1.
Classical mechanics
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In physics, classical mechanics is one of the two major sub-fields of mechanics, along with quantum mechanics. Classical mechanics is concerned with the set of physical laws describing the motion of bodies under the influence of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology. Classical mechanics describes the motion of objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars. Within classical mechanics are fields of study that describe the behavior of solids, liquids and gases, Classical mechanics also provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When both quantum and classical mechanics cannot apply, such as at the level with high speeds. Since these aspects of physics were developed long before the emergence of quantum physics and relativity, however, a number of modern sources do include relativistic mechanics, which in their view represents classical mechanics in its most developed and accurate form. Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and these advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newtons work, particularly through their use of analytical mechanics. The following introduces the concepts of classical mechanics. For simplicity, it often models real-world objects as point particles, the motion of a point particle is characterized by a small number of parameters, its position, mass, and the forces applied to it. Each of these parameters is discussed in turn, in reality, the kind of objects that classical mechanics can describe always have a non-zero size. Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the degrees of freedom. However, the results for point particles can be used to such objects by treating them as composite objects. The center of mass of a composite object behaves like a point particle, Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space, non-relativistic mechanics also assumes that forces act instantaneously. The position of a point particle is defined with respect to a fixed reference point in space called the origin O, in space. A simple coordinate system might describe the position of a point P by means of a designated as r. In general, the point particle need not be stationary relative to O, such that r is a function of t, the time
2.
Physics
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Physics is the natural science that involves the study of matter and its motion and behavior through space and time, along with related concepts such as energy and force. One of the most fundamental disciplines, the main goal of physics is to understand how the universe behaves. Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy, Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry, and the boundaries of physics are not rigidly defined. New ideas in physics often explain the mechanisms of other sciences while opening new avenues of research in areas such as mathematics. Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs, the United Nations named 2005 the World Year of Physics. Astronomy is the oldest of the natural sciences, the stars and planets were often a target of worship, believed to represent their gods. While the explanations for these phenomena were often unscientific and lacking in evidence, according to Asger Aaboe, the origins of Western astronomy can be found in Mesopotamia, and all Western efforts in the exact sciences are descended from late Babylonian astronomy. The most notable innovations were in the field of optics and vision, which came from the works of many scientists like Ibn Sahl, Al-Kindi, Ibn al-Haytham, Al-Farisi and Avicenna. The most notable work was The Book of Optics, written by Ibn Al-Haitham, in which he was not only the first to disprove the ancient Greek idea about vision, but also came up with a new theory. In the book, he was also the first to study the phenomenon of the pinhole camera, many later European scholars and fellow polymaths, from Robert Grosseteste and Leonardo da Vinci to René Descartes, Johannes Kepler and Isaac Newton, were in his debt. Indeed, the influence of Ibn al-Haythams Optics ranks alongside that of Newtons work of the same title, the translation of The Book of Optics had a huge impact on Europe. From it, later European scholars were able to build the devices as what Ibn al-Haytham did. From this, such important things as eyeglasses, magnifying glasses, telescopes, Physics became a separate science when early modern Europeans used experimental and quantitative methods to discover what are now considered to be the laws of physics. Newton also developed calculus, the study of change, which provided new mathematical methods for solving physical problems. The discovery of new laws in thermodynamics, chemistry, and electromagnetics resulted from greater research efforts during the Industrial Revolution as energy needs increased, however, inaccuracies in classical mechanics for very small objects and very high velocities led to the development of modern physics in the 20th century. Modern physics began in the early 20th century with the work of Max Planck in quantum theory, both of these theories came about due to inaccuracies in classical mechanics in certain situations. Quantum mechanics would come to be pioneered by Werner Heisenberg, Erwin Schrödinger, from this early work, and work in related fields, the Standard Model of particle physics was derived. Areas of mathematics in general are important to this field, such as the study of probabilities, in many ways, physics stems from ancient Greek philosophy
3.
Mass
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In physics, mass is a property of a physical body. It is the measure of a resistance to acceleration when a net force is applied. It also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram, Mass is not the same as weight, even though mass is often determined by measuring the objects weight using a spring scale, rather than comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity and this is because weight is a force, while mass is the property that determines the strength of this force. In Newtonian physics, mass can be generalized as the amount of matter in an object, however, at very high speeds, special relativity postulates that energy is an additional source of mass. Thus, any body having mass has an equivalent amount of energy. In addition, matter is a defined term in science. There are several distinct phenomena which can be used to measure mass, active gravitational mass measures the gravitational force exerted by an object. Passive gravitational mass measures the force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force, according to Newtons second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A bodys mass also determines the degree to which it generates or is affected by a gravitational field and this is sometimes referred to as gravitational mass. The standard International System of Units unit of mass is the kilogram, the kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the prototype kilogram. As of January 2013, there are proposals for redefining the kilogram yet again. In this context, the mass has units of eV/c2, the electronvolt and its multiples, such as the MeV, are commonly used in particle physics. The atomic mass unit is 1/12 of the mass of a carbon-12 atom, the atomic mass unit is convenient for expressing the masses of atoms and molecules. Outside the SI system, other units of mass include, the slug is an Imperial unit of mass, the pound is a unit of both mass and force, used mainly in the United States
4.
Acceleration
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Acceleration, in physics, is the rate of change of velocity of an object with respect to time. An objects acceleration is the net result of any and all forces acting on the object, the SI unit for acceleration is metre per second squared. Accelerations are vector quantities and add according to the parallelogram law, as a vector, the calculated net force is equal to the product of the objects mass and its acceleration. For example, when a car starts from a standstill and travels in a line at increasing speeds. If the car turns, there is an acceleration toward the new direction, in this example, we can call the forward acceleration of the car a linear acceleration, which passengers in the car might experience as a force pushing them back into their seats. When changing direction, we call this non-linear acceleration, which passengers might experience as a sideways force. If the speed of the car decreases, this is an acceleration in the direction from the direction of the vehicle. Passengers may experience deceleration as a force lifting them forwards, mathematically, there is no separate formula for deceleration, both are changes in velocity. Each of these accelerations might be felt by passengers until their velocity matches that of the car, an objects average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t, instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. The SI unit of acceleration is the metre per second squared, or metre per second per second, as the velocity in metres per second changes by the acceleration value, every second. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, in this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a relative to a free-fall condition, is measured by an instrument called an accelerometer. As speeds approach the speed of light, relativistic effects become increasingly large and these components are called the tangential acceleration and the normal or radial acceleration. Geometrical analysis of space curves, which explains tangent, normal and binormal, is described by the Frenet–Serret formulas. Uniform or constant acceleration is a type of motion in which the velocity of an object changes by an amount in every equal time period. A frequently cited example of uniform acceleration is that of an object in free fall in a gravitational field. The acceleration of a body in the absence of resistances to motion is dependent only on the gravitational field strength g
5.
Force
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In physics, a force is any interaction that, when unopposed, will change the motion of an object. In other words, a force can cause an object with mass to change its velocity, force can also be described intuitively as a push or a pull. A force has both magnitude and direction, making it a vector quantity and it is measured in the SI unit of newtons and represented by the symbol F. The original form of Newtons second law states that the net force acting upon an object is equal to the rate at which its momentum changes with time. In an extended body, each part usually applies forces on the adjacent parts, such internal mechanical stresses cause no accelation of that body as the forces balance one another. Pressure, the distribution of small forces applied over an area of a body, is a simple type of stress that if unbalanced can cause the body to accelerate. Stress usually causes deformation of materials, or flow in fluids. In part this was due to an understanding of the sometimes non-obvious force of friction. A fundamental error was the belief that a force is required to maintain motion, most of the previous misunderstandings about motion and force were eventually corrected by Galileo Galilei and Sir Isaac Newton. With his mathematical insight, Sir Isaac Newton formulated laws of motion that were not improved-on for nearly three hundred years, 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, 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. 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 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 famous for formulating a treatment of buoyant forces inherent in fluids. Aristotle provided a discussion of the concept of a force as an integral part of Aristotelian cosmology. In Aristotles view, the sphere contained four elements that come to rest at different natural places therein. Aristotle believed that objects on Earth, those composed mostly of the elements earth and water, to be in their natural place on the ground. He distinguished between the tendency of objects to find their natural place, which led to natural motion, and unnatural or forced motion
6.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is
7.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space
8.
Concurrent lines
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In geometry, three or more lines in a plane or higher-dimensional space are said to be concurrent if they intersect at a single point. The point where the three altitudes meet is the orthocenter, angle bisectors are rays running from each vertex of the triangle and bisecting the associated angle. They all meet at the incenter, medians connect each vertex of a triangle to the midpoint of the opposite side. The three medians meet at the centroid, perpendicular bisectors are lines running out of the midpoints of each side of a triangle at 90 degree angles. The three perpendicular bisectors meet at the circumcenter, other sets of lines associated with a triangle are concurrent as well. For example, Any median is concurrent with two other area bisectors each of which is parallel to a side, a cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint of one of the three sides. The three cleavers concur at the center of the Spieker circle, which is the incircle of the medial triangle, a splitter of a triangle is a line segment having one endpoint at one of the three vertices of the triangle and bisecting the perimeter. The three splitters concur at the Nagel point of the triangle, Any line through a triangle that splits both the triangles area and its perimeter in half goes through the triangles incenter, and each triangle has one, two, or three of these lines. Thus if there are three of them, they concur at the incenter, the Tarry point of a triangle is the point of concurrency of the lines through the vertices of the triangle perpendicular to the corresponding sides of the triangles first Brocard triangle. The Napoleon points and generalizations of them are points of concurrency, a generalization of this notion is the Jacobi point. The de Longchamps point is the point of concurrence of several lines with the Euler line, in the case in which the original triangle has no angle greater than 120°, this point is also the Fermat point. The two bimedians of a quadrilateral and the segment joining the midpoints of the diagonals are concurrent and are all bisected by their point of intersection. In a tangential quadrilateral, the four angle bisectors concur at the center of the incircle, other concurrencies of a tangential quadrilateral are given here. In a cyclic quadrilateral, four segments, each perpendicular to one side. These line segments are called the maltitudes, which is an abbreviation for midpoint altitude and their common point is called the anticenter. If the successive sides of a hexagon are a, b, c, d, e, f. If a hexagon has a conic, then by Brianchons theorem its principal diagonals are concurrent. Concurrent lines arise in the dual of Pappuss hexagon theorem, for each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side
9.
Equation
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In mathematics, an equation is a statement of an equality containing one or more variables. Solving the equation consists of determining which values of the make the equality true. Variables are also called unknowns and the values of the unknowns which satisfy the equality are called solutions of the equation, there are two kinds of equations, identity equations and conditional equations. An identity equation is true for all values of the variable, a conditional equation is true for only particular values of the variables. Each side of an equation is called a member of the equation, each member will contain one or more terms. The equation, A x 2 + B x + C = y has two members, A x 2 + B x + C and y, the left member has three terms and the right member one term. The variables are x and y and the parameters are A, B, an equation is analogous to a scale into which weights are placed. When equal weights of something are place into the two pans, the two weights cause the scale to be in balance and are said to be equal. If a quantity of grain is removed from one pan of the balance, likewise, to keep an equation in balance, the same operations of addition, subtraction, multiplication and division must be performed on both sides of an equation for it to remain an equality. In geometry, equations are used to describe geometric figures and this is the starting idea of algebraic geometry, an important area of mathematics. Algebra studies two main families of equations, polynomial equations and, among them the case of linear equations. Polynomial equations have the form P =0, where P is a polynomial, linear equations have the form ax + b =0, where a and b are parameters. To solve equations from either family, one uses algorithmic or geometric techniques, algebra also studies Diophantine equations where the coefficients and solutions are integers. The techniques used are different and come from number theory and these equations are difficult in general, one often searches just to find the existence or absence of a solution, and, if they exist, to count the number of solutions. Differential equations are equations that involve one or more functions and their derivatives and they are solved by finding an expression for the function that does not involve derivatives. Differential equations are used to model processes that involve the rates of change of the variable, and are used in such as physics, chemistry, biology. The = symbol, which appears in equation, was invented in 1557 by Robert Recorde. An equation is analogous to a scale, balance, or seesaw
10.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
11.
Manifold
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In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, each point of a manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. One-dimensional manifolds include lines and circles, but not figure eights, two-dimensional manifolds are also called surfaces. Although a manifold locally resembles Euclidean space, globally it may not, for example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of map projections of the region into the Euclidean plane. When a region appears in two neighbouring charts, the two representations do not coincide exactly and a transformation is needed to pass from one to the other, Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. One important class of manifolds is the class of differentiable manifolds and this differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured, symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity. After a line, the circle is the simplest example of a topological manifold, Topology ignores bending, so a small piece of a circle is treated exactly the same as a small piece of a line. Consider, for instance, the top part of the circle, x2 + y2 =1. Any point of this arc can be described by its x-coordinate. So, projection onto the first coordinate is a continuous, and invertible, mapping from the arc to the open interval. Such functions along with the regions they map are called charts. Similarly, there are charts for the bottom, left, and right parts of the circle, together, these parts cover the whole circle and the four charts form an atlas for the circle. The top and right charts, χtop and χright respectively, overlap in their domain, Each map this part into the interval, though differently. Let a be any number in, then, T = χ r i g h t = χ r i g h t =1 − a 2 Such a function is called a transition map. The top, bottom, left, and right charts show that the circle is a manifold, charts need not be geometric projections, and the number of charts is a matter of some choice. These two charts provide a second atlas for the circle, with t =1 s Each chart omits a single point, either for s or for t and it can be proved that it is not possible to cover the full circle with a single chart. Viewed using calculus, the transition function T is simply a function between open intervals, which gives a meaning to the statement that T is differentiable
12.
Lie group
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In mathematics, a Lie group /ˈliː/ is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of transformation groups. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie’s student Arthur Tresse, an extension of Galois theory to the case of continuous symmetry groups was one of Lies principal motivations. Lie groups are smooth manifolds and as such can be studied using differential calculus. Lie groups play an role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various geometries by specifying an appropriate transformation group that leaves certain geometric properties invariant and this idea later led to the notion of a G-structure, where G is a Lie group of local symmetries of a manifold. On a global level, whenever a Lie group acts on an object, such as a Riemannian or a symplectic manifold. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry, Linear actions of Lie groups are especially important, and are studied in representation theory. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, a real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication μ, G × G → G μ = x y means that μ is a mapping of the product manifold G×G into G. These two requirements can be combined to the requirement that the mapping ↦ x −1 y be a smooth mapping of the product manifold into G. The 2×2 real invertible matrices form a group under multiplication, denoted by GL or by GL2 and this is a four-dimensional noncompact real Lie group. This group is disconnected, it has two connected components corresponding to the positive and negative values of the determinant, the rotation matrices form a subgroup of GL, denoted by SO. It is a Lie group in its own right, specifically, using the rotation angle φ as a parameter, this group can be parametrized as follows, SO =. Addition of the angles corresponds to multiplication of the elements of SO, thus both multiplication and inversion are differentiable maps. The orthogonal group also forms an example of a Lie group. All of the examples of Lie groups fall within the class of classical groups. Hilberts fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples, if the underlying manifold is allowed to be infinite-dimensional, then one arrives at the notion of an infinite-dimensional Lie group
13.
Analytical mechanics
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In theoretical physics and mathematical physics, analytical mechanics, or theoretical mechanics is a collection of closely related alternative formulations of classical mechanics. It was developed by scientists and mathematicians during the 18th century and onward. A scalar is a quantity, whereas a vector is represented by quantity, the equations of motion are derived from the scalar quantity by some underlying principle about the scalars variation. Analytical mechanics takes advantage of a systems constraints to solve problems, the constraints limit the degrees of freedom the system can have, and can be used to reduce the number of coordinates needed to solve for the motion. The formalism is well suited to arbitrary choices of coordinates, known in the context as generalized coordinates and it does not always work for non-conservative forces or dissipative forces like friction, in which case one may revert to Newtonian mechanics or use the Udwadia–Kalaba equation. Two dominant branches of mechanics are Lagrangian mechanics and Hamiltonian mechanics. There are other such as Hamilton–Jacobi theory, Routhian mechanics. All equations of motion for particles and fields, in any formalism, one result is Noethers theorem, a statement which connects conservation laws to their associated symmetries. Analytical mechanics does not introduce new physics and is not more general than Newtonian mechanics, rather it is a collection of equivalent formalisms which have broad application. In fact the principles and formalisms can be used in relativistic mechanics and general relativity. Analytical mechanics is used widely, from physics to applied mathematics. The methods of analytical mechanics apply to particles, each with a finite number of degrees of freedom. They can be modified to describe continuous fields or fluids, which have infinite degrees of freedom, the definitions and equations have a close analogy with those of mechanics. Generalized coordinates and constraints In Newtonian mechanics, one customarily uses all three Cartesian coordinates, or other 3D coordinate system, to refer to a position during its motion. In physical systems, however, some structure or other system usually constrains the motion from taking certain directions. In the Lagrangian and Hamiltonian formalisms, the constraints are incorporated into the motions geometry and these are known as generalized coordinates, denoted qi. Difference between curvillinear and generalized coordinates Generalized coordinates incorporate constraints on the system, there is one generalized coordinate qi for each degree of freedom, i. e. each way the system can change its configuration, as curvilinear lengths or angles of rotation. Generalized coordinates are not the same as curvilinear coordinates, DAlemberts principle The foundation which the subject is built on is DAlemberts principle
14.
Lagrangian mechanics
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Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. No new physics is introduced in Lagrangian mechanics compared to Newtonian mechanics, Newtons laws can include non-conservative forces like friction, however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system, dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler-Lagrange equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noethers theorem. Lagrangian mechanics is important not just for its applications. It can also be applied to systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics. Lagrangian mechanics applies to the dynamics of particles, fields are described using a Lagrangian density, Lagranges equations are also used in optimisation problems of dynamic systems. In mechanics, Lagranges equations of the second kind are used more than those of the first kind. Suppose we have a bead sliding around on a wire, or a simple pendulum. This choice eliminates the need for the constraint force to enter into the resultant system of equations, there are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment. For a wide variety of systems, if the size and shape of a massive object are negligible. For a system of N point particles with masses m1, m2, MN, each particle has a position vector, denoted r1, r2. Cartesian coordinates are often sufficient, so r1 =, r2 =, in three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system. These are all points in space to locate the particles. The velocity of particle is how fast the particle moves along its path of motion. In Newtonian mechanics, the equations of motion are given by Newtons laws, the second law net force equals mass times acceleration, Σ F = m d2r/dt2, applies to each particle. For an N particle system in 3d, there are 3N second order differential equations in the positions of the particles to solve for
15.
Acoustics
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Acoustics is the interdisciplinary science that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of society with the most obvious being the audio. Hearing is one of the most crucial means of survival in the animal world, accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more. Likewise, animal species such as songbirds and frogs use sound, art, craft, science and technology have provoked one another to advance the whole, as in many other fields of knowledge. Robert Bruce Lindsays Wheel of Acoustics is a well accepted overview of the fields in acoustics. The word acoustic is derived from the Greek word ἀκουστικός, meaning of or for hearing, ready to hear and that from ἀκουστός, heard, audible, which in turn derives from the verb ἀκούω, I hear. The Latin synonym is sonic, after which the term used to be a synonym for acoustics. Frequencies above and below the range are called ultrasonic and infrasonic. If, for example, a string of a length would sound particularly harmonious with a string of twice the length. In modern parlance, if a string sounds the note C when plucked, a string twice as long will sound a C an octave lower. In one system of tuning, the tones in between are then given by 16,9 for D,8,5 for E,3,2 for F,4,3 for G,6,5 for A. Aristotle understood that sound consisted of compressions and rarefactions of air which falls upon, a very good expression of the nature of wave motion. The physical understanding of acoustical processes advanced rapidly during and after the Scientific Revolution, mainly Galileo Galilei but also Marin Mersenne, independently, discovered the complete laws of vibrating strings. Experimental measurements of the speed of sound in air were carried out successfully between 1630 and 1680 by a number of investigators, prominently Mersenne, meanwhile, Newton derived the relationship for wave velocity in solids, a cornerstone of physical acoustics. The eighteenth century saw advances in acoustics as mathematicians applied the new techniques of calculus to elaborate theories of sound wave propagation. Also in the 19th century, Wheatstone, Ohm, and Henry developed the analogy between electricity and acoustics, the twentieth century saw a burgeoning of technological applications of the large body of scientific knowledge that was by then in place. The first such application was Sabine’s groundbreaking work in architectural acoustics, Underwater acoustics was used for detecting submarines in the first World War
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Center of mass
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The distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates. Calculations in mechanics are simplified when formulated with respect to the center of mass. It is a point where entire mass of an object may be assumed to be concentrated to visualise its motion. In other words, the center of mass is the equivalent of a given object for application of Newtons laws of motion. In the case of a rigid body, the center of mass is fixed in relation to the body. The center of mass may be located outside the body, as is sometimes the case for hollow or open-shaped objects. In the case of a distribution of separate bodies, such as the planets of the Solar System, in orbital mechanics, the equations of motion of planets are formulated as point masses located at the centers of mass. The center of mass frame is a frame in which the center of mass of a system is at rest with respect to the origin of the coordinate system. The concept of center of mass in the form of the center of gravity was first introduced by the ancient Greek physicist, mathematician, and engineer Archimedes of Syracuse. He worked with simplified assumptions about gravity that amount to a uniform field, in work on floating bodies he demonstrated that the orientation of a floating object is the one that makes its center of mass as low as possible. He developed mathematical techniques for finding the centers of mass of objects of uniform density of various well-defined shapes, Newtons second law is reformulated with respect to the center of mass in Eulers first law. The center of mass is the point at the center of a distribution of mass in space that has the property that the weighted position vectors relative to this point sum to zero. In analogy to statistics, the center of mass is the location of a distribution of mass in space. Solving this equation for R yields the formula R =1 M ∑ i =1 n m i r i, solve this equation for the coordinates R to obtain R =1 M ∭ Q ρ r d V, where M is the total mass in the volume. If a continuous mass distribution has density, which means ρ is constant. The center of mass is not generally the point at which a plane separates the distribution of mass into two equal halves, in analogy with statistics, the median is not the same as the mean. The coordinates R of the center of mass of a system, P1 and P2, with masses m1. The percentages of mass at each point can be viewed as projective coordinates of the point R on this line, another way of interpreting the process here is the mechanical balancing of moments about an arbitrary point
17.
Velocity
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The velocity of an object is the rate of change of its position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of its speed and direction of motion, Velocity is an important concept in kinematics, the branch of classical mechanics that describes the motion of bodies. Velocity is a vector quantity, both magnitude and direction are needed to define it. The scalar absolute value of velocity is called speed, being a coherent derived unit whose quantity is measured in the SI system as metres per second or as the SI base unit of. For example,5 metres per second is a scalar, whereas 5 metres per second east is a vector, if there is a change in speed, direction or both, then the object has a changing velocity and is said to be undergoing an acceleration. To have a constant velocity, an object must have a constant speed in a constant direction, constant direction constrains the object to motion in a straight path thus, a constant velocity means motion in a straight line at a constant speed. For example, a car moving at a constant 20 kilometres per hour in a path has a constant speed. Hence, the car is considered to be undergoing an acceleration, Speed describes only how fast an object is moving, whereas velocity gives both how fast and in what direction the object is moving. If a car is said to travel at 60 km/h, its speed has been specified, however, if the car is said to move at 60 km/h to the north, its velocity has now been specified. The big difference can be noticed when we consider movement around a circle and this is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Velocity is defined as the rate of change of position with respect to time, average velocity can be calculated as, v ¯ = Δ x Δ t. The average velocity is less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, from this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time is the displacement, x. In calculus terms, the integral of the velocity v is the displacement function x. In the figure, this corresponds to the area under the curve labeled s. Since the derivative of the position with respect to time gives the change in position divided by the change in time, although velocity is defined as the rate of change of position, it is often common to start with an expression for an objects acceleration. As seen by the three green tangent lines in the figure, an objects instantaneous acceleration at a point in time is the slope of the tangent to the curve of a v graph at that point. In other words, acceleration is defined as the derivative of velocity with respect to time, from there, we can obtain an expression for velocity as the area under an a acceleration vs. time graph
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Jerk (physics)
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Jerk is a vector, and there is no generally used term to describe its scalar magnitude. According to the result of analysis of jerk, the SI units are m/s3. Where a → is acceleration, v → is velocity, r → is position, there is no universal agreement on the symbol for jerk, but j is commonly used. Newtons notation for the derivative is also applied. The fourth derivative of position, equivalent to the first derivative of jerk, is jounce, because of involving third derivatives, in mathematics differential equations of the form J =0 are called jerk equations. This motivates mathematical interest in jerk systems, systems involving a fourth or higher derivative are accordingly called hyperjerk systems. In balancing some given force the postcentral gyrus establishes a control loop to achieve equilibrium by adjusting the muscular tension according to the sensed position of the actuator. As an everyday example, driving in a car can show effects of acceleration, the more experienced drivers accelerate smoothly, but beginners provide a jerky ride. High-powered sports cars offer the feeling of being pressed into the cushioning, note that there would be no jerk if the car started to move backwards with the same acceleration. Every experienced driver knows how to start and how to stop braking with low jerk, see also below in the motion profile, segment 7, Deceleration ramp-down. X itself, zeroth derivative The most prominent force F associated with the position of a particle relates through Hookes law to the stiffness k r of a spring. This is a force opposing the increase in displacement, the drag coefficient depends on the scalable shape of the object and on the Reynolds number, which itself depends on the speed. The acceleration a is according to Newtons second law F = m ⋅ a bound to a force F by the proportionality given by the mass m. It is often reported that NASA in designing the Hubble Telescope not only limited the jerk in their requirement specification, but also the next higher derivative, the jounce. For a recoil force on accelerating charged particles emitting radiation, which is proportional to their jerk, a more advanced theory, applicable in a relativistic and quantum environment, accounting for self-energy is provided in Wheeler–Feynman absorber theory. In real world environments, because of deformation, granularity at least at the Planck scale, i. e. quanta-effects, extrapolating from the idealized settings, the effect of jerk in real situations can be qualitatively described, explained and predicted. The jump-discontinuity in acceleration may be modeled by a Dirac delta in the jerk, assume a path along a circular arc with radius r, which tangentially connects to a straight line. The whole path is continuous and its pieces are smooth, see below for a more concrete application
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Angular velocity
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This speed can be measured in the SI unit of angular velocity, radians per second, or in terms of degrees per second, degrees per hour, etc. Angular velocity is usually represented by the symbol omega, the direction of the angular velocity vector is perpendicular to the plane of rotation, in a direction that is usually specified by the right-hand rule. The angular velocity of a particle is measured around or relative to a point, called the origin. As shown in the diagram, if a line is drawn from the origin to the particle, then the velocity of the particle has a component along the radius, if there is no radial component, then the particle moves in a circle. On the other hand, if there is no cross-radial component, a radial motion produces no change in the direction of the particle relative to the origin, so, for the purpose of finding the angular velocity, the radial component can be ignored. Therefore, the rotation is completely produced by the perpendicular motion around the origin, the angular velocity in two dimensions is a pseudoscalar, a quantity that changes its sign under a parity inversion. The positive direction of rotation is taken, by convention, to be in the direction towards the y axis from the x axis, if the parity is inverted, but the orientation of a rotation is not, then the sign of the angular velocity changes. There are three types of angular velocity involved in the movement on an ellipse corresponding to the three anomalies, in three dimensions, the angular velocity becomes a bit more complicated. The angular velocity in case is generally thought of as a vector, or more precisely. It now has not only a magnitude, but a direction as well, the magnitude is the angular speed, and the direction describes the axis of rotation that Eulers rotation theorem guarantees must exist. The right-hand rule indicates the direction of the angular velocity pseudovector. Let u be a vector along the instantaneous rotation axis. This is the definition of a vector space, the only property that presents difficulties to prove is the commutativity of the addition. This can be proven from the fact that the velocity tensor W is skew-symmetric, therefore, R = e W t is a rotation matrix and in a time dt is an infinitesimal rotation matrix. Therefore, it can be expanded as R = I + W ⋅ d t +122 +, in such a frame, each vector is a particular case of the previous case, in which the module of the vector is constant. Though it just a case of a moving particle, this is a very important one for its relationship with the rigid body study. There are two ways to describe the angular velocity of a rotating frame, the angular velocity vector. Both entities are related and they can be calculated from each other, in a consistent way with the general definition, the angular velocity of a frame is defined as the angular velocity of each of the three vectors of the frame
20.
Moment of inertia tensor
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It depends on the bodys mass distribution and the axis chosen, with larger moments requiring more torque to change the bodys rotation. It is a property, the moment of inertia of a composite system is the sum of the moments of inertia of its component subsystems. One of its definitions is the moment of mass with respect to distance from an axis r, I = ∫ Q r 2 d m. For bodies constrained to rotate in a plane, it is sufficient to consider their moment of inertia about a perpendicular to the plane. When a body is rotating, or free to rotate, around an axis, the amount of torque needed to cause any given angular acceleration is proportional to the moment of inertia of the body. Moment of inertia may be expressed in units of kilogram metre squared in SI units, moment of inertia plays the role in rotational kinetics that mass plays in linear kinetics - both characterize the resistance of a body to changes in its motion. The moment of inertia depends on how mass is distributed around an axis of rotation, for a point-like mass, the moment of inertia about some axis is given by mr2, where r is the distance to the axis, and m is the mass. For an extended body, the moment of inertia is just the sum of all the pieces of mass multiplied by the square of their distances from the axis in question. For an extended body of a shape and uniform density. In 1673 Christiaan Huygens introduced this parameter in his study of the oscillation of a body hanging from a pivot, the term moment of inertia was introduced by Leonhard Euler in his book Theoria motus corporum solidorum seu rigidorum in 1765, and it is incorporated into Eulers second law. Comparison of this frequency to that of a simple pendulum consisting of a single point of mass provides a mathematical formulation for moment of inertia of an extended body. Moment of inertia appears in momentum, kinetic energy, and in Newtons laws of motion for a rigid body as a physical parameter that combines its shape. There is a difference in the way moment of inertia appears in planar. The moment of inertia of a flywheel is used in a machine to resist variations in applied torque to smooth its rotational output. Moment of inertia I is defined as the ratio of the angular momentum L of a system to its angular velocity ω around a principal axis, if the angular momentum of a system is constant, then as the moment of inertia gets smaller, the angular velocity must increase. This occurs when spinning figure skaters pull in their arms or divers curl their bodies into a tuck position during a dive. For a simple pendulum, this yields a formula for the moment of inertia I in terms of the mass m of the pendulum and its distance r from the pivot point as. Thus, moment of inertia depends on both the mass m of a body and its geometry, or shape, as defined by the distance r to the axis of rotation
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Momentum
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In classical mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass and velocity of an object, quantified in kilogram-meters per second. It is dimensionally equivalent to impulse, the product of force and time, Newtons second law of motion states that the change in linear momentum of a body is equal to the net impulse acting on it. If the truck were lighter, or moving slowly, then it would have less momentum. Linear momentum is also a quantity, meaning that if a closed system is not affected by external forces. In classical mechanics, conservation of momentum is implied by Newtons laws. It also holds in special relativity and, with definitions, a linear momentum conservation law holds in electrodynamics, quantum mechanics, quantum field theory. It is ultimately an expression of one of the symmetries of space and time. Linear momentum depends on frame of reference, observers in different frames would find different values of linear momentum of a system. But each would observe that the value of linear momentum does not change with time, momentum has a direction as well as magnitude. Quantities that have both a magnitude and a direction are known as vector quantities, because momentum has a direction, it can be used to predict the resulting direction of objects after they collide, as well as their speeds. Below, the properties of momentum are described in one dimension. The vector equations are almost identical to the scalar equations, the momentum of a particle is traditionally represented by the letter p. It is the product of two quantities, the mass and velocity, p = m v, the units of momentum are the product of the units of mass and velocity. In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meters/second, in cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeters/second. Being a vector, momentum has magnitude and direction, for example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg m/s due north measured from the ground. The momentum of a system of particles is the sum of their momenta, if two particles have masses m1 and m2, and velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2. If all the particles are moving, the center of mass will generally be moving as well, if the center of mass is moving at velocity vcm, the momentum is, p = m v cm. This is known as Eulers first law, if a force F is applied to a particle for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t
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Impulse (physics)
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In classical mechanics, impulse is the integral of a force, F, over the time interval, t, for which it acts. Since force is a quantity, impulse is also a vector in the same direction. Impulse applied to an object produces an equivalent vector change in its linear momentum, the SI unit of impulse is the newton second, and the dimensionally equivalent unit of momentum is the kilogram meter per second. The corresponding English engineering units are the pound-second and the slug-foot per second, a resultant force causes acceleration and a change in the velocity of the body for as long as it acts. Conversely, a force applied for a long time produces the same change in momentum—the same impulse—as a larger force applied briefly. This is often called the impulse-momentum theorem, as a result, an impulse may also be regarded as the change in momentum of an object to which a resultant force is applied. Impulse has the units and dimensions as momentum. In the International System of Units, these are kg·m/s = N·s, in English engineering units, they are slug·ft/s = lbf·s. The term impulse is also used to refer to a force or impact. This type of impulse is often idealized so that the change in momentum produced by the force happens with no change in time and this sort of change is a step change, and is not physically possible. However, this is a model for computing the effects of ideal collisions. The application of Newtons second law for variable mass allows impulse, in the case of rockets, the impulse imparted can be normalized by unit of propellant expended, to create a performance parameter, specific impulse. This fact can be used to derive the Tsiolkovsky rocket equation, which relates the vehicles propulsive change in velocity to the specific impulse. Wave–particle duality defines the impulse of a wave collision, the preservation of momentum in the collision is then called phase matching. Applications include, Compton effect nonlinear optics Acousto-optic modulator Electron phonon scattering Serway, Raymond A. Jewett, John W. Physics for Scientists, Physics for Scientists and Engineers, Mechanics, Oscillations and Waves, Thermodynamics
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Angular momentum
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In physics, angular momentum is the rotational analog of linear momentum. It is an important quantity in physics because it is a conserved quantity – the angular momentum of a system remains constant unless acted on by an external torque. The definition of momentum for a point particle is a pseudovector r×p. This definition can be applied to each point in continua like solids or fluids, unlike momentum, angular momentum does depend on where the origin is chosen, since the particles position is measured from it. The angular momentum of an object can also be connected to the angular velocity ω of the object via the moment of inertia I. However, while ω always points in the direction of the rotation axis, Angular momentum is additive, the total angular momentum of a system is the vector sum of the angular momenta. For continua or fields one uses integration, torque can be defined as the rate of change of angular momentum, analogous to force. Applications include the gyrocompass, control moment gyroscope, inertial systems, reaction wheels, flying discs or Frisbees. In general, conservation does limit the motion of a system. In quantum mechanics, angular momentum is an operator with quantized eigenvalues, Angular momentum is subject to the Heisenberg uncertainty principle, meaning only one component can be measured with definite precision, the other two cannot. Also, the spin of elementary particles does not correspond to literal spinning motion, Angular momentum is a vector quantity that represents the product of a bodys rotational inertia and rotational velocity about a particular axis. Angular momentum can be considered an analog of linear momentum. Thus, where momentum is proportional to mass m and linear speed v, p = m v, angular momentum is proportional to moment of inertia I. Unlike mass, which only on amount of matter, moment of inertia is also dependent on the position of the axis of rotation. Unlike linear speed, which occurs in a line, angular speed occurs about a center of rotation. Therefore, strictly speaking, L should be referred to as the angular momentum relative to that center and this simple analysis can also apply to non-circular motion if only the component of the motion which is perpendicular to the radius vector is considered. In that case, L = r m v ⊥, where v ⊥ = v sin θ is the component of the motion. It is this definition, × to which the moment of momentum refers
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Torque
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Torque, moment, or moment of force is rotational force. Just as a force is a push or a pull. Loosely speaking, torque is a measure of the force on an object such as a bolt or a flywheel. For example, pushing or pulling the handle of a wrench connected to a nut or bolt produces a torque that loosens or tightens the nut or bolt, the symbol for torque is typically τ, the lowercase Greek letter tau. When it is called moment of force, it is denoted by M. The SI unit for torque is the newton metre, for more on the units of torque, see Units. This article follows US physics terminology in its use of the word torque, in the UK and in US mechanical engineering, this is called moment of force, usually shortened to moment. In US physics and UK physics terminology these terms are interchangeable, unlike in US mechanical engineering, Torque is defined mathematically as the rate of change of angular momentum of an object. The definition of states that one or both of the angular velocity or the moment of inertia of an object are changing. Moment is the term used for the tendency of one or more applied forces to rotate an object about an axis. For example, a force applied to a shaft causing acceleration, such as a drill bit accelerating from rest. By contrast, a force on a beam produces a moment, but since the angular momentum of the beam is not changing. Similarly with any force couple on an object that has no change to its angular momentum and this article follows the US physics terminology by calling all moments by the term torque, whether or not they cause the angular momentum of an object to change. The concept of torque, also called moment or couple, originated with the studies of Archimedes on levers, the term torque was apparently introduced into English scientific literature by James Thomson, the brother of Lord Kelvin, in 1884. A force applied at an angle to a lever multiplied by its distance from the levers fulcrum is its torque. A force of three newtons applied two metres from the fulcrum, for example, exerts the same torque as a force of one newton applied six metres from the fulcrum. More generally, the torque on a particle can be defined as the product, τ = r × F, where r is the particles position vector relative to the fulcrum. Alternatively, τ = r F ⊥, where F⊥ is the amount of force directed perpendicularly to the position of the particle, any force directed parallel to the particles position vector does not produce a torque
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Mechanical energy
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In the physical sciences, mechanical energy is the sum of potential energy and kinetic energy. It is the associated with the motion and position of an object. The principle of conservation of energy states that in an isolated system that is only subject to conservative forces the mechanical energy is constant. In elastic collisions, the energy is conserved but in inelastic collisions. The equivalence between lost mechanical energy and an increase in temperature was discovered by James Prescott Joule and it is defined as the objects ability to do work and is increased as the object is moved in the opposite direction of the direction of the force. On the contrary, when a force acts upon an object. Though energy cannot be created or destroyed in an isolated system, the pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the positions of its swing. However, when taking the forces into account, the system loses mechanical energy with each swing because of the work done by the pendulum to oppose these non-conservative forces. This equivalence between mechanical energy and heat is especially important when considering colliding objects, in an elastic collision, mechanical energy is conserved – the sum of the mechanical energies of the colliding objects is the same before and after the collision. After an inelastic collision, however, the energy of the system will have changed. Usually, the energy before the collision is greater than the mechanical energy after the collision. In inelastic collisions, some of the energy of the colliding objects is transformed into kinetic energy of the constituent particles. This increase in energy of the constituent particles is perceived as an increase in temperature. The collision can be described by saying some of the energy of the colliding objects has been converted into an equal amount of heat. Thus, the energy of the system remains unchanged though the mechanical energy of the system has reduced. A satellite of mass m at a distance r from the centre of Earth possesses both kinetic energy, K, and gravitational energy, U. These devices can be placed in categories, An electric motor converts electrical energy into mechanical energy
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Work (physics)
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In physics, a force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force. For example, when a ball is held above the ground and then dropped, the SI unit of work is the joule. The SI unit of work is the joule, which is defined as the work expended by a force of one newton through a distance of one metre. The dimensionally equivalent newton-metre is sometimes used as the unit for work, but this can be confused with the unit newton-metre. Usage of N⋅m is discouraged by the SI authority, since it can lead to confusion as to whether the quantity expressed in newton metres is a torque measurement, or a measurement of energy. Non-SI units of work include the erg, the foot-pound, the foot-poundal, the hour, the litre-atmosphere. Due to work having the physical dimension as heat, occasionally measurement units typically reserved for heat or energy content, such as therm, BTU. The work done by a constant force of magnitude F on a point that moves a distance s in a line in the direction of the force is the product W = F s. For example, if a force of 10 newtons acts along a point that travels 2 meters and this is approximately the work done lifting a 1 kg weight from ground level to over a persons head against the force of gravity. Notice that the work is doubled either by lifting twice the weight the distance or by lifting the same weight twice the distance. Work is closely related to energy, the work-energy principle states that an increase in the kinetic energy of a rigid body is caused by an equal amount of positive work done on the body by the resultant force acting on that body. Conversely, a decrease in energy is caused by an equal amount of negative work done by the resultant force. From Newtons second law, it can be shown that work on a free, rigid body, is equal to the change in energy of the velocity and rotation of that body. The work of forces generated by a function is known as potential energy. These formulas demonstrate that work is the associated with the action of a force, so work subsequently possesses the physical dimensions. The work/energy principles discussed here are identical to Electric work/energy principles, constraint forces determine the movement of components in a system, constraining the object within a boundary. Constraint forces ensure the velocity in the direction of the constraint is zero and this only applies for a single particle system. For example, in an Atwood machine, the rope does work on each body, there are, however, cases where this is not true
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Potential energy
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In physics, potential energy is energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule, the term potential energy was introduced by the 19th century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotles concept of potentiality. Potential energy is associated with forces that act on a body in a way that the work done by these forces on the body depends only on the initial and final positions of the body in space. These forces, that are called potential forces, can be represented at every point in space by vectors expressed as gradients of a scalar function called potential. Potential energy is the energy of an object. It is the energy by virtue of a position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force that works against the force field of the potential. This work is stored in the field, which is said to be stored as potential energy. If the external force is removed the field acts on the body to perform the work as it moves the body back to the initial position. Suppose a ball which mass is m, and it is in h position in height, if the acceleration of free fall is g, the weight of the ball is mg. There are various types of energy, each associated with a particular type of force. Chemical potential energy, such as the energy stored in fossil fuels, is the work of the Coulomb force during rearrangement of mutual positions of electrons and nuclei in atoms and molecules. Thermal energy usually has two components, the energy of random motions of particles and the potential energy of their mutual positions. Forces derivable from a potential are also called conservative forces, the work done by a conservative force is W = − Δ U where Δ U is the change in the potential energy associated with the force. The negative sign provides the convention that work done against a force field increases potential energy, common notations for potential energy are U, V, also Ep. Potential energy is closely linked with forces, in this case, the force can be defined as the negative of the vector gradient of the potential field. If the work for a force is independent of the path, then the work done by the force is evaluated at the start
28.
Power (physics)
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In physics, power is the rate of doing work. It is the amount of energy consumed per unit time, having no direction, it is a scalar quantity. In the SI system, the unit of power is the joule per second, known as the watt in honour of James Watt, another common and traditional measure is horsepower. Being the rate of work, the equation for power can be written, because this integral depends on the trajectory of the point of application of the force and torque, this calculation of work is said to be path dependent. As a physical concept, power requires both a change in the universe and a specified time in which the change occurs. This is distinct from the concept of work, which is measured in terms of a net change in the state of the physical universe. The output power of a motor is the product of the torque that the motor generates. The power involved in moving a vehicle is the product of the force of the wheels. The dimension of power is divided by time. The SI unit of power is the watt, which is equal to one joule per second, other units of power include ergs per second, horsepower, metric horsepower, and foot-pounds per minute. One horsepower is equivalent to 33,000 foot-pounds per minute, or the required to lift 550 pounds by one foot in one second. Other units include dBm, a logarithmic measure with 1 milliwatt as reference, food calories per hour, Btu per hour. This shows how power is an amount of energy consumed per unit time. If ΔW is the amount of work performed during a period of time of duration Δt and it is the average amount of work done or energy converted per unit of time. The average power is simply called power when the context makes it clear. The instantaneous power is then the value of the average power as the time interval Δt approaches zero. P = lim Δ t →0 P a v g = lim Δ t →0 Δ W Δ t = d W d t. In the case of constant power P, the amount of work performed during a period of duration T is given by, W = P t
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Conservative force
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A conservative force is a force with the property that the work done in moving a particle between two points is independent of the taken path. Equivalently, if a particle travels in a loop, the net work done by a conservative force is zero. A conservative force is dependent only on the position of the object, if a force is conservative, it is possible to assign a numerical value for the potential at any point. When an object moves from one location to another, the changes the potential energy of the object by an amount that does not depend on the path taken. If the force is not conservative, then defining a scalar potential is not possible, gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force. Other examples of conservative forces are, force in elastic spring, the last two forces are called central forces as they act along the line joining the centres of two charged/magnetized bodies. Thus, all forces are conservative forces. Informally, a force can be thought of as a force that conserves mechanical energy. Suppose a particle starts at point A, and there is a force F acting on it, then the particle is moved around by other forces, and eventually ends up at A again. Though the particle may still be moving, at that instant when it passes point A again, if the net work done by F at this point is 0, then F passes the closed path test. Any force that passes the closed path test for all possible closed paths is classified as a conservative force, the gravitational force, spring force, magnetic force and electric force are examples of conservative forces, while friction and air drag are classical examples of non-conservative forces. For non-conservative forces, the energy that is lost has to go somewhere else. Usually the energy is turned into heat, for example the heat generated by friction, in addition to heat, friction also often produces some sound energy. The water drag on a moving boat converts the mechanical energy into not only heat and sound energy. These and other losses are irreversible because of the second law of thermodynamics. A direct consequence of the closed path test is that the work done by a force on a particle moving between any two points does not depend on the path taken by the particle. This is illustrated in the figure to the right, The work done by the force on an object depends only on its change in height because the gravitational force is conservative. The work done by a force is equal to the negative of change in potential energy during that process
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Generalized coordinates
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These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart, the generalized velocities are the time derivatives of the generalized coordinates of the system. An example of a generalized coordinate is the angle that locates a point moving on a circle, if these parameters are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system. However, it can occur that a useful set of generalized coordinates may be dependent. For a system of N particles in 3D real coordinate space, any of the position vectors can be denoted rk where k =1,2. A holonomic constraint is a constraint equation of the form for particle k f =0 which connects all the 3 spatial coordinates of that particle together, the constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, when t is a constant, any one coordinate will be determined from the coordinates, e. g. if xk. One constraint equation counts as one constraint, if there are C constraints, each has an equation, so there will be C constraint equations. There is not necessarily one constraint equation for each particle, so far, the configuration of the system is defined by 3N quantities, but C coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is n = 3N − C and it is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context, denoted qj and it is convenient to collect them into an n-tuple q = which is a point in the configuration space of the system. They are all independent of one other, and each is a function of time, geometrically they can be lengths along straight lines, or arc lengths along curves, or angles, not necessarily Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals the number of degrees of freedom, n. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, the corresponding time derivatives of q are the generalized velocities, q ˙ = d q d t =. Since we are free to specify the values of the generalized coordinates and velocities separately. A mechanical system can involve constraints on both the generalized coordinates and their derivatives, constraints of this type are known as non-holonomic. First-order non-holonomic constraints have the form g =0, An example of such a constraint is a wheel or knife-edge that constrains the direction of the velocity vector. Non-holonomic constraints can also involve next-order derivatives such as generalized accelerations
31.
Generalized velocities
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These parameters must uniquely define the configuration of the system relative to the reference configuration. This is done assuming that this can be done with a single chart, the generalized velocities are the time derivatives of the generalized coordinates of the system. An example of a generalized coordinate is the angle that locates a point moving on a circle, if these parameters are independent of one another, the number of independent generalized coordinates is defined by the number of degrees of freedom of the system. However, it can occur that a useful set of generalized coordinates may be dependent. For a system of N particles in 3D real coordinate space, any of the position vectors can be denoted rk where k =1,2. A holonomic constraint is a constraint equation of the form for particle k f =0 which connects all the 3 spatial coordinates of that particle together, the constraint may change with time, so time t will appear explicitly in the constraint equations. At any instant of time, when t is a constant, any one coordinate will be determined from the coordinates, e. g. if xk. One constraint equation counts as one constraint, if there are C constraints, each has an equation, so there will be C constraint equations. There is not necessarily one constraint equation for each particle, so far, the configuration of the system is defined by 3N quantities, but C coordinates can be eliminated, one coordinate from each constraint equation. The number of independent coordinates is n = 3N − C and it is ideal to use the minimum number of coordinates needed to define the configuration of the entire system, while taking advantage of the constraints on the system. These quantities are known as generalized coordinates in this context, denoted qj and it is convenient to collect them into an n-tuple q = which is a point in the configuration space of the system. They are all independent of one other, and each is a function of time, geometrically they can be lengths along straight lines, or arc lengths along curves, or angles, not necessarily Cartesian coordinates or other standard orthogonal coordinates. There is one for each degree of freedom, so the number of generalized coordinates equals the number of degrees of freedom, n. A degree of freedom corresponds to one quantity that changes the configuration of the system, for example the angle of a pendulum, the corresponding time derivatives of q are the generalized velocities, q ˙ = d q d t =. Since we are free to specify the values of the generalized coordinates and velocities separately. A mechanical system can involve constraints on both the generalized coordinates and their derivatives, constraints of this type are known as non-holonomic. First-order non-holonomic constraints have the form g =0, An example of such a constraint is a wheel or knife-edge that constrains the direction of the velocity vector. Non-holonomic constraints can also involve next-order derivatives such as generalized accelerations
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Rigid body
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In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In other words, the distance between any two points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, in classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules are often seen as rigid bodies, the position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, if the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, however, typically a different, mathematically more convenient, but equivalent approach is used. Thus, the position of a body has two components, linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a body, such as linear and angular velocity, acceleration, momentum, impulse. This reference point may define the origin of a coordinate system fixed to the body, there are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix. In general, when a body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translation and rotation, indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation of the body starting from a hypothetic reference position. Velocity and angular velocity are measured with respect to a frame of reference, the linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a point fixed to the body. During purely translational motion, all points on a body move with the same velocity. However, when motion involves rotation, the velocity of any two points on the body will generally not be the same. Two points of a body will have the same instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of rotation. Angular velocity is a quantity that describes the angular speed at which the orientation of the rigid body is changing. All points on a rigid body experience the same velocity at all times
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Pseudovector
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Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a vector, also known as a polar vector. In three dimensions the pseudovector p is associated with the product of two polar vectors a and b, p = a × b. The vector p calculated this way is a pseudovector, one example is the normal to an oriented plane. An oriented plane can be defined by two vectors, a and b, which can be said to span the plane. The vector a × b is a normal to the plane and this has consequences in computer graphics where it has to be considered when transforming surface normals. A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field, in mathematics pseudovectors are equivalent to three-dimensional bivectors, from which the transformation rules of pseudovectors can be derived. More generally in geometric algebra pseudovectors are the elements of the algebra with dimension n −1. The label pseudo can be generalized to pseudoscalars and pseudotensors. Physical examples of pseudovectors include magnetic field, torque, vorticity, consider the pseudovector angular momentum L = r × p. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. The distinction between vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems, consider an electric current loop in the z =0 plane that inside the loop generates a magnetic field oriented in the z direction. This system is symmetric under mirror reflections through this plane, with the field unchanged by the reflection. The definition of a vector in physics is more specific than the definition of vector. This important requirement is what distinguishes a vector from any other triplet of physical quantities The discussion so far only relates to proper rotations, however, one can also consider improper rotations, i. e. a mirror-reflection possibly followed by a proper rotation. Suppose everything in the universe undergoes an improper rotation described by the rotation matrix R, if the vector v is a polar vector, it will be transformed to v′ = Rv. If it is a pseudovector, it will be transformed to v′ = −Rv, the symbol det denotes determinant, this formula works because the determinant of proper and improper rotation matrices are +1 and -1, respectively. Suppose v1 and v2 are known pseudovectors, and v3 is defined to be their sum, if the universe is transformed by a rotation matrix R, then v3 is transformed to v 3 ′ = v 1 ′ + v 2 ′ = + = =. So v3 is also a pseudovector, on the other hand, suppose v1 is known to be a polar vector, v2 is known to be a pseudovector, and v3 is defined to be their sum, v3 = v1 + v2
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Tensor
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In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors. Elementary examples of such include the dot product, the cross product. Geometric vectors, often used in physics and engineering applications, given a coordinate basis or fixed frame of reference, a tensor can be represented as an organized multidimensional array of numerical values. The order of a tensor is the dimensionality of the array needed to represent it, or equivalently, for example, a linear map is represented by a matrix in a basis, and therefore is a 2nd-order tensor. A vector is represented as a 1-dimensional array in a basis, scalars are single numbers and are thus 0th-order tensors. Because they express a relationship between vectors, tensors themselves must be independent of a choice of coordinate system. The precise form of the transformation law determines the type of the tensor, the tensor type is a pair of natural numbers, where n is the number of contravariant indices and m is the number of covariant indices. The total order of a tensor is the sum of two numbers. The concept enabled an alternative formulation of the differential geometry of a manifold in the form of the Riemann curvature tensor. There are several approaches to defining tensors, although seemingly different, the approaches just describe the same geometric concept using different languages and at different levels of abstraction. For example, an operator is represented in a basis as a two-dimensional square n × n array. The numbers in the array are known as the scalar components of the tensor or simply its components. They are denoted by giving their position in the array, as subscripts and superscripts. For example, the components of an order 2 tensor T could be denoted Tij , whether an index is displayed as a superscript or subscript depends on the transformation properties of the tensor, described below. The total number of required to identify each component uniquely is equal to the dimension of the array. However, the term generally has another meaning in the context of matrices. Just as the components of a change when we change the basis of the vector space. Each tensor comes equipped with a law that details how the components of the tensor respond to a change of basis
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Newton's laws of motion
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Newtons laws of motion are three physical laws that, together, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. More precisely, the first law defines the force qualitatively, the second law offers a measure of the force. These three laws have been expressed in different ways, over nearly three centuries, and can be summarised as follows. The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation. Newtons laws are applied to objects which are idealised as single point masses, in the sense that the size and 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, even a planet can be idealised as a particle for analysis of its orbital motion around a star, in their original form, Newtons laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newtons laws of motion for rigid bodies called Eulers laws of motion, if a body is represented as an assemblage of discrete particles, each governed by Newtons laws of motion, then Eulers laws can be derived from Newtons laws. Eulers laws can, however, be taken as axioms describing the laws of motion for extended bodies, Newtons laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. 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 Newtons death. In the given mass, acceleration, momentum, and force are assumed to be externally defined quantities. This is the most common, but not the 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 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 first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F =0 ⇔ d v d t =0. Consequently, An object that is at rest will stay at rest unless a force acts upon it, an object that is in motion will not change its velocity unless a force acts upon it. This is known as uniform motion, an object continues to do whatever it happens to be doing unless a force is exerted upon it. If it is at rest, it continues in a state of rest, if an object is moving, it continues to move without turning or changing its speed
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Top
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A spinning top is a toy designed to spin rapidly on the ground, the motion of which causes it to remain precisely balanced on its tip because of its rotational inertia. Such toys have existed since antiquity, the motion of a top is produced in the most simple forms by twirling the stem using the fingers. More sophisticated tops are spun by holding the axis firmly while pulling a string or twisting a stick or pushing an auger, in the kinds with an auger, an internal weight rotates, producing an overall circular motion. The top is one of the oldest recognizable toys found on archaeological sites, Spinning tops originated independently in cultures all over the world. Besides toys, tops have also historically used for gambling. Some role-playing games use tops to augment dice in generating randomized results, a thumbtack may also be made to spin on the same principles. The action of a top is described by equations of rigid body dynamics, in the sleep period, and only in it, provided it is ever reached, less friction means longer sleep time There have been many developments within the technology of the top. Bearing tops, with a tip made of a hard ceramic. In addition, plastic and metal have largely supplanted the use of wood in tops, fixed tip tops are featured in National Championships in Chico, California and in the World Championships in Orlando, Florida. Asymmetric tops of any shape can also be created and designed to balance. The bully and the top in the title are challenged by Shepherds ongoing protagonist Ralph. Bauernroulette Diabolo Gee-haw whammy diddle Yo-yo ForeverSpin Spindle Lagrange, Euler, an Elementary Treatment of the Theory of Spinning Tops and Gyroscopic Motion. Reprinted by Michigan Historical Reprint Series, london Society for Promoting Christian Knowledge,1870. Reprinted by Project Gutemberg ebook,2010, provatidis, Christopher, G. Revisiting the Spinning Top, International Journal of Materials and Mechanical Engineering, Vol.1, No. 4, pp. 71–88, open access at http, //www. ijm-me. org/Download. aspx. ID=2316 A forum discussing all things related to the art of Top Spinning, iTopSpin. com Top
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Unit vector
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In mathematics, a unit vector in a normed vector space is a vector of length 1. A unit vector is denoted by a lowercase letter with a circumflex, or hat. The term direction vector is used to describe a unit vector being used to represent spatial direction, two 2D direction vectors, d1 and d2 are illustrated. 2D spatial directions represented this way are equivalent numerically to points on the unit circle, the same construct is used to specify spatial directions in 3D. As illustrated, each direction is equivalent numerically to a point on the unit sphere. The normalized vector or versor û of a vector u is the unit vector in the direction of u, i. e. u ^ = u ∥ u ∥ where ||u|| is the norm of u. The term normalized vector is used as a synonym for unit vector. Unit vectors are often chosen to form the basis of a vector space, every vector in the space may be written as a linear combination of unit vectors. By definition, in a Euclidean space the dot product of two vectors is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the product of two arbitrary unit vectors is a 3rd vector orthogonal to both of them having length equal to the sine of the smaller subtended angle. Unit vectors may be used to represent the axes of a Cartesian coordinate system and they are often denoted using normal vector notation rather than standard unit vector notation. In most contexts it can be assumed that i, j, the notations, or, with or without hat, are also used, particularly in contexts where i, j, k might lead to confusion with another quantity. When a unit vector in space is expressed, with Cartesian notation, as a combination of i, j, k. The value of each component is equal to the cosine of the angle formed by the vector with the respective basis vector. This is one of the used to describe the orientation of a straight line, segment of straight line, oriented axis. It is important to note that ρ ^ and φ ^ are functions of φ, when differentiating or integrating in cylindrical coordinates, these unit vectors themselves must also be operated on. For a more complete description, see Jacobian matrix, to minimize degeneracy, the polar angle is usually taken 0 ≤ θ ≤180 ∘. It is especially important to note the context of any ordered triplet written in spherical coordinates, here, the American physics convention is used
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Kinetic energy
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In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes, the same amount of work is done by the body in decelerating from its current speed to a state of rest. In classical mechanics, the energy of a non-rotating object of mass m traveling at a speed v is 12 m v 2. In relativistic mechanics, this is an approximation only when v is much less than the speed of light. The standard unit of energy is the joule. The adjective kinetic has its roots in the Greek word κίνησις kinesis, the dichotomy between kinetic energy and potential energy can be traced back to Aristotles concepts of actuality and potentiality. The principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, Willem s Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, 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 lEffet des Machines outlining the mathematics of kinetic energy. William Thomson, later Lord Kelvin, is given the credit for coining the term kinetic energy c, 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 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, 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, 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 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
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Hooke's law
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Hookes law is a principle of physics that states that the force needed to extend or compress a spring by some distance X is proportional to that distance. That is, F = kX, where k is a constant factor characteristic of the spring, its stiffness, the law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram and he published the solution of his anagram in 1678 as, ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law already in 1660, an elastic body or material for which this equation can be assumed is said to be linear-elastic or Hookean. Hookes law is only a linear approximation to the real response of springs. Many materials will deviate from Hookes law well before those elastic limits are reached. On the other hand, Hookes law is an approximation for most solid bodies, as long as the forces. For this reason, Hookes law is used in all branches of science and engineering. It is also the principle behind the spring scale, the manometer. The modern theory of elasticity generalizes Hookes law to say that the strain of an object or material is proportional to the stress applied to it. In this general form, Hookes law makes it possible to deduce the relation between strain and stress for complex objects in terms of properties of the materials it is made of. Consider a simple helical spring that has one end attached to some fixed object, 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 end of the spring was displaced from its relaxed position. Hookes law states that F = k X or, equivalently, X = F k where k is a real number. Moreover, the formula holds when the spring is compressed. According to this formula, the graph of the applied force F as a function of the displacement X will be a line passing through the origin. Hookes law for a spring is often stated under the convention that F is the force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes F = − k X since the direction of the force is opposite to that of the displacement