of matter are in constant random motion as long as the temperature is above absolute zero. Thus the molecules and atoms that make up the human body are vibrating, colliding, and moving. This motion can be detected as temperature; higher temperatures, which represent greater kinetic energy in the particles, feel warm to humans who sense the thermal energy transferring from the object being touched to their nerves. Similarly, when lower temperature objects are touched, the senses perceive the transfer of heat away from the body as a feeling of cold. Subatomic particles Within the standard atomic orbital model, electrons exist in a region around the nucleus of each atom. This region is called the electron cloud. According to Bohr's model of the atom, electrons have a high velocity, and the larger the nucleus they are orbiting the faster they would need to move. If electrons were to move about the electron cloud in strict paths the same way planets orbit the Sun, then electrons would be required to do so at speeds that would far exceed the speed of light. However, there is no reason that one must confine oneself to this strict conceptualization (that electrons move in paths the same way macroscopic objects do), rather one can conceptualize electrons to be 'particles' that capriciously exist within the bounds of the electron cloud. Inside the atomic nucleus, the protons and neutrons are also probably moving around due to the electrical repulsion of the protons and the presence of angular momentum of both particles. Light Light moves at a speed of 299,792,458 m/s, or , in a vacuum. The speed of light in vacuum (or ) is also the speed of all massless particles and associated fields in a vacuum, and it is the upper limit on the speed at which energy, matter, information or causation can travel. The speed of light in vacuum is thus the upper limit for speed for all physical systems. In addition, the speed of light is an invariant quantity: it has the same value, irrespective of the position or speed of the observer. This property makes the speed of light c a natural measurement unit for speed and a fundamental constant of nature. In 2019, the speed of light was redefined alongside all seven SI base units using what it calls "the explicit-constant formulation", where each "unit is defined indirectly by specifying explicitly an exact value for a well-recognized fundamental constant", as was done for the speed of light. A new, but completely equivalent, wording of the metre's definition was proposed: "The metre, symbol m, is the unit of length; its magnitude is set by fixing the numerical value of the speed of light in vacuum to be equal to exactly when it is expressed in the SI unit ." This implicit change to the speed of light was one of the changes that was incorporated in the 2019 revision of the SI, also termed the New SI. Superluminal motion Some motion appears to an observer to exceed the speed of light. Bursts of energy moving out along the relativistic jets emitted from these objects can have a proper motion that appears greater than the speed of light. All of these sources are thought to contain a black hole, responsible for the ejection of mass at high velocities. Light echoes can also produce apparent superluminal motion. This occurs owing to how motion is often calculated at long distances; oftentimes calculations fail to account for the fact that the speed of light is finite. When measuring the movement of distant objects across the sky, there is a large time delay between what has been observed and what has occurred, due to the large distance the light from the distant object has to travel to reach us. The error in the above naive calculation comes from the fact that when an object has a component of velocity directed towards the Earth, as the object moves closer to the Earth that time delay becomes smaller. This means that the apparent speed as calculated above is greater than the actual speed. Correspondingly, if the object is moving away from the Earth, the above calculation underestimates the actual speed. Types of motion Simple harmonic motion – motion in which the body oscillates in such a way that the restoring force acting on it is directly proportional to the body's displacement. Mathematically Force is directly proportional to the negative of displacement. Negative sign signifies the restoring nature of the force. (e.g., that of a pendulum). Linear motion – motion that follows a straight linear path, and whose displacement is exactly the same as its trajectory. [Also known as rectilinear motion] Reciprocal motion Brownian motion – the random movement of very small particles Circular motion Rotatory motion – a motion about a fixed point. (e.g. Ferris wheel). Curvilinear motion – It is defined as the motion along a curved path that may be planar or in three dimensions. Rolling motion – (as of the wheel of a bicycle) Oscillatory – (swinging from side to side) Vibratory motion Combination (or simultaneous) motions – Combination of two or more above listed motions Projectile motion – uniform horizontal motion + vertical accelerated motion Fundamental motions Linear motion Circular motion Oscillation Wave Relative motion Rotary motion See also References External links Feynman's lecture on motion Mechanics Physical phenomena

Mass–energy equivalence

In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicist Albert Einstein's formula: . In a reference frame where the system is moving, its relativistic energy and relativistic mass (instead of rest mass) obey the same formula. The formula defines the energy of a particle in its rest frame as the product of mass with the speed of light squared. Because the speed of light is a large number in everyday units (approximately ), the formula implies that a small amount of "rest mass", measured when the system is at rest, corresponds to an enormous amount of energy, which is independent of the composition of the matter. Rest mass, also called invariant mass, is a fundamental physical property that is independent of momentum, even at extreme speeds approaching the speed of light. Its value is the same in all inertial frames of reference. Massless particles such as photons have zero invariant mass, but massless free particles have both momentum and energy. The equivalence principle implies that when mass is lost in chemical reactions or nuclear reactions, a corresponding amount of energy will be released. The energy can be released to the environment (outside of the system being considered) as radiant energy, such as light, or as thermal energy. The principle is fundamental to many fields of physics, including nuclear and particle physics. Mass–energy equivalence arose from special relativity as a paradox described by the French polymath Henri Poincaré (1854–1912). Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the symmetries of space and time. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his annus mirabilis papers, published on 21 November 1905. The formula and its relationship to momentum, as described by the energy–momentum relation, were later developed by other physicists. Description Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equal or they differ only by a constant factor, the speed of light squared. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. These energies tend to be much smaller than the mass of the object multiplied by , which is on the order of 1017 joules for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a nuclear reaction is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as the difference. In analyzing these extreme events, Einstein's formula can be used with as the energy released (removed), and as the change in mass. In relativity, all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box by the amount equal to their energy divided by . For an observer in the rest frame, removing energy is the same as removing mass and the formula indicates how much mass is lost when energy is removed. In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by . Mass in special relativity An object moves at different speeds in different frames of reference, depending on the motion of the observer. This implies the kinetic energy, in both Newtonian mechanics and relativity, is 'frame dependent', so that the amount of relativistic energy that an object is measured to have depends on the observer. The relativistic mass of an object is given by the relativistic energy divided by . Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonymous; the only difference between them is the units. The rest mass or invariant mass of an object is defined as the mass an object has in its rest frame, when it is not moving with respect to the observer. Physicists typically use the term mass, though experiments have shown an object's gravitational mass depends on its total energy and not just its rest mass. The rest mass is the same for all inertial frames, as it is independent of the motion of the observer, it is the smallest possible value of the relativistic mass of the object. Because of the attraction between components of a system, which results in potential energy, the rest mass is almost never additive; in general, the mass of an object is not the sum of the masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as observed from the center of momentum frame, and potential energy. The masses add up only if the constituents are at rest (as observed from the center of momentum frame) and do not attract or repel, so that they do not have any extra kinetic or potential energy. Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum. Relativistic mass Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. The relativistic mass of a moving object is larger than the relativistic mass of an object at rest, because a moving object has kinetic energy. If the object moves slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the classical inertial mass (as it appears in Newton's laws of motion). If the object moves quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. Massless particles also have relativistic mass derived from their kinetic energy, equal to their relativistic energy divided by , or . The speed of light is one in a system where length and time are measured in natural units and the relativistic mass and energy would be equal in value and dimension. As it is just another name for the energy, the use of the term relativistic mass is redundant and physicists generally reserve mass to refer to rest mass, or invariant mass, as opposed to relativistic mass. A consequence of this terminology is that the mass is not conserved in special relativity, whereas the conservation of momentum and conservation of energy are both fundamental laws. Conservation of mass and energy Conservation of energy is a universal principle in physics and holds for any interaction, along with the conservation of momentum. The classical conservation of mass, in contrast, is violated in certain relativistic settings. This concept has been experimentally proven in a number of ways, including the conversion of mass into kinetic energy in nuclear reactions and other interactions between elementary particles. While modern physics has discarded the expression 'conservation of mass', in older terminology a relativistic mass can also be defined to be equivalent to the energy of a moving system, allowing for a conservation of relativistic mass. Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or radiant energy. Massless particles Massless particles have zero rest mass. The Planck–Einstein relation for the energy for photons is given by the equation , where is the Planck constant and is the photon frequency. This frequency and thus the relativistic energy are frame-dependent. If an observer runs away from a photon in the direction the photon travels from a source, and it catches up with the observer, the observer sees it as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon would be seen to have. As an observer approaches the speed of light with regard to the source, the redshift of the photon increases, according to the relativistic Doppler effect. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, because of the massless nature of photons, which does not permit any intrinsic energy. Composite systems For closed systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is bound by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up. This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together, and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the Solar System is slightly less than the sum of its individual masses. For an isolated system of particles moving in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by ) in the center of momentum frame. The center of momentum frame is defined so that the system has zero total momentum; the term center of mass frame is also sometimes used, where the center of mass frame is a special case of the center of momentum frame where the center of mass is put at the origin. A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system's total energy and invariant mass are the same in any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects, including solids, contributes to their total masses, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object. Similarly, even photons, if trapped in an isolated container, would contribute their energy to the mass of the container. Such extra mass, in theory, could be weighed in the same way as any other type of rest mass, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the consequences of relativity. It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass. Relation to gravity Physics has two concepts of mass, the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the gravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newtonian gravity, the weak equivalence principle is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the weak equivalence principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is one of the pillars of the general theory of relativity. The prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the Eddington experiment, was made during the Solar eclipse of May 29, 1919. During the solar eclipse, the English astronomer and physicist Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960. In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck's relation. Efficiency In some reactions, matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy. Such conversions between types of energy happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their original mass, though the mass lost is not due to the destruction of any smaller constituents. Nuclear fission allows a tiny fraction of the energy associated with the mass to be converted into usable energy such as radiation; in the decay of the uranium, for instance, about 0.1% of the mass of the original atom is lost. In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light, but none of the theoretically known methods are practical. One way to harness all the energy associated with mass is to annihilate matter with antimatter. Antimatter is rare in the universe, however, and the known mechanisms of production require more usable energy than would be released in annihilation. CERN estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation. As most of the mass which comprises ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful forms requires that the protons and neutrons be converted to lighter particles, or particles with no mass at all. In the Standard Model of particle physics, the number of protons plus neutrons is nearly exactly conserved. Despite this, Gerard 't Hooft showed that there is a process that converts protons and neutrons to antielectrons and neutrinos. This is the weak SU(2) instanton proposed by the physicists Alexander Belavin, Alexander Markovich Polyakov, Albert Schwarz, and Yu. S. Tyupkin. This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the Big Bang. Many extensions of the standard model contain magnetic monopoles, and in some models of grand unification, these monopoles catalyze proton decay, a process known as the Callan–Rubakov effect. This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles, whose production is expected to be inefficient. Another method of completely annihilating matter uses the gravitational field of black holes. The British theoretical physicist Stephen Hawking theorized it is possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes. Extension for systems in motion Unlike a system's energy in an inertial frame, the relativistic energy of a system depends on both the rest mass and the total momentum of the system. The extension of Einstein's equation to these systems is given by: or where the term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation is called the energy–momentum relation and reduces to when the momentum term is zero. For photons where , the equation reduces to . Low-speed approximation Using the Lorentz factor, , the energy–momentum can be rewritten as and expanded as a power series: For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because is small. For low speeds, all but the first two terms can be ignored: In classical mechanics, both the term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured and so the term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields: . The difference between the two approximations is given by , a number very small for everyday objects. In 2018 NASA announced the Parker Solar Probe was the fastest ever, with a speed of . The difference between the approximations for the Parker Solar Probe in 2018 is , which accounts for an energy correction of four parts per hundred million. The gravitational constant, in contrast, has a standard relative uncertainty of about . Applications Application to nuclear physics The nuclear binding energy is the minimum energy that is required to disassemble the nucleus of an atom into its component parts. The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the strong nuclear force. The difference between the two masses is called the mass defect and is related to the binding energy through Einstein's formula. The principle is used in modeling nuclear fission reactions, and it implies that a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power. A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by ), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. Thus, if a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation. Thus, a 21.5 kiloton nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If a transparent window passing only electromagnetic radiation were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass that absorbed the X-rays (and other "heat") would gain this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation. Practical examples Einstein used the centimetre–gram–second system of units (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: . In the SI system (expressing the ratio in joules per kilogram using the value of in metres per second): (≈ 9.0 × 1016 joules per kilogram). So the energy equivalent of one kilogram of mass is 89.9 petajoules 25.0 billion kilowatt-hours (≈ 25,000 GW·h) 21.5 trillion kilocalories (≈ 21 Pcal) 85.2 trillion BTUs 0.0852 quads or the energy released by combustion of the following: 21 500 kilotons of TNT-equivalent energy (≈ 21 Mt) litres or US gallons of automotive gasoline Any time energy is released, the process can be evaluated from an perspective. For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing gram of mass. Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged: A spring's mass increases whenever it is put into compression or tension. Its mass increase arises from the increased potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring. Raising the temperature of an object (increasing its thermal energy) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of platinum and iridium. If its temperature is allowed to change by 1 °C, its mass changes by 1.5 picograms (1 pg = ). A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of energy of rotation, which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, the Earth itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 1024 Joules, which is over 107 kg. History While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields. Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps. Developments prior to Einstein Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it. During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various ether theories. In 1873 the Russian physicist and mathematician Nikolay Umov pointed out a relation between mass and energy for ether in the form of , where . The writings of the English engineer Samuel Tolver Preston, and a 1903 paper by the Italian industrialist and geologist Olinto De Pretto, presented a mass–energy relation. Italian mathematician and math historian Umberto Bartocci observed that there were only three degrees of separation linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work. Preston and De Pretto, following physicist Georges-Louis Le Sage, imagined that the universe was filled with an ether of tiny particles that always move at speed . Each of these particles has a kinetic energy of up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of , since German polymath Gottfried Leibniz introduced kinetic energy without it, and the is largely conventional in prerelativistic physics. By assuming that every particle has a mass that is the sum of the masses of the ether particles, the authors concluded that all matter contains an amount of kinetic energy either given by or depending on the convention. A particle ether was usually considered unacceptably speculative science at the time, and since these authors did not formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames. In 1905, independently of Einstein, French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics. Electromagnetic mass There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists J. J. Thomson in 1881 and Oliver Heaviside in 1889, and George Frederick Charles Searle in 1897, German physicists Wilhelm Wien in 1900 and Max Abraham in 1902, and the Dutch physicist Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the electrostatic field. This concept was called electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass: , where Another way of deriving a type of electromagnetic mass was based on the concept of radiation pressure. In 1900, French polymath Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes. Austrian physicist Friedrich Hasenöhrl showed in 1904 that electromagnetic cavity radiation contributes the "apparent mass" to the cavity's mass. He argued that this implies mass dependence on temperature as well. Einstein: mass–energy equivalence Einstein did not write the exact formula in his 1905 Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy by emitting light, its mass diminishes by . This formulation relates only a change in mass to a change in energy without requiring the absolute relationship. The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity. He has stated that the laws of conservation of energy and conservation of mass are "one and the same". Einstein elaborated in a 1946 essay that "the principle of the conservation of mass… proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy conservation principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone." Mass–velocity relationship In developing special relativity, Einstein found that the kinetic energy of a moving body is with the velocity, the rest mass, and the Lorentz factor. He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle: Without this second term, there would be an additional contribution in the energy when the particle is not moving. Einstein's view on mass Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on , he treated as what would now be called the rest mass, and it has been noted that in his later years he did not like the idea of "relativistic mass". In older physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass. Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. One view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. Another view, attributed to Norwegian physicist Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection. Einstein's 1905 derivation Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles: . Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his Annus Mirabilis papers. Here, Einstein used to represent the speed of light in vacuum and to represent the energy lost by a body in the form of radiation. Consequently, the equation was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy in the form of radiation, its mass diminishes by ." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion. Einstein used a body emitting two light pulses in opposite directions, having energies of before and after the emission as seen in its rest frame. As seen from a moving frame, becomes and becomes . Einstein obtained, in modern notation: . He then argued that can only differ from the kinetic energy by an additive constant, which gives . Neglecting effects higher than third order in after a Taylor series expansion of the right side of this yields: Einstein concluded that the emission reduces the body's mass by , and that the mass of a body is a measure of its energy content. The correctness of Einstein's 1905 derivation of was criticized by German theoretical physicist Max Planck in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist Herbert Ives in 1952 and the Israeli physicist Max Jammer in 1961, asserting that Einstein's derivation is based on begging the question. Other scholars, such as American and Chilean philosophers John Stachel and Roberto Torretti, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct. American physics writer Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons. Relativistic center-of-mass theorem of 1906 Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work." In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the perpetual motion problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's , because mass conservation appears as a special case of the energy conservation law. Further developments There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be (where is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula , with being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased. Max Planck rewrote Einstein's mass–energy relationship as in June 1907, where is the pressure and the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body. Subsequently, in October 1907, this was rewritten as and given a quantum interpretation by German physicist Johannes Stark, who assumed its validity and correctness. In December 1907, Einstein expressed the equivalence in the form and concluded: "A mass is equivalent, as regards inertia, to a quantity of energy . […] It appears far more natural to consider every inertial mass as a store of energy." American physical chemists Gilbert N. Lewis and Richard C. Tolman used two variations of the formula in 1909: and , with being the relativistic energy (the energy of an object when the object is moving), is the rest energy (the energy when not moving), is the relativistic mass (the rest mass and the extra mass gained when moving), and is the rest mass. The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: and , with being the total energy (rest energy plus kinetic energy) of a moving material point, its rest energy, the relativistic mass, and the invariant mass. In 1911, German physicist Max von Laue gave a more comprehensive proof of from the stress–energy tensor, which was later generalized by German mathematician Felix Klein in 1918. Einstein returned to the topic once again after World War II and this time he wrote in the title of his article intended as an explanation for a general reader by analogy. Alternative version An alternative version of Einstein's thought experiment was proposed by American theoretical physicist Fritz Rohrlich in 1990, who based his reasoning on the Doppler effect. Like Einstein, he considered a body at rest with mass . If the body is examined in a frame moving with nonrelativistic velocity , it is no longer at rest and in the moving frame it has momentum . Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy . In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. However, if the same process is considered in a frame that moves with velocity to the left, the pulse moving to the left is redshifted, while the pulse moving to the right is blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor . The momentum of the light is its energy divided by , and it is increased by a factor of . So the right-moving light is carrying an extra momentum given by: The left-moving light carries a little less momentum, by the same amount . So the total right-momentum in both light pulses is twice . This is the right-momentum that the object lost. The momentum of the object in the moving frame after the emission is reduced to this amount: So the change in the object's mass is equal to the total energy lost divided by . Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass. Radioactivity and nuclear energy It was quickly noted after the discovery of radioactivity in 1897 that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist Ernest Rutherford and British radiochemist Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter." Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking moonshine." This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of lithium-7 plus protons giving rise to two alpha particles, allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945, the equation became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of Time magazine prominently featured a picture of Einstein next to an image of a mushroom cloud emblazoned with the equation. Einstein himself had only a minor role in the Manhattan Project: he had cosigned a letter to the U.S. president in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem. While is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200 MeV (which was directly possible, using a quantitative Geiger counter, at that time). The physicist and Manhattan Project participant Robert Serber noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his equation , plays some essential role in the theory of fission. Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists Lise Meitner and Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic fission. To do this, they used packing fraction, or nuclear binding energy values for elements. These, together with use of allowed them to realize on the spot that the basic fission process was energetically possible. Einstein's equation written According to the Einstein Papers Project at the California Institute of Technology and Hebrew University of Jerusalem, there remain only four known copies of this equation as written by Einstein. One of these is a letter written in German to Ludwik Silberstein, which was in Silberstein's archives, and sold at auction for $1.2 million, RR Auction of Boston, Massachusetts said on May 21, 2021. See also Notes References External links Einstein on the Inertia of Energy – MathPages Einstein-on film explaining a mass energy equivalence Mass and Energy – Conversations About Science with Theoretical Physicist Matt Strassler The Equivalence of Mass and Energy – Entry in the Stanford Encyclopedia of Philosophy 1905 introductions 1905 in science 1905 in Germany Albert Einstein Energy (physics) Equations Mass Special relativity

Force

A force is an influence that can cause an object to change its velocity unless counterbalanced by other forces. The concept of force makes the everyday notion of pushing or pulling mathematically precise. Because the magnitude and direction of a force are both important, force is a vector quantity. The SI unit of force is the newton (N), and force is often represented by the symbol . Force plays an important role in classical mechanics. The concept of force is central to all three of Newton's laws of motion. Types of forces often encountered in classical mechanics include elastic, frictional, contact or "normal" forces, and gravitational. The rotational version of force is torque, which produces changes in the rotational speed of an object. In an extended body, each part often applies forces on the adjacent parts; the distribution of such forces through the body is the internal mechanical stress. In equilibrium these stresses cause no acceleration of the body as the forces balance one another. If these are not in equilibrium they can cause deformation of solid materials, or flow in fluids. In modern physics, which includes relativity and quantum mechanics, the laws governing motion are revised to rely on fundamental interactions as the ultimate origin of force. However, the understanding of force provided by classical mechanics is useful for practical purposes. Development of the concept Philosophers in antiquity used the concept of force in the study of stationary and moving objects and simple machines, but thinkers such as Aristotle and Archimedes retained fundamental errors in understanding force. In part, this was due to an incomplete understanding of the sometimes non-obvious force of friction and a consequently inadequate view of the nature of natural motion. A fundamental error was the belief that a force is required to maintain motion, even at a constant velocity. Most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Newton formulated laws of motion that were not improved for over two hundred years. By the early 20th century, Einstein developed a theory of relativity that correctly predicted the action of forces on objects with increasing momenta near the speed of light and also provided insight into the forces produced by gravitation and inertia. With modern insights into quantum mechanics and technology that can accelerate particles close to the speed of light, particle physics has devised a Standard Model to describe forces between particles smaller than atoms. The Standard Model predicts that exchanged particles called gauge bosons are the fundamental means by which forces are emitted and absorbed. Only four main interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational. High-energy particle physics observations made during the 1970s and 1980s confirmed that the weak and electromagnetic forces are expressions of a more fundamental electroweak interaction. Pre-Newtonian concepts Since antiquity the concept of force has been recognized as integral to the functioning of each of the simple machines. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance for the same amount of work. Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotle's view, the terrestrial sphere contained four elements that come to rest at different "natural places" therein. Aristotle believed that motionless objects on Earth, those composed mostly of the elements earth and water, were in their natural place when on the ground, and that they stay that way if left alone. He distinguished between the innate tendency of objects to find their "natural place" (e.g., for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. An archer causes the arrow to move at the start of the flight, and it then sails through the air even though no discernible efficient cause acts upon it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path carries the projectile to its target. This explanation requires a continuous medium such as air to sustain the motion. Though Aristotelian physics was criticized as early as the 6th century, its shortcomings would not be corrected until the 17th century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion. He showed that the bodies were accelerated by gravity to an extent that was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Galileo's idea that force is needed to change motion rather than to sustain it, further improved upon by Isaac Beeckman, René Descartes, and Pierre Gassendi, became a key principle of Newtonian physics. In the early 17th century, before Newton's Principia, the term "force" was applied to many physical and non-physical phenomena, e.g., for an acceleration of a point. The product of a point mass and the square of its velocity was named (live force) by Leibniz. The modern concept of force corresponds to Newton's (accelerating force). Newtonian mechanics Sir Isaac Newton described the motion of all objects using the concepts of inertia and force. In 1687, Newton published his magnum opus, Philosophiæ Naturalis Principia Mathematica. In this work Newton set out three laws of motion that have dominated the way forces are described in physics to this day. The precise ways in which Newton's laws are expressed have evolved in step with new mathematical approaches. First law Newton's first law of motion states that the natural behavior of an object at rest is to continue being at rest, and the natural behavior of an object moving at constant speed in a straight line is to continue moving at that constant speed along that straight line. The latter follows from the former because of the principle that the laws of physics are the same for all inertial observers, i.e., all observers who do not feel themselves to be in motion. An observer moving in tandem with an object will see it as being at rest. So, its natural behavior will be to remain at rest with respect to that observer, which means that an observer who sees it moving at constant speed in a straight line will see it continuing to do so. Second law According to the first law, motion at constant speed in a straight line does not need a cause. It is change in motion that requires a cause, and Newton's second law gives the quantitative relationship between force and change of motion. Newton's second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. If the mass of the object is constant, this law implies that the acceleration of an object is directly proportional to the net force acting on the object, is in the direction of the net force, and is inversely proportional to the mass of the object. A modern statement of Newton's second law is a vector equation: where is the momentum of the system, and is the net (vector sum) force. If a body is in equilibrium, there is zero net force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an unbalanced force acting on an object it will result in the object's momentum changing over time. In common engineering applications the mass in a system remains constant allowing as simple algebraic form for the second law. By the definition of momentum, where m is the mass and is the velocity. If Newton's second law is applied to a system of constant mass, m may be moved outside the derivative operator. The equation then becomes By substituting the definition of acceleration, the algebraic version of Newton's second law is derived: Third law Whenever one body exerts a force on another, the latter simultaneously exerts an equal and opposite force on the first. In vector form, if is the force of body 1 on body 2 and that of body 2 on body 1, then This law is sometimes referred to as the action-reaction law, with called the action and the reaction. Newton's Third Law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. The third law means that all forces are interactions between different bodies. and thus that there is no such thing as a unidirectional force or a force that acts on only one body. In a system composed of object 1 and object 2, the net force on the system due to their mutual interactions is zero: More generally, in a closed system of particles, all internal forces are balanced. The particles may accelerate with respect to each other but the center of mass of the system will not accelerate. If an external force acts on the system, it will make the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system. Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved in any closed system. In a system of two particles, if is the momentum of object 1 and the momentum of object 2, then Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained. Defining "force" Some textbooks use Newton's second law as a definition of force. However, for the equation for a constant mass to then have any predictive content, it must be combined with further information. Moreover, inferring that a force is present because a body is accelerating is only valid in an inertial frame of reference. The question of which aspects of Newton's laws to take as definitions and which to regard as holding physical content has been answered in various ways, which ultimately do not affect how the theory is used in practice. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll. Combining forces Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules than physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. If both of these pieces of information are not known for each force, the situation is ambiguous. Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. When two forces act on a point particle, the resulting force, the resultant (also called the net force), can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector that is equal in magnitude and direction to the transversal of the parallelogram. The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the net force. As well as being added, forces can also be resolved into independent components at right angles to each other. A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. Orthogonal components are independent of each other because forces acting at ninety degrees to each other have no effect on the magnitude or direction of the other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Orthogonal force vectors can be three-dimensional with the third component being at right angles to the other two. Equilibrium When all the forces that act upon an object are balanced, then the object is said to be in a state of equilibrium. Hence, equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector sum of all forces is zero). When dealing with an extended body, it is also necessary that the net torque be zero. A body is in static equilibrium with respect to a frame of reference if it at rest and not accelerating, whereas a body in dynamic equilibrium is moving at a constant speed in a straight line, i.e., moving but not accelerating. What one observer sees as static equilibrium, another can see as dynamic equilibrium and vice versa. Static Static equilibrium was understood well before the invention of classical mechanics. Objects that are not accelerating have zero net force acting on them. The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, an object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, a force is applied by the surface that resists the downward force with equal upward force (called a normal force). The situation produces zero net force and hence no acceleration. Pushing against an object that rests on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object. A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force", which equals the object's weight. Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion. Dynamic Dynamic equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo concluded that motion in a constant velocity was completely equivalent to rest. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. When this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity. Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamic equilibrium: when all the forces on an object balance but it still moves at a constant velocity. A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in zero net force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. When kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion. Examples of forces in classical mechanics Some forces are consequences of the fundamental ones. In such situations, idealized models can be used to gain physical insight. For example, each solid object is considered a rigid body. Gravitational force or Gravity What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Today, this acceleration due to gravity towards the surface of the Earth is usually designated as and has a magnitude of about 9.81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of will experience a force: For an object in free-fall, this force is unopposed and the net force on the object is its weight. For objects not in free-fall, the force of gravity is opposed by the reaction forces applied by their supports. For example, a person standing on the ground experiences zero net force, since a normal force (a reaction force) is exerted by the ground upward on the person that counterbalances his weight that is directed downward. Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's laws of planetary motion. Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the Moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. Further, Newton realized that the acceleration of a body due to gravity is proportional to the mass of the other attracting body. Combining these ideas gives a formula that relates the mass and the radius of the Earth to the gravitational acceleration: where the vector direction is given by , is the unit vector directed outward from the center of the Earth. In this equation, a dimensional constant is used to describe the relative strength of gravity. This constant has come to be known as the Newtonian constant of gravitation, though its value was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the first measurement of using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing could allow one to solve for the Earth's mass given the above equation. Newton realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. Succinctly stated, Newton's law of gravitation states that the force on a spherical object of mass due to the gravitational pull of mass is where is the distance between the two objects' centers of mass and is the unit vector pointed in the direction away from the center of the first object toward the center of the second object. This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the 20th century. During that time, sophisticated methods of perturbation analysis were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed. Electromagnetic The electrostatic force was first described in 1784 by Coulomb as a force that existed intrinsically between two charges. The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the superposition principle. Coulomb's law unifies all these observations into one succinct statement. Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. Thus the electric field anywhere in space is defined as where is the magnitude of the hypothetical test charge. Similarly, the idea of the magnetic field was introduced to express how magnets can influence one another at a distance. The Lorentz force law gives the force upon a body with charge due to electric and magnetic fields: where is the electromagnetic force, is the electric field at the body's location, is the magnetic field, and is the velocity of the particle. The magnetic contribution to the Lorentz force is the cross product of the velocity vector with the magnetic field. The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs. These "Maxwell's equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed that he calculated to be the speed of light. This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum. Normal When objects are in contact, the force directly between them is called the normal force, the component of the total force in the system exerted normal to the interface between the objects. The normal force is closely related to Newton's third law. The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force on an object crashing into an immobile surface. Friction Friction is a force that opposes relative motion of two bodies. At the macroscopic scale, the frictional force is directly related to the normal force at the point of contact. There are two broad classifications of frictional forces: static friction and kinetic friction. The static friction force will exactly oppose forces applied to an object parallel to a surface up to the limit specified by the coefficient of static friction multiplied by the normal force. In other words, the magnitude of the static friction force satisfies the inequality: The kinetic friction force is typically independent of both the forces applied and the movement of the object. Thus, the magnitude of the force equals: where is the coefficient of kinetic friction. The coefficient of kinetic friction is normally less than the coefficient of static friction. Tension Tension forces can be modeled using ideal strings that are massless, frictionless, unbreakable, and do not stretch. They can be combined with ideal pulleys, which allow ideal strings to switch physical direction. Ideal strings transmit tension forces instantaneously in action–reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object. By connecting the same string multiple times to the same object through the use of a configuration that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. Such machines allow a mechanical advantage for a corresponding increase in the length of displaced string needed to move the load. These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine. Spring A simple elastic force acts to return a spring to its natural length. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. If is the displacement, the force exerted by an ideal spring equals: where is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the force to act in opposition to the applied load. Centripetal For an object in uniform circular motion, the net force acting on the object equals: where is the mass of the object, is the velocity of the object and is the distance to the center of the circular path and is the unit vector pointing in the radial direction outwards from the center. This means that the net force felt by the object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. More generally, the net force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force, which accelerates the object by either slowing it down or speeding it up, and the radial (centripetal) force, which changes its direction. Continuum mechanics Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. In real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows: where is the volume of the object in the fluid and is the scalar function that describes the pressure at all locations in space. Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight. A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction: where: is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and is the velocity of the object. More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as where is the relevant cross-sectional area for the volume for which the stress tensor is being calculated. This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor accounts for forces that cause all strains (deformations) including also tensile stresses and compressions. Fictitious There are forces that are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force and the Coriolis force. These forces are considered fictitious because they do not exist in frames of reference that are not accelerating. Because these forces are not genuine they are also referred to as "pseudo forces". In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry. Concepts derived from force Rotation and torque Forces that cause extended objects to rotate are associated with torques. Mathematically, the torque of a force is defined relative to an arbitrary reference point as the cross product: where is the position vector of the force application point relative to the reference point. Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. As a consequence of Newton's first law of motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. Likewise, Newton's second law of motion can be used to derive an analogous equation for the instantaneous angular acceleration of the rigid body: where is the moment of inertia of the body is the angular acceleration of the body. This provides a definition for the moment of inertia, which is the rotational equivalent for mass. In more advanced treatments of mechanics, where the rotation over a time interval is described, the moment of inertia must be substituted by the tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation. Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque: where is the angular momentum of the particle. Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques, and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques. Yank The yank is defined as the rate of change of force The term is used in biomechanical analysis, athletic assessment and robotic control. The second ("tug"), third ("snatch"), fourth ("shake"), and higher derivatives are rarely used. Kinematic integrals Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. For example, integrating with respect to time gives the definition of impulse: which by Newton's Second Law must be equivalent to the change in momentum (yielding the Impulse momentum theorem). Similarly, integrating with respect to position gives a definition for the work done by a force: which is equivalent to changes in kinetic energy (yielding the work energy theorem). Power P is the rate of change dW/dt of the work W, as the trajectory is extended by a position change in a time interval dt: so with the velocity. Potential energy Instead of a force, often the mathematically related concept of a potential energy field is used. For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. Restating mathematically the definition of energy (via the definition of work), a potential scalar field is defined as that field whose gradient is equal and opposite to the force produced at every point: Forces can be classified as conservative or nonconservative. Conservative forces are equivalent to the gradient of a potential while nonconservative forces are not. Conservation A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. The force, therefore, is related directly to the difference in potential energy between two different locations in space, and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area. Conservative forces include gravity, the electromagnetic force, and the spring force. Each of these forces has models that are dependent on a position often given as a radial vector emanating from spherically symmetric potentials. Examples of this follow: For gravity: where is the gravitational constant, and is the mass of object n. For electrostatic forces: where is electric permittivity of free space, and is the electric charge of object n. For spring forces: where is the spring constant. For certain physical scenarios, it is impossible to model forces as being due to a simple gradient of potentials. This is often due a macroscopic statistical average of microstates. For example, static friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model that is independent of any macroscale position vector. Nonconservative forces other than friction include other contact forces, tension, compression, and drag. For any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials. The connection between macroscopic nonconservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. According to the Second law of thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases. Units The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg·m·s−2.The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g·cm·s−2. A newton is thus equal to 100,000 dynes. The gravitational foot-pound-second English unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9.80665 m·s−2. The pound-force provides an alternative unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force. An alternative unit of force in a different foot–pound–second system, the absolute fps system, is the poundal, defined as the force required to accelerate a one-pound mass at a rate of one foot per second squared. The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass that accelerates at 1 m·s−2 when subjected to a force of 1 kgf. The kilogram-force is not a part of the modern SI system, and is generally deprecated, sometimes used for expressing aircraft weight, jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. See also Ton-force. Revisions of the force concept At the beginning of the 20th century, new physical ideas emerged to explain experimental results in astronomical and submicroscopic realms. As discussed below, relativity alters the definition of momentum and quantum mechanics reuses the concept of "force" in microscopic contexts where Newton's laws do not apply directly. Special theory of relativity In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). When an object's velocity increases, so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's Second Law, remains valid because it is a mathematical definition. But for momentum to be conserved at relativistic relative velocity, , momentum must be redefined as: where is the rest mass and the speed of light. The expression relating force and acceleration for a particle with constant non-zero rest mass moving in the direction at velocity is: where is called the Lorentz factor. The Lorentz factor increases steeply as the relative velocity approaches the speed of light. Consequently, the greater and greater force must be applied to produce the same acceleration at extreme velocity. The relative velocity cannot reach . If is very small compared to , then is very close to 1 and is a close approximation. Even for use in relativity, one can restore the form of through the use of four-vectors. This relation is correct in relativity when is the four-force, is the invariant mass, and is the four-acceleration. The general theory of relativity incorporates a more radical departure from the Newtonian way of thinking about force, specifically gravitational force. This reimagining of the nature of gravity is described more fully below. Quantum mechanics Quantum mechanics is a theory of physics originally developed in order to understand microscopic phenomena: behavior at the scale of molecules, atoms or subatomic particles. Generally and loosely speaking, the smaller a system is, the more an adequate mathematical model will require understanding quantum effects. The conceptual underpinning of quantum physics is different from that of classical physics. Instead of thinking about quantities like position, momentum, and energy as properties that an object has, one considers what result might appear when a measurement of a chosen type is performed. Quantum mechanics allows the physicist to calculate the probability that a chosen measurement will elicit a particular result. The expectation value for a measurement is the average of the possible results it might yield, weighted by their probabilities of occurrence. In quantum mechanics, interactions are typically described in terms of energy rather than force. The Ehrenfest theorem provides a connection between quantum expectation values and the classical concept of force, a connection that is necessarily inexact, as quantum physics is fundamentally different from classical. In quantum physics, the Born rule is used to calculate the expectation values of a position measurement or a momentum measurement. These expectation values will generally change over time; that is, depending on the time at which (for example) a position measurement is performed, the probabilities for its different possible outcomes will vary. The Ehrenfest theorem says, roughly speaking, that the equations describing how these expectation values change over time have a form reminiscent of Newton's second law, with a force defined as the negative derivative of the potential energy. However, the more pronounced quantum effects are in a given situation, the more difficult it is to derive meaningful conclusions from this resemblance. Quantum mechanics also introduces two new constraints that interact with forces at the submicroscopic scale and which are especially important for atoms. Despite the strong attraction of the nucleus, the uncertainty principle limits the minimum extent of an electron probability distribution and the Pauli exclusion principle prevents electrons from sharing the same probability distribution. This gives rise to an emergent pressure known as degeneracy pressure. The dynamic equilibrium between the degeneracy pressure and the attractive electromagnetic force give atoms, molecules, liquids, and solids stability. Quantum field theory In modern particle physics, forces and the acceleration of particles are explained as a mathematical by-product of exchange of momentum-carrying gauge bosons. With the development of quantum field theory and general relativity, it was realized that force is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). The conservation of momentum can be directly derived from the homogeneity or symmetry of space and so is usually considered more fundamental than the concept of a force. Thus the currently known fundamental forces are considered more accurately to be "fundamental interactions". While sophisticated mathematical descriptions are needed to predict, in full detail, the result of such interactions, there is a conceptually simple way to describe them through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time, which normally increases up or to the right in the diagram. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and antineutrino, an interaction mediated by the same gauge boson that is responsible for the weak nuclear force. Fundamental interactions All of the known forces of the universe are classified into four fundamental interactions. The strong and the weak forces act only at very short distances, and are responsible for the interactions between subatomic particles, including nucleons and compound nuclei. The electromagnetic force acts between electric charges, and the gravitational force acts between masses. All other forces in nature derive from these four fundamental interactions operating within quantum mechanics, including the constraints introduced by the Schrödinger equation and the Pauli exclusion principle. For example, friction is a manifestation of the electromagnetic force acting between atoms of two surfaces. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces. Centrifugal forces are acceleration forces that arise simply from the acceleration of rotating frames of reference. The fundamental theories for forces developed from the unification of different ideas. For example, Newton's universal theory of gravitation showed that the force responsible for objects falling near the surface of the Earth is also the force responsible for the falling of celestial bodies about the Earth (the Moon) and around the Sun (the planets). Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through a theory of electromagnetism. In the 20th century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons. This Standard Model of particle physics assumes a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory, which was subsequently confirmed by observation. Gravitational Newton's law of gravitation is an example of action at a distance: one body, like the Sun, exerts an influence upon any other body, like the Earth, no matter how far apart they are. Moreover, this action at a distance is instantaneous. According to Newton's theory, the one body shifting position changes the gravitational pulls felt by all other bodies, all at the same instant of time. Albert Einstein recognized that this was inconsistent with special relativity and its prediction that influences cannot travel faster than the speed of light. So, he sought a new theory of gravitation that would be relativistically consistent. Mercury's orbit did not match that predicted by Newton's law of gravitation. Some astrophysicists predicted the existence of an undiscovered planet (Vulcan) that could explain the discrepancies. When Einstein formulated his theory of general relativity (GR) he focused on Mercury's problematic orbit and found that his theory added a correction, which could account for the discrepancy. This was the first time that Newton's theory of gravity had been shown to be inexact. Since then, general relativity has been acknowledged as the theory that best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved spacetime – defined as the shortest spacetime path between two spacetime events. From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of spacetime can be observed and the force is inferred from the object's curved path. Thus, the straight line path in spacetime is seen as a curved line in space, and it is called the ballistic trajectory of the object. For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Its spacetime trajectory is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). The time derivative of the changing momentum of the object is what we label as "gravitational force". Electromagnetic Maxwell's equations and the set of techniques built around them adequately describe a wide range of physics involving force in electricity and magnetism. This classical theory already includes relativity effects. Understanding quantized electromagnetic interactions between elementary particles requires quantum electrodynamics (or QED). In QED, photons are fundamental exchange particles, describing all interactions relating to electromagnetism including the electromagnetic force. Strong nuclear There are two "nuclear forces", which today are usually described as interactions that take place in quantum theories of particle physics. The strong nuclear force is the force responsible for the structural integrity of atomic nuclei, and gains its name from its ability to overpower the electromagnetic repulsion between protons. The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The strong force only acts directly upon elementary particles. A residual is observed between hadrons (notably, the nucleons in atomic nuclei), known as the nuclear force. Here the strong force acts indirectly, transmitted as gluons that form part of the virtual pi and rho mesons, the classical transmitters of the nuclear force. The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called color confinement. Weak nuclear Unique among the fundamental interactions, the weak nuclear force creates no bound states. The weak force is due to the exchange of the heavy W and Z bosons. Since the weak force is mediated by two types of bosons, it can be divided into two types of interaction or "vertices" — charged current, involving the electrically charged W+ and W− bosons, and neutral current, involving electrically neutral Z0 bosons. The most familiar effect of weak interaction is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. This is a type of charged-current interaction. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed, which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately . Such temperatures occurred in the plasma collisions in the early moments of the Big Bang. See also References External links Natural philosophy Classical mechanics Vector physical quantities Temporal rates

Electromagnetism

In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interactions of atoms and molecules. Electromagnetism can be thought of as a combination of electrostatics and magnetism, which are distinct but closely intertwined phenomena. Electromagnetic forces occur between any two charged particles. Electric forces cause an attraction between particles with opposite charges and repulsion between particles with the same charge, while magnetism is an interaction that occurs between charged particles in relative motion. These two forces are described in terms of electromagnetic fields. Macroscopic charged objects are described in terms of Coulomb's law for electricity and Ampère's force law for magnetism; the Lorentz force describes microscopic charged particles. The electromagnetic force is responsible for many of the chemical and physical phenomena observed in daily life. The electrostatic attraction between atomic nuclei and their electrons holds atoms together. Electric forces also allow different atoms to combine into molecules, including the macromolecules such as proteins that form the basis of life. Meanwhile, magnetic interactions between the spin and angular momentum magnetic moments of electrons also play a role in chemical reactivity; such relationships are studied in spin chemistry. Electromagnetism also plays several crucial roles in modern technology: electrical energy production, transformation and distribution; light, heat, and sound production and detection; fiber optic and wireless communication; sensors; computation; electrolysis; electroplating; and mechanical motors and actuators. Electromagnetism has been studied since ancient times. Many ancient civilizations, including the Greeks and the Mayans, created wide-ranging theories to explain lightning, static electricity, and the attraction between magnetized pieces of iron ore. However, it was not until the late 18th century that scientists began to develop a mathematical basis for understanding the nature of electromagnetic interactions. In the 18th and 19th centuries, prominent scientists and mathematicians such as Coulomb, Gauss and Faraday developed namesake laws which helped to explain the formation and interaction of electromagnetic fields. This process culminated in the 1860s with the discovery of Maxwell's equations, a set of four partial differential equations which provide a complete description of classical electromagnetic fields. Maxwell's equations provided a sound mathematical basis for the relationships between electricity and magnetism that scientists had been exploring for centuries, and predicted the existence of self-sustaining electromagnetic waves. Maxwell postulated that such waves make up visible light, which was later shown to be true. Gamma-rays, x-rays, ultraviolet, visible, infrared radiation, microwaves and radio waves were all determined to be electromagnetic radiation differing only in their range of frequencies. In the modern era, scientists continue to refine the theory of electromagnetism to account for the effects of modern physics, including quantum mechanics and relativity. The theoretical implications of electromagnetism, particularly the requirement that observations remain consistent when viewed from various moving frames of reference (relativistic electromagnetism) and the establishment of the speed of light based on properties of the medium of propagation (permeability and permittivity), helped inspire Einstein's theory of special relativity in 1905. Quantum electrodynamics (QED) modifies Maxwell's equations to be consistent with the quantized nature of matter. In QED, changes in the electromagnetic field are expressed in terms of discrete excitations, particles known as photons, the quanta of light. History Ancient world Investigation into electromagnetic phenomena began about 5,000 years ago. There is evidence that the ancient Chinese, Mayan, and potentially even Egyptian civilizations knew that the naturally magnetic mineral magnetite had attractive properties, and many incorporated it into their art and architecture. Ancient people were also aware of lightning and static electricity, although they had no idea of the mechanisms behind these phenomena. The Greek philosopher Thales of Miletus discovered around 600 B.C.E. that amber could acquire an electric charge when it was rubbed with cloth, which allowed it to pick up light objects such as pieces of straw. Thales also experimented with the ability of magnetic rocks to attract one other, and hypothesized that this phenomenon might be connected to the attractive power of amber, foreshadowing the deep connections between electricity and magnetism that would be discovered over 2,000 years later. Despite all this investigation, ancient civilizations had no understanding of the mathematical basis of electromagnetism, and often analyzed its impacts through the lens of religion rather than science (lightning, for instance, was considered to be a creation of the gods in many cultures). 19th century Electricity and magnetism were originally considered to be two separate forces. This view changed with the publication of James Clerk Maxwell's 1873 A Treatise on Electricity and Magnetism in which the interactions of positive and negative charges were shown to be mediated by one force. There are four main effects resulting from these interactions, all of which have been clearly demonstrated by experiments: Electric charges or one another with a force inversely proportional to the square of the distance between them: opposite charges attract, like charges repel. Magnetic poles (or states of polarization at individual points) attract or repel one another in a manner similar to positive and negative charges and always exist as pairs: every north pole is yoked to a south pole. An electric current inside a wire creates a corresponding circumferential magnetic field outside the wire. Its direction (clockwise or counter-clockwise) depends on the direction of the current in the wire. A current is induced in a loop of wire when it is moved toward or away from a magnetic field, or a magnet is moved towards or away from it; the direction of current depends on that of the movement. In April 1820, Hans Christian Ørsted observed that an electrical current in a wire caused a nearby compass needle to move. At the time of discovery, Ørsted did not suggest any satisfactory explanation of the phenomenon, nor did he try to represent the phenomenon in a mathematical framework. However, three months later he began more intensive investigations. Soon thereafter he published his findings, proving that an electric current produces a magnetic field as it flows through a wire. The CGS unit of magnetic induction (oersted) is named in honor of his contributions to the field of electromagnetism. His findings resulted in intensive research throughout the scientific community in electrodynamics. They influenced French physicist André-Marie Ampère's developments of a single mathematical form to represent the magnetic forces between current-carrying conductors. Ørsted's discovery also represented a major step toward a unified concept of energy. This unification, which was observed by Michael Faraday, extended by James Clerk Maxwell, and partially reformulated by Oliver Heaviside and Heinrich Hertz, is one of the key accomplishments of 19th-century mathematical physics. It has had far-reaching consequences, one of which was the understanding of the nature of light. Unlike what was proposed by the electromagnetic theory of that time, light and other electromagnetic waves are at present seen as taking the form of quantized, self-propagating oscillatory electromagnetic field disturbances called photons. Different frequencies of oscillation give rise to the different forms of electromagnetic radiation, from radio waves at the lowest frequencies, to visible light at intermediate frequencies, to gamma rays at the highest frequencies. Ørsted was not the only person to examine the relationship between electricity and magnetism. In 1802, Gian Domenico Romagnosi, an Italian legal scholar, deflected a magnetic needle using a Voltaic pile. The factual setup of the experiment is not completely clear, nor if current flowed across the needle or not. An account of the discovery was published in 1802 in an Italian newspaper, but it was largely overlooked by the contemporary scientific community, because Romagnosi seemingly did not belong to this community. An earlier (1735), and often neglected, connection between electricity and magnetism was reported by a Dr. Cookson. The account stated:A tradesman at Wakefield in Yorkshire, having put up a great number of knives and forks in a large box ... and having placed the box in the corner of a large room, there happened a sudden storm of thunder, lightning, &c. ... The owner emptying the box on a counter where some nails lay, the persons who took up the knives, that lay on the nails, observed that the knives took up the nails. On this the whole number was tried, and found to do the same, and that, to such a degree as to take up large nails, packing needles, and other iron things of considerable weight ... E. T. Whittaker suggested in 1910 that this particular event was responsible for lightning to be "credited with the power of magnetizing steel; and it was doubtless this which led Franklin in 1751 to attempt to magnetize a sewing-needle by means of the discharge of Leyden jars." A fundamental force The electromagnetic force is the second strongest of the four known fundamental forces and has unlimited range. All other forces, known as non-fundamental forces. (e.g., friction, contact forces) are derived from the four fundamental forces. At high energy, the weak force and electromagnetic force are unified as a single interaction called the electroweak interaction. Most of the forces involved in interactions between atoms are explained by electromagnetic forces between electrically charged atomic nuclei and electrons. The electromagnetic force is also involved in all forms of chemical phenomena. Electromagnetism explains how materials carry momentum despite being composed of individual particles and empty space. The forces we experience when "pushing" or "pulling" ordinary material objects result from intermolecular forces between individual molecules in our bodies and in the objects. The effective forces generated by the momentum of electrons' movement is a necessary part of understanding atomic and intermolecular interactions. As electrons move between interacting atoms, they carry momentum with them. As a collection of electrons becomes more confined, their minimum momentum necessarily increases due to the Pauli exclusion principle. The behavior of matter at the molecular scale, including its density, is determined by the balance between the electromagnetic force and the force generated by the exchange of momentum carried by the electrons themselves. Classical electrodynamics In 1600, William Gilbert proposed, in his De Magnete, that electricity and magnetism, while both capable of causing attraction and repulsion of objects, were distinct effects. Mariners had noticed that lightning strikes had the ability to disturb a compass needle. The link between lightning and electricity was not confirmed until Benjamin Franklin's proposed experiments in 1752 were conducted on 10May 1752 by Thomas-François Dalibard of France using a iron rod instead of a kite and he successfully extracted electrical sparks from a cloud. One of the first to discover and publish a link between human-made electric current and magnetism was Gian Romagnosi, who in 1802 noticed that connecting a wire across a voltaic pile deflected a nearby compass needle. However, the effect did not become widely known until 1820, when Ørsted performed a similar experiment. Ørsted's work influenced Ampère to conduct further experiments, which eventually gave rise to a new area of physics: electrodynamics. By determining a force law for the interaction between elements of electric current, Ampère placed the subject on a solid mathematical foundation. A theory of electromagnetism, known as classical electromagnetism, was developed by several physicists during the period between 1820 and 1873, when James Clerk Maxwell's treatise was published, which unified previous developments into a single theory, proposing that light was an electromagnetic wave propagating in the luminiferous ether. In classical electromagnetism, the behavior of the electromagnetic field is described by a set of equations known as Maxwell's equations, and the electromagnetic force is given by the Lorentz force law. One of the peculiarities of classical electromagnetism is that it is difficult to reconcile with classical mechanics, but it is compatible with special relativity. According to Maxwell's equations, the speed of light in vacuum is a universal constant that is dependent only on the electrical permittivity and magnetic permeability of free space. This violates Galilean invariance, a long-standing cornerstone of classical mechanics. One way to reconcile the two theories (electromagnetism and classical mechanics) is to assume the existence of a luminiferous aether through which the light propagates. However, subsequent experimental efforts failed to detect the presence of the aether. After important contributions of Hendrik Lorentz and Henri Poincaré, in 1905, Albert Einstein solved the problem with the introduction of special relativity, which replaced classical kinematics with a new theory of kinematics compatible with classical electromagnetism. (For more information, see History of special relativity.) In addition, relativity theory implies that in moving frames of reference, a magnetic field transforms to a field with a nonzero electric component and conversely, a moving electric field transforms to a nonzero magnetic component, thus firmly showing that the phenomena are two sides of the same coin. Hence the term "electromagnetism". (For more information, see Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism.) Today few problems in electromagnetism remain unsolved. These include: the lack of magnetic monopoles, Abraham–Minkowski controversy, the location in space of the electromagnetic field energy, and the mechanism by which some organisms can sense electric and magnetic fields. Extension to nonlinear phenomena The Maxwell equations are linear, in that a change in the sources (the charges and currents) results in a proportional change of the fields. Nonlinear dynamics can occur when electromagnetic fields couple to matter that follows nonlinear dynamical laws. This is studied, for example, in the subject of magnetohydrodynamics, which combines Maxwell theory with the Navier–Stokes equations. Another branch of electromagnetism dealing with nonlinearity is nonlinear optics. Quantities and units Here is a list of common units related to electromagnetism: ampere (electric current, SI unit) coulomb (electric charge) farad (capacitance) henry (inductance) ohm (resistance) siemens (conductance) tesla (magnetic flux density) volt (electric potential) watt (power) weber (magnetic flux) In the electromagnetic CGS system, electric current is a fundamental quantity defined via Ampère's law and takes the permeability as a dimensionless quantity (relative permeability) whose value in vacuum is unity. As a consequence, the square of the speed of light appears explicitly in some of the equations interrelating quantities in this system. Formulas for physical laws of electromagnetism (such as Maxwell's equations) need to be adjusted depending on what system of units one uses. This is because there is no one-to-one correspondence between electromagnetic units in SI and those in CGS, as is the case for mechanical units. Furthermore, within CGS, there are several plausible choices of electromagnetic units, leading to different unit "sub-systems", including Gaussian, "ESU", "EMU", and Heaviside–Lorentz. Among these choices, Gaussian units are the most common today, and in fact the phrase "CGS units" is often used to refer specifically to CGS-Gaussian units. Applications The study of electromagnetism informs electric circuits, magnetic circuits, and semiconductor devices' construction. See also Abraham–Lorentz force Aeromagnetic surveys Computational electromagnetics Double-slit experiment Electrodynamic droplet deformation Electromagnet Electromagnetic induction Electromagnetic wave equation Electromagnetic scattering Electromechanics Geophysics Introduction to electromagnetism Magnetostatics Magnetoquasistatic field Optics Relativistic electromagnetism Wheeler–Feynman absorber theory References Further reading Web sources Textbooks General coverage External links Magnetic Field Strength Converter Electromagnetic Force – from Eric Weisstein's World of Physics Fundamental interactions

Rotational energy

Rotational energy or angular kinetic energy is kinetic energy due to the rotation of an object and is part of its total kinetic energy. Looking at rotational energy separately around an object's axis of rotation, the following dependence on the object's moment of inertia is observed: where The mechanical work required for or applied during rotation is the torque times the rotation angle. The instantaneous power of an angularly accelerating body is the torque times the angular velocity. For free-floating (unattached) objects, the axis of rotation is commonly around its center of mass. Note the close relationship between the result for rotational energy and the energy held by linear (or translational) motion: In the rotating system, the moment of inertia, I, takes the role of the mass, m, and the angular velocity, , takes the role of the linear velocity, v. The rotational energy of a rolling cylinder varies from one half of the translational energy (if it is massive) to the same as the translational energy (if it is hollow). An example is the calculation of the rotational kinetic energy of the Earth. As the Earth has a sidereal rotation period of 23.93 hours, it has an angular velocity of . The Earth has a moment of inertia, I = . Therefore, it has a rotational kinetic energy of . Part of the Earth's rotational energy can also be tapped using tidal power. Additional friction of the two global tidal waves creates energy in a physical manner, infinitesimally slowing down Earth's angular velocity ω. Due to the conservation of angular momentum, this process transfers angular momentum to the Moon's orbital motion, increasing its distance from Earth and its orbital period (see tidal locking for a more detailed explanation of this process). See also Flywheel List of energy storage projects Rigid rotor Rotational spectroscopy Notes References Resnick, R. and Halliday, D. (1966) PHYSICS, Section 12-5, John Wiley & Sons Inc. Forms of energy Rotation

Energy–momentum relation

"In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativisti(...TRUNCATED)