Newton's laws of motion
Newton's laws of motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, its motion in response to those forces. More the first law defines the force qualitatively, the second law offers a quantitative measure of the force, the third asserts that a single isolated force doesn't exist; these three laws have been expressed in several ways, over nearly three centuries, can be summarised as follows: The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, first published in 1687. Newton used them to investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. A fourth law is also described in the bibliography, which states that forces add up like vectors, that is, that forces obey the principle of superposition.
Newton's laws are applied to objects which are idealised as single point masses, in the sense that the size and shape of the object's body are neglected to focus on its motion more easily. This can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way a planet can be idealised as a particle for analysis of its orbital motion around a star. In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion Euler's laws can be derived from Newton's laws. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.
Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining. Other authors do treat the first law as a corollary of the second; the explicit concept of an inertial frame of reference was not developed until long after Newton's death. In the given interpretation mass, acceleration and force are assumed to be externally defined quantities; this is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is still useful as an approximation when the speeds involved are much slower than the speed of light; the first law states that if the net force is zero the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F = 0 ⇔ d v d t = 0.
An object, at rest will stay at rest unless a force acts upon it. An object, in motion will not change its velocity unless a force acts upon it; this is known as uniform motion. An object continues to do. If it is at rest, it continues in a state of rest. If an object is moving, it continues to move without changing its speed; this is evident in space probes. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely. Newton placed the first law of motion to establish frames of reference for which the other laws are applicable; the first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed. Newton's first law is referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero.
In this sense, the first law can be restated as: In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it. Newton's first and second laws are valid only in an inertial reference frame. Any reference frame, in uniform motion with respect to an inertial frame is an in
Physics Today is the membership magazine of the American Institute of Physics, established in 1948. It is provided to the members including the American Physical Society. Although its content is scientifically rigorous and up to date, it is not a true scholarly journal in the sense of being a primary vehicle for communicating new results. Rather, it is more of a hybrid magazine that informs readers about important developments in the form of overview articles written by experts, shorter review articles written internally by staff, discusses the latest issues and events of importance to the science community such as science politics; the physics community's main vessel for new results are the Physical Review suite of scientific journals published by the American Physical Society and Applied Physics Letters published by the American Institute of Physics. The magazine provides a historical resource of events associated with physics, including debunking the physics behind the Star Wars program of the 1980s, the state of physics in China and the Soviet Union during the 1950s and 1970s.
According to the Journal Citation Reports, the journal has a 2017 impact factor of 4.370. Official website
Hooke's law is a law of physics that states that the force needed to extend or compress a spring by some distance x scales linearly with respect to that distance. That is: F s = k x, where k is a constant factor characteristic of the spring: its stiffness, x is small compared to the total possible deformation of the spring; the law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram, he published the solution of his anagram in 1678 as: sic vis. Hooke states in the 1678 work that he was aware of the law in 1660. Hooke's equation holds in many other situations where an elastic body is deformed, such as wind blowing on a tall building, a musician plucking a string of a guitar, the filling of a party balloon. An elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces.
It must fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached. On the other hand, Hooke's law is an accurate approximation for most solid bodies, as long as the forces and deformations are small enough. For this reason, Hooke's law is extensively used in all branches of science and engineering, is the foundation of many disciplines such as seismology, molecular mechanics and acoustics, it is the fundamental principle behind the spring scale, the manometer, the balance wheel of the mechanical clock. The modern theory of elasticity generalizes Hooke's law to say that the strain of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map that can be represented by a matrix of real numbers.
In this general form, Hooke's law makes it possible to deduce the relation between strain and stress for complex objects in terms of intrinsic properties of the materials it is made of. For example, one can deduce that a homogeneous rod with uniform cross section will behave like a simple spring when stretched, with a stiffness k directly proportional to its cross-section area and inversely proportional to its length. Consider a simple helical spring that has one end attached to some fixed object, while the free end is being pulled by a force whose magnitude is F s. Suppose that the spring has reached a state of equilibrium, where its length is not changing anymore. Let x be the amount by which the free end of the spring was displaced from its "relaxed" position. Hooke's law states that F s = k x or, equivalently, x = F s k where k is a positive real number, characteristic of the spring. Moreover, the same formula holds when the spring is compressed, with F s and x both negative in that case.
According to this formula, the graph of the applied force F s as a function of the displacement x will be a straight line passing through the origin, whose slope is k. Hooke's law for a spring is stated under the convention that F s is the restoring force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F s = − k x since the direction of the restoring force is opposite to that of the displacement. Hooke's spring law applies to any elastic object, of arbitrary complexity, as long as both the deformation and the stress can be expressed by a single number that can be both positive and negative. For example, when a block of rubber attached to two parallel plates is deformed by shearing, rather than stretching or compression, the shearing force F s and the sideways displacement of the plates x obey Hooke's law. Hooke's law applies when a straight steel bar or concrete beam, supported at both ends, is bent by a weight F placed at some intermediate point.
The displacement x in this case is the deviation of the beam, measured in the transversal direction, relative to its unloaded shape. The law applies when a stretched steel wire is twisted by pulling on a lever attached to one end. In this case the stress F s can be taken as the force applied to the lever, x as the distance traveled by it along its circular path. Or, one can let F s be the torque applied by the lever to the end of the wire, x be the angle by which that end turns. In either case F s is proportional to x In the case of a helical spring, stretched or compressed along its axis, the applied force and the resulting elongation or compression have the same direction (which is the directi
Frequency is the number of occurrences of a repeating event per unit of time. It is referred to as temporal frequency, which emphasizes the contrast to spatial frequency and angular frequency; the period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute, its period—the time interval between beats—is half a second. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals, radio waves, light. For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. In physics and engineering disciplines, such as optics and radio, frequency is denoted by a Latin letter f or by the Greek letter ν or ν; the relation between the frequency and the period T of a repeating event or oscillation is given by f = 1 T.
The SI derived unit of frequency is the hertz, named after the German physicist Heinrich Hertz. One hertz means. If a TV has a refresh rate of 1 hertz the TV's screen will change its picture once a second. A previous name for this unit was cycles per second; the SI unit for period is the second. A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. 60 rpm equals one hertz. As a matter of convenience and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Short and fast waves, like audio and radio, are described by their frequency instead of period; these used conversions are listed below: Angular frequency denoted by the Greek letter ω, is defined as the rate of change of angular displacement, θ, or the rate of change of the phase of a sinusoidal waveform, or as the rate of change of the argument to the sine function: y = sin = sin = sin d θ d t = ω = 2 π f Angular frequency is measured in radians per second but, for discrete-time signals, can be expressed as radians per sampling interval, a dimensionless quantity.
Angular frequency is larger than regular frequency by a factor of 2π. Spatial frequency is analogous to temporal frequency, but the time axis is replaced by one or more spatial displacement axes. E.g.: y = sin = sin d θ d x = k Wavenumber, k, is the spatial frequency analogue of angular temporal frequency and is measured in radians per meter. In the case of more than one spatial dimension, wavenumber is a vector quantity. For periodic waves in nondispersive media, frequency has an inverse relationship to the wavelength, λ. In dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: f = v λ. In the special case of electromagnetic waves moving through a vacuum v = c, where c is the speed of light in a vacuum, this expression becomes: f = c λ; when waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. Measurement of frequency can done in the following ways, Calculating the frequency of a repeating event is accomplished by counting the number of times that event occurs within a specific time period dividing the count by the length of the time period.
For example, if 71 events occur within 15 seconds the frequency is: f = 71 15 s ≈ 4.73 Hz If the number of counts is not large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. The latter method introduces a random error into the count of between zero and one count, so on average half a count; this is called gating error and causes an average error in the calculated frequency of Δ f = 1 2 T
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are sensitive to initial conditions. "Chaos" is an interdisciplinary theory stating that within the apparent randomness of chaotic complex systems, there are underlying patterns, constant feedback loops, self-similarity, self-organization, reliance on programming at the initial point known as sensitive dependence on initial conditions. The butterfly effect describes how a small change in one state of a deterministic nonlinear system can result in large differences in a state, e.g. a butterfly flapping its wings in Brazil can cause a hurricane in Texas. Small differences in initial conditions, such as those due to rounding errors in numerical computation, yield diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general; this happens though these systems are deterministic, meaning that their future behavior is determined by their initial conditions, with no random elements involved.
In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or chaos; the theory was summarized by Edward Lorenz as: Chaos: When the present determines the future, but the approximate present does not determine the future. Chaotic behavior exists in many natural systems, such as climate, it occurs spontaneously in some systems with artificial components, such as road traffic. This behavior can be studied through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps. Chaos theory has applications in several disciplines, including meteorology, sociology, environmental science, computer science, economics, biology and philosophy; the theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, self-assembly processes. Chaos theory concerns deterministic systems. Chaotic systems are predictable for a while and then'appear' to become random.
The amount of time that the behavior of a chaotic system can be predicted depends on three things: How much uncertainty can be tolerated in the forecast, how its current state can be measured, a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: about 1 millisecond. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, doubling the forecast time more than squares the proportional uncertainty in the forecast; this means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random. In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a used definition formulated by Robert L. Devaney says that, to classify a dynamical system as chaotic, it must have these properties: it must be sensitive to initial conditions, it must be topologically transitive, it must have dense periodic orbits.
In some cases, the last two properties in the above have been shown to imply sensitivity to initial conditions. In these cases, while it is the most significant property, "sensitivity to initial conditions" need not be stated in the definition. If attention is restricted to intervals, the second property implies the other two. An alternative, in general weaker, definition of chaos uses only the first two properties in the above list. In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, an intrinsic property of evolution operators of all stochastic and deterministic differential equations; this picture of dynamical chaos works not only for deterministic models but for models with external noise, an important generalization from the physical point of view, because in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics, e.g. the butterfly effect, is a consequence of the Goldstone's theorem in the application to the spontaneous topological supersymmetry breaking.
Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily approximated by other points with different future paths, or trajectories. Thus, an arbitrarily small change, or perturbation, of the current trajectory may lead to different future behavior. Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D. C. entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system would have been vastly different. A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system beyond a certain time the system is no longer predict
In physics, a shock wave, or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure and density of the medium. For the purpose of comparison, in supersonic flows, additional increased expansion may be achieved through an expansion fan known as a Prandtl–Meyer expansion fan; the accompanying expansion wave may approach and collide and recombine with the shock wave, creating a process of destructive interference. The sonic boom associated with the passage of a supersonic aircraft is a type of sound wave produced by constructive interference. Unlike solitons, the energy and speed of a shock wave alone dissipates quickly with distance; when a shock wave passes through matter, energy is preserved but entropy increases. This change in the matter's properties manifests itself as a decrease in the energy which can be extracted as work, as a drag force on supersonic objects.
Shock waves can be: Normal At 90° to the shock medium's flow direction. Oblique At an angle to the direction of flow. Bow Occurs upstream of the front of a blunt object when the upstream flow velocity exceeds Mach 1; some other terms Shock front: The boundary over which the physical conditions undergo an abrupt change because of a shock wave. Contact front: In a shock wave caused by a driver gas, the boundary between the driver and the driven gases; the Contact Front trails the Shock Front. The abruptness of change in the features of the medium, that characterize shock waves, can be viewed as a phase transition: the pressure-time diagram of a supersonic object propagating shows how the transition induced by a shock wave is analogous to a dynamic phase transition; when an object moves faster than the information can propagate into the surrounding fluid the fluid near the disturbance cannot react or "get out of the way" before the disturbance arrives. In a shock wave the properties of the fluid change instantaneously.
Measurements of the thickness of shock waves in air have resulted in values around 200 nm, on the same order of magnitude as the mean free gas molecule path. In reference to the continuum, this implies the shock wave can be treated as either a line or a plane if the flow field is two-dimensional or three-dimensional, respectively. Shock waves are formed when a pressure front moves at supersonic speeds and pushes on the surrounding air. At the region where this occurs, sound waves travelling against the flow reach a point where they cannot travel any further upstream and the pressure progressively builds in that region. Shock waves are not conventional sound waves. Shock waves in air are heard as "snap" noise. Over longer distances, a shock wave can change from a nonlinear wave into a linear wave, degenerating into a conventional sound wave as it heats the air and loses energy; the sound wave is heard as the familiar "thud" or "thump" of a sonic boom created by the supersonic flight of aircraft.
The shock wave is one of several different ways in which a gas in a supersonic flow can be compressed. Some other methods are isentropic compressions, including Prandtl–Meyer compressions; the method of compression of a gas results in different temperatures and densities for a given pressure ratio which can be analytically calculated for a non-reacting gas. A shock wave compression results in a loss of total pressure, meaning that it is a less efficient method of compressing gases for some purposes, for instance in the intake of a scramjet; the appearance of pressure-drag on supersonic aircraft is due to the effect of shock compression on the flow. In elementary fluid mechanics utilizing ideal gases, a shock wave is treated as a discontinuity where entropy increases over a nearly infinitesimal region. Since no fluid flow is discontinuous, a control volume is established around the shock wave, with the control surfaces that bound this volume parallel to the shock wave; the two surfaces are separated by a small depth such that the shock itself is contained between them.
At such control surfaces, mass flux and energy are constant. It is assumed the system is adiabatic and no work is being done; the Rankine–Hugoniot conditions arise from these considerations. Taking into account the established assumptions, in a system where the downstream properties are becoming subsonic: the upstream and downstream flow properties of the fluid are considered isentropic. Since the total amount of energy within the system is constant, the stagnation enthalpy remains constant over both regions. Though, entropy is increasing; when analyzing shock waves in a flow field, which are still attached to the body, the shock wave, deviating at some arbitrary angle from the flow direction is termed oblique shock. These shocks require a component vector analysis of the flow.
Richard Sheldon Palais is a mathematician working in geometry who introduced the Principle of Symmetric Criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, the Palais–Smale compactness condition. From 1965 to 1967 Palais was a Sloan Fellow. In 1970 he was an invited speaker at the International Congress of Mathematicians in Nice. From 1965 to 1982 he was an editor for the Journal of Differential Geometry and from 1966 to 1969 an editor for the Transactions of the American Mathematical Society. In 2010 he received a Lester R. Ford Award. In 2012 he became a fellow of the American Mathematical Society, he obtained his Ph. D. from Harvard University in 1956 under the joint supervision of Andrew M. Gleason and George Mackey, his doctoral students include Edward Bierstone, Leslie Lamport, Jill P. Mesirov, Chuu-lian Terng, Karen Uhlenbeck. as editor: Seminar on the Atiyah-Singer Index Theorem, Annals of Mathematical Studies, no. 4, Princeton Univ. Press, 1964 as author: A Global Formulation of the Lie Theory of Transformation Groups, Memoirs AMS 1957 The classification of G-Spaces, Memoirs AMS 1960 Foundations of Global Nonlinear Analysis, Benjamin 1968 The geometrization of physics, Tsinghua University Press 1981 Real algebraic differential topology, Publish or Perish 1981 with Chuu-Lian Terng: Critical point theory and submanifold geometry, Lecture Notes in Mathematics, vol.1353, Springer 1988 Richard Palais and Stephen Smale, A generalized Morse theory, Research Announcement, Bulletin of the American Mathematical Society 70, 165-172 R. Palais, Morse Theory on Hilbert Manifolds, Topology 2, 299-340.
R. Palais and Nonlinear Waves and Solitons, in The Princeton Companion to Mathematics, T. Gower Ed. Princeton Univ. Press 2008, 234-239 R. Palais, The Symmetries of Solitons, Bulletin. Amer. Math. Soc. New Series 34, No. 4, 339-403, R. Palais, The Visualization of Mathematics: Towards a Mathematical Exploratorium, Notices Amer. Math. Soc. 46, No. 6 (June–July 1999, R. Palais, A Simple Proof of the Banach Contraction Principle, The Journal for Fixed Point Theory and its Applications, 2 221–223, A nearly complete list of all papers authored or co-authored by Richard Palais is available for downloading as PDF files at http://vmm.math.uci.edu/PalaisPapers Richard Palais at the Mathematics Genealogy Project Home page Curriculum Vitae Homepage of 3D-XplorMath, Mathematical Visualization software developed by R. Palais