Forces are often described as pushes or pulls. They can be due to phenomena such as gravity, magnetism, or anything else that causes a mass to accelerate. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials.
Classical mechanics
$\vec{F} = \frac{\mathrm{d}}{\mathrm{d}t}(m \vec{v})$
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In physics, a force is a push or pull that can cause an object with mass to accelerate. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Early Ideas on Motion The Greek philosophers, and Aristotle in particular were the first to propose that there are abstract principles governing nature Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object [1] Force has both magnitude and direction, making it a vector quantity. Direction is the information contained in the relative position of one point with respect to another point without the Distance information According to Newton's Second Law, an object will accelerate in proportion to the net force acting upon it and in inverse proportion to the object's mass. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the See also Vector addition A net force, F net = F 1 + F 2 + … (also known as a resultant force Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object An equivalent formulation is that the net force on an object is equal to the rate of change of momentum it experiences. A time derivative is a Derivative of a function with respect to Time, usually interpreted as the Rate of change of the value of the function In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product [2] Forces acting on three-dimensional objects may also cause them to rotate or deform, or result in a change in pressure. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation In Materials science, deformation is a change in the shape or size of an object due to an applied force. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The tendency of a force to cause rotation about an axis is termed torque. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about Deformation and pressure are the result of stress forces within an object. Stress is a measure of the average amount of Force exerted per unit Area. [3][4]

Since antiquity, scientists have used the concept of force in the study of stationary and moving objects. Statics is the branch of Mechanics concerned with the analysis of loads ( Force, torque/moment) on Physical systems in Static equilibrium In physics the term dynamics customarily refers to the time evolution of physical processes These studies culminated with the descriptions made by the third century BC philosopher Archimedes of how simple machines functioned. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer In Physics, especially Mechanics, a simple machine is a mechanical device that changes the direction or magnitude of a Force. The rules Archimedes determined for how forces interact in simple machines are still a part of modern physics. [5] Earlier descriptions of forces by Aristotle incorporated fundamental misunderstandings, which would not be resolved until the seventeenth century when Isaac Newton correctly described how forces behaved. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements [4] Newtonian descriptions of forces remained unchanged for nearly three hundred years.

Current understanding of quantum mechanics and the standard model of particle physics associate forces with the fundamental interactions accompanying the emission or absorption of gauge bosons. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature Only four fundamental interactions are known: in order of decreasing strength, they are: strong, electromagnetic, weak, and gravitational. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and In Physics, the electromagnetic force is the force that the Electromagnetic field exerts on electrically charged particles The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass [3] High-energy particle physics observations, in the 1970s and 1980s, confirmed that the weak and electromagnetic forces are expressions of a unified electroweak interaction. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the [6] Einstein in his Theory of General Relativity explained that gravity is an attribute of the curvature of space-time, though perceived as a force. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS

## Pre-Newtonian concepts

Aristotle famously described a force as anything which causes an object to undergo "unnatural motion"

Since antiquity, the concept of force has been recognized as integral to the functioning of each of the simple machines. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. In Physics, especially Mechanics, a simple machine is a mechanical device that changes the direction or magnitude of a Force. The mechanical advantage given by a simple machine allowed for less force to be used in exchange for that force acting over a greater distance. In Physics and Engineering, mechanical advantage (MA is the factor by which a mechanism multiplies the force put into it Analysis of the characteristics of forces ultimately culminated in the work of Archimedes who was especially famous for formulating a treatment of buoyant forces inherent in fluids. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is FLUID ( F ast L ight '''U'''ser '''I'''nterface D esigner is a graphical editor that is used to produce FLTK Source code [5]

Aristotle provided a philosophical discussion of the concept of a force as an integral part of Aristotelian cosmology. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language Physics (or "Physica" or "Physicae Auscultationes" meaning "lessons" is a key text in the philosophy of Aristotle. In Aristotle's view, the natural world held four elements that existed in "natural states". Nature, in the broadest sense is equivalent to the natural world, physical universe, material world or material universe. Many ancient philosophies used a set of archetypal classical "elements" to explain patterns in Nature. Aristotle believed that it was the natural state of objects with mass on Earth, such as the elements water and earth, to be motionless on the ground and that they tended towards that state if left alone. EARTH was a short-lived Japanese vocal trio which released 6 singles and 1 album between 2000 and 2001 He distinguished between the innate tendency of objects to find their "natural place" (e. g. , for heavy bodies to fall), which led to "natural motion", and unnatural or forced motion, which required continued application of a force. [7] This theory, based on the everyday experience of how objects move, such as the constant application of a force needed to keep a cart moving, had conceptual trouble accounting for the behavior of projectiles, such as the flight of arrows. A projectile is any object propelled through space by the exertion of a force which ceases after launch The place where forces were applied to projectiles was only at the start of the flight, and while the projectile sailed through the air, no discernible force acts on it. Aristotle was aware of this problem and proposed that the air displaced through the projectile's path provided the needed force to continue the projectile moving. This explanation demands that air is needed for projectiles and that, for example, in a vacuum, no projectile would move after the initial push. This vacuum means "absence of matter" or "an empty area or space" for the cleaning appliance see Vacuum cleaner. Additional problems with the explanation include the fact that air resists the motion of the projectiles. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a [8]

These shortcomings would not be fully explained and corrected until the seventeenth century work of Galileo Galilei, who was influenced by the late medieval idea that objects in forced motion carried an innate force of impetus. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher The theory of impetus was an antiquated auxiliary or secondary theory of Aristotelian dynamics, put forth initially to explain Projectile motion against Gravity Galileo constructed an experiment in which stones and cannonballs were both rolled down an incline to disprove the Aristotelian theory of motion early in the seventeenth century. The Greek Philosopher Aristotle ( 384 BC – 322 BC) developed many theories on the nature of Physics that are completely different He showed that the bodies were accelerated by gravity to an extent which was independent of their mass and argued that objects retain their velocity unless acted on by a force, for example friction. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In Physics, velocity is defined as the rate of change of Position. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e [9]

## Newtonian mechanics

Isaac Newton is the first person known to explicitly state the first, and the only, mathematical definition of force—as the time-derivative of momentum: F = dp / dt. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements In 1687, Newton went on to publish his Philosophiae Naturalis Principia Mathematica, which used concepts of inertia, force, and conservation to describe the motion of all objects. The Philosophiæ Naturalis Principia Mathematica ( Latin: "mathematical principles of natural philosophy" often Principia The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving In Physics, a conservation law states that a particular measurable property of an isolated Physical system does not change as the system evolves [10][4] In this work, Newton set out three laws of motion that to this day are the way forces are described in physics. [10]

Though Sir Isaac Newton's most famous equation is F=ma, he actually wrote down a different form for his second law of motion that used differential calculus. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Differential Calculus, a field in Mathematics, is the study of how functions change when their inputs change

### Newton's first law

Main article: Newton's first law

Newton's first law of motion states that objects continue to move in a state of constant velocity unless acted upon by an external net force or resultant force. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the See also Vector addition A net force, F net = F 1 + F 2 + … (also known as a resultant force [10] This law is an extension of Galileo's insight that constant velocity was associated with a lack of net force (see a more detailed description of this below). Newton proposed that every object with mass has an innate inertia that functions as the fundamental equilibrium "natural state" in place of the Aristotelian idea of the "natural state of rest". The vis insita or innate force of matter is a power of resisting by which every body as much as in it lies endeavors to preserve in its present state whether it be of rest or of moving That is, the first law contradicts the intuitive Aristotelian belief that a net force is required to keep an object moving with constant velocity. By making rest physically indistinguishable from non-zero constant velocity, Newton's first law directly connects inertia with the concept of relative velocities. Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial Specifically, in systems where objects are moving with different velocities, it is impossible to determine which object is "in motion" and which object is "at rest". In other words, to phrase matters more technically, the laws of physics are the same in every inertial frame of reference, that is, in all frames related by a Galilean transformation. In Physics, an inertial frame of reference is a Frame of reference which belongs to a set of frames in which Physical laws hold in the same and simplest The Galilean transformation is used to transform between the coordinates of two Reference frames which differ only by constant relative motion within the constructs of Newtonian

For example, while traveling in a moving vehicle at a constant velocity, the laws of physics do not change from being at rest. In Physics, velocity is defined as the rate of change of Position. A person can throw a ball straight up in the air and catch it as it falls down without worrying about applying a force in the direction the vehicle is moving. This is true even though another person who is observing the moving vehicle pass by also observes the ball follow a curving parabolic path in the same direction as the motion of the vehicle. In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular It is the inertia of the ball associated with its constant velocity in the direction of the vehicle's motion that ensures the ball continues to move forward even as it is thrown up and falls back down. From the perspective of the person in the car, the vehicle and every thing inside of it is at rest: It is the outside world that is moving with a constant speed in the opposite direction. Since there is no experiment that can distinguish whether it is the vehicle that is at rest or the outside world that is at rest, the two situations are considered to be physically indistinguishable. The equivalence principle Inertia therefore applies equally well to constant velocity motion as it does to rest.

The concept of inertia can be further generalized to explain the tendency of objects to continue in many different forms of constant motion, even those that are not strictly constant velocity. The rotational inertia of planet Earth is what fixes the constancy of the length of a day and the length of a year. This article is about the moment of inertia of a rotating object. A day (symbol d is a unit of Time equivalent to 24 Hours and the duration of a single Rotation of planet Earth with respect to the A year (from Old English gēr) is the time between two recurrences of an event related to the Orbit of the Earth around the Sun Albert Einstein extended the principle of inertia further when he explained that reference frames subject to constant acceleration, such as those free-falling toward a gravitating object, were physically equivalent to inertial reference frames. This is why, for example, astronauts experience weightlessness when in free-fall orbit around the Earth, and why Newton's Laws of Motion are more easily discernible in such environments. Weightlessness is a phenomenon experienced by people during Free-fall. If an astronaut places an object with mass in mid-air next to herself, it will remain stationary with respect to the astronaut due to its inertia. This is the same thing that would occur if the astronaut and the object were in intergalactic space with no net force of gravity acting on their shared reference frame. This principle of equivalence was one of the foundational underpinnings for the development of the general theory of relativity. The equivalence principle General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 [11]

### Newton's second law

Main article: Newton's second law

A modern statement of Newton's second law is a vector differential equation:[12]

$\vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m \vec{v})}{dt}$

where $\vec{p}$ is the momentum of the system. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product The $\vec{F}$ in the equation represents the net (vector sum) force; in equilibrium there is zero net force by definition, but (balanced) forces may be present nevertheless. In contrast, the second law states an unbalanced force acting on an object will result in the object's momentum changing over time. [10]

By assuming mass to be constant, Newton's second law can be expressed approximately as force equaling the product of mass m multiplied by acceleration $\vec{a}$. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Acceleration is the rate of change of velocity over time:

$\vec{F} =m\vec{a},$

sometimes called the "second most famous formula in physics". [13] Newton never stated explicitly the F=ma formula for which he is often credited.

Newton's second law asserts the proportionality of acceleration and mass to force. Accelerations can be defined through kinematic measurements. Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without However, while kinematics are well-described through reference frame analysis in advanced physics, there are still deep questions that remain as to what is the proper definition of mass. General relativity offers an equivalence between space-time and mass, but lacking a coherent theory of quantum gravity, it is unclear as to how or whether this connection is relevant on microscales. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Quantum gravity is the field of Theoretical physics attempting to unify Quantum mechanics, which describes three of the fundamental forces of nature With some justification, Newton's second law can be taken as a quantitative definition of mass by writing the law as an equality, the relative units of force and mass are fixed.

The use of Newton's second law as a definition of force has been disparaged in some of the more rigorous textbooks,[3][14] because it is essentially a mathematical truism. A truism is a claim that is so obvious or self-evident as to be hardly worth mentioning except as a reminder or as a rhetorical or literary device The equality between the abstract idea of a "force" and the abstract idea of a "changing momentum vector" ultimately has no observational significance because one cannot be defined without simultaneously defining the other. What a "force" or "changing momentum" is must either be referred to an intuitive understanding of our direct perception, or be defined implicitly through a set of self-consistent mathematical formulas. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of "force" include Ernst Mach, Clifford Truesdell and Walter Noll. Ernst Mach (max ( February 18, 1838 &ndash February 19, 1916) was an Austrian Physicist and Philosopher and Clifford Ambrose Truesdell III, ( February 18, 1919 &ndash January 14, 2000) was an American Mathematician, Natural philosopher Walter Noll (born January 7 1925 is a mathematician and Professor Emeritus at Carnegie Mellon University. [15]

Newton's second law can be used to measure the strength of forces. For instance, knowledge of the masses of planets along with the accelerations of their orbits allows scientists to calculate the gravitational forces on planets. A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star

### Newton's third law

Main article: Newton's third law

Newton's third law is a result of applying symmetry to situations where forces can be attributed to the presence of different objects. Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or For any two objects (call them 1 and 2), Newton's third law states that

$\vec{F}_{\mathrm{1 on 2}}=-\vec{F}_{\mathrm{2 on 1}}.$

This law implies that forces always occur in action-reaction pairs. [10] Any force that is applied to object 1 due to the action of object 2 is automatically accompanied by a force applied to object 2 due to the action of object 1. [16] If object 1 and object 2 are considered to be in the same system, then the net force on the system due to the interactions between objects 1 and 2 is zero since

$\vec{F}_{\mathrm{1 on 2}}+\vec{F}_{\mathrm{2 on 1}}=0$.

This means that in a closed system of particles, there are no internal forces that are unbalanced. A Closed system is a System in the state of being isolated from the environment That is, action-reaction pairs of forces shared between any two objects in a closed system will not cause the center of mass of the system to accelerate. The constituent objects only accelerate with respect to each other, the system itself remains unaccelerated. Alternatively, if an external force acts on the system, then the center of mass will experience an acceleration proportional to the magnitude of the external force divided by the mass of the system. [3]

Combining Newton's second and third laws, it is possible to show that the linear momentum of a system is conserved. In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Using

$\vec{F}_{\mathrm{1 on 2}} = \frac{d\vec{p}_{\mathrm{1 on 2}}}{dt} = -\vec{F}_{\mathrm{2 on 1}} = -\frac{d\vec{p}_{\mathrm{2 on 1}}}{dt}$

and integrating with respect to time, the equation:

$\Delta{\vec{p}_{\mathrm{1 on 2}}} = - \Delta{\vec{p}_{\mathrm{2 on 1}}}$

is obtained. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space For a system which includes objects 1 and 2,

$\sum{\Delta{\vec{p}}}=\Delta{\vec{p}_{\mathrm{1 on 2}}} + \Delta{\vec{p}_{\mathrm{2 on 1}}} = 0$

which is the conservation of linear momentum. [17] Using the similar arguments, it is possible to generalizing this to a system of an arbitrary number of particles. This shows that exchanging momentum between constituent objects will not affect the net momentum of a system. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define a system such that net momentum is never lost nor gained. [3]

## Descriptions

Free-body diagrams of an object on a flat surface and an inclined plane. A free body diagram is a pictorial representation often used by physicists and engineers to analyze the forces acting on a Free body. This article deals with the physical structure For related terms see Canal inclined plane, Cable railway, Funicular, or Fixed-wing Forces are resolved and added together to determine their magnitudes and the resultant.

Since forces are perceived as pushes or pulls, this can provide an intuitive understanding for describing forces. [4] As with other physical concepts (e. g. temperature), the intuitive understanding of forces is quantified using precise operational definitions that are consistent with direct observations and compared to a standard measurement scale. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature An operational definition is a demonstration of a process &mdash such as a Variable, term, or object &mdash relative in terms of the specific Process In Psychology and the Cognitive sciences perception is the process of attaining awareness or understanding of sensory Information. Measurement is the process of estimating the magnitude of some attribute of an object such as its length or weight relative to some standard ( unit of measurement) such as Through experimentation, it is determined that laboratory measurements of forces are fully consistent with the conceptual definition of force offered by Newtonian mechanics. A conceptual definition is an element of the scientific Research process, in which a specific concept is defined as a measurable occurrence

Forces act in a particular direction and have sizes dependent upon how strong the push or pull is. Because of these characteristics, forces are classified as "vector quantities". This means that forces follow a different set of mathematical rules that physical quantities that do not have direction (denoted scalar quantities). For example, when determining what happens when two forces act on the same object, it is necessary to know both the magnitude and the direction of both forces to calculate the result. In Mathematics, the resultant of two Monic polynomials P and Q over a field k is defined as the product If both of these pieces of information are not known, for each force, the situation is ambiguous. For example, if you know that two people are pulling on the same rope with a certain amount of strength but you do not know which direction either person is pulling, it is impossible to determine what the acceleration of the rope will be. The two people could be pulling against each other as in tug of war or the two people could be pulling in the same direction. Tug of war, tug o' war, or tug war, also known as rope pulling, is a Sport that directly puts two teams against each other in a test of strength In this simple one-dimensional example, without knowing the direction of the forces it is impossible to decide whether the net force is the result of adding the two force magnitudes or subtracting one from the other. In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Associating forces with vectors avoids such problems.

Historically, forces were first quantitatively investigated in conditions of static equilibrium where several forces canceled each other out. Such experiments demonstrate the crucial properties that forces are additive vector quantities: they have magnitude and direction. The magnitude of a mathematical object is its size a property by which it can be larger or smaller than other objects of the same kind in technical terms an Ordering Direction is the information contained in the relative position of one point with respect to another point without the Distance information [4] When two forces act on an object, the resulting force, the resultant, is the vector sum of the original forces. This is called the superposition principle. In Physics and Systems theory, the superposition principle, also known as superposition property, states that for all Linear systems The magnitude of the resultant varies from the difference of the magnitudes of the two forces to their sum, depending on the angle between their lines of action. The resultant force can be determined by following the parallelogram rule of vector addition: the addition of two vectors represented by sides of a parallelogram, gives an equivalent resultant vector which is equal in magnitude and direction to the transversal of the parallelogram. In Mathematics, the simplest form of the parallelogram law belongs to elementary Geometry. [3]

Free-body diagrams can be used as a convenient way to keep track of forces acting on a system. A free body diagram is a pictorial representation often used by physicists and engineers to analyze the forces acting on a Free body. Ideally, these diagrams are drawn with the angles and relative magnitudes of the force vectors preserved so that graphical vector addition can be done to determine the resultant. [18]

As well as being added, forces can also be resolved into independent components at right angles to each other. In Geometry and Trigonometry, a right angle is an angle of 90 degrees corresponding to a quarter turn (that is a quarter of a full circle A horizontal force pointing northeast can therefore be split into two forces, one pointing north, and one pointing east. Summing these component forces using vector addition yields the original force. Resolving force vectors into components of a set of basis vectors is often a more mathematically clean way to describe forces than using magnitudes and directions. Basis vector redirects here For basis vector in the context of crystals see Crystal structure. [19] This is because, for orthogonal components, the components of the vector sum are uniquely determined by the scalar addition of the components of the individual vectors. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i Orthogonal components are independent of each other; forces acting at ninety degrees to each other have no effect on each other. Choosing a set of orthogonal basis vectors is often done by considering what set of basis vectors will make the mathematics most convenient. Choosing a basis vector that is in the same direction as one of the forces is desirable, since that force would then have only one non-zero component. Force vectors can also be three-dimensional, with the third component at right-angles to the two other components. [3]

### Equilibria

Equilibrium occurs when the resultant force acting on an object is zero (that is, the vector sum of all forces is zero). There are two kinds of equilibrium: static equilibrium and dynamic equilibrium. A dynamic equilibrium occurs when two opposing Processes proceed at the same rate

#### Static equilibrium

Main article: statics

Static equilibrium was understood well before the invention of classical mechanics. Statics is the branch of Mechanics concerned with the analysis of loads ( Force, torque/moment) on Physical systems in Static equilibrium Objects which are at rest have zero net force acting on them. [20]

The simplest case of static equilibrium occurs when two forces are equal in magnitude but opposite in direction. For example, any object on a level surface is pulled (attracted) downward toward the center of the Earth by the force of gravity. At the same time, surface forces resist the downward force with equal upward force (called the normal force) and result in the object having a non-zero weight. In Physics, the normal force F_n\ (or in some books N) is the component perpendicular to the surface of contact of the Contact force In the Physical sciences weight is a Measurement of the gravitational Force acting on an object The situation is one of zero net force and no acceleration. [4]

Pushing against an object on a frictional surface can result in a situation where the object does not move because the applied force is opposed by static friction, generated between the object and the table surface. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e For a situation with no movement, the static friction force exactly balances the applied force resulting in no acceleration. The static friction increases or decreases in response to the applied force up to an upper limit determined by the characteristics of the contact between the surface and the object. [4]

A static equilibrium between two forces is the most usual way of measuring forces, using simple devices such as weighing scales and spring balances. A weighing scale (usually just "scale" in common usage except in Australian English where "scales" is more common is a Measuring instrument for A spring scale is a Weighing scale used to measure Force, such as the force of Gravity, exerted on a Mass or the force of a person's grip For example, an object suspended on a vertical spring scale experiences the force of gravity acting on the object balanced by a force applied by the "spring reaction force" which is equal to the object's weight. A spring scale is a Weighing scale used to measure Force, such as the force of Gravity, exerted on a Mass or the force of a person's grip Using such tools, some quantitative force laws were discovered: that the force of gravity is proportional to volume for objects of constant density (widely exploited for millennia to define standard weights); Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas pressure; and Hooke's law for springs. The density of a material is defined as its Mass per unit Volume: \rho = \frac{m}{V} Different materials usually have different In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Boyle's law (sometimes referred to as the Boyle-Mariotte law) is one of several Gas laws and a special case of the Ideal gas law. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the These were all formulated and experimentally verified before Isaac Newton expounded his three laws of motion. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the [3][4]

#### Dynamical equilibrium

Main article: Dynamics (physics)
Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's description of forces. In physics the term dynamics customarily refers to the time evolution of physical processes Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher

Dynamical equilibrium was first described by Galileo who noticed that certain assumptions of Aristotelian physics were contradicted by observations and logic. Galileo Galilei (15 February 1564 &ndash 8 January 1642 was a Tuscan ( Italian) Physicist, Mathematician, Astronomer, and Philosopher Observation is either an activity of a living being (such as a Human) which senses and assimilates the Knowledge of a Phenomenon, or the recording of data Logic is the study of the principles of valid demonstration and Inference. Galileo realized that simple velocity addition demands that the concept of an "absolute rest frame" did not exist. Galilean invariance or Galilean relativity is a Principle of relativity which states that the fundamental laws of physics are the same in all Inertial In Special relativity the rest frame of a particle is the coordinate system ( Frame of reference) in which the particle is at rest Galileo concluded that motion in a constant velocity was completely equivalent to rest. In Physics, velocity is defined as the rate of change of Position. This was contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally approached. Simple experiments showed that Galileo's understanding of the equivalence of constant velocity and rest to be correct. For example, if a mariner dropped a cannonball from the crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe, the falling cannonball would land behind the foot of the mast of a moving ship. However, when this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the cannonball knows to travel with the ship despite being separated from it. Since there is no forward horizontal force being applied on the cannonball as it falls, the only conclusion left is that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no force is required to keep the cannonball moving at the constant forward velocity. [9]

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant force). This is the definition of dynamical equilibrium: when all the forces on an object balance but it still moves at a constant velocity.

A simple case of dynamical equilibrium occurs in constant velocity motion across a surface with kinetic friction. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In such a situation, a force is applied in the direction of motion while the kinetic friction force exactly opposes the applied force. This results in a net zero force, but since the object started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle misinterpreted this motion as being caused by the applied force. However, when kinetic friction is taken into consideration it is clear that there is no net force causing constant velocity motion. [3]

### Feynman diagrams

Main article: Feynman diagrams
A Feynman diagram for the decay of a neutron into a proton. Motivation and history When calculating Scattering cross sections in Particle physics, the interaction between particles can be described The W boson is between two vertices indicating a repulsion. The W and Z bosons are the Elementary particles that mediate the Weak force.

In modern particle physics, forces and the acceleration of particles are explained as the exchange of momentum-carrying gauge bosons. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature With the development of quantum field theory and general relativity, it was realized that “force” is a redundant concept arising from conservation of momentum (4-momentum in relativity and momentum of virtual particles in quantum electrodynamics). In quantum field theory (QFT the forces between particles are mediated by other particles General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Special relativity, four-momentum is the generalization of the classical three-dimensional Momentum to four-dimensional Spacetime. In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. The conservation of momentum, from Noether's theorem, can be directly derived from the symmetry of space and so is usually considered more fundamental than the concept of a force. Noether's theorem (also known as Noether's first theorem) states that any differentiable symmetry of the action of a physical system has Symmetry in physics refers to features of a Physical system that exhibit the property of Symmetry —that is under certain transformations, aspects of these Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another Thus the currently known fundamental forces are considered more accurately to be “fundamental interactions”. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms [6] When particle A emits (creates) or absorbs (annihilates) particle B, a force accelerates particle A in response to the momentum of particle B, thereby conserving momentum as a whole. This description applies for all forces arising from fundamental interactions. While sophisticated mathematical descriptions are needed to predict, in full detail, the nature of such interactions, there is a conceptually simple way to describe such interactions through the use of Feynman diagrams. In a Feynman diagram, each matter particle is represented as a straight line (see world line) traveling through time which normally increases up or to the right in the diagram. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. Matter and anti-matter particles are identical except for their direction of propagation through the Feynman diagram. World lines of particles intersect at interaction vertices, and the Feynman diagram represents any force arising from an interaction as occurring at the vertex with an associated instantaneous change in the direction of the particle world lines. Gauge bosons are emitted away from the vertex as wavy lines (similar to waves) and, in the case of virtual particle exchange, are absorbed at an adjacent vertex. In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty When the gauge bosons are represented in a Feynman diagram as existing between two interacting particles, this represents a repulsive force. When the gauge bosons are represented in a Feynman diagram as existing surrounding the two interacting particles, this represents an attractive force. [21]

The utility of Feynman diagrams is that other types of physical phenomena that are part of the general picture of fundamental interactions but are conceptually separate from forces can also be described using the same rules. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms For example, a Feynman diagram can describe in succinct detail how a neutron decays into an electron, proton, and neutrino: an interaction mediated by the same gauge boson that is responsible for the weak nuclear force. This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. In Nuclear physics, beta decay is a type of Radioactive decay in which a Beta particle (an Electron or a Positron) is emitted The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The proton ( Greek πρῶτον / proton "first" is a Subatomic particle with an Electric charge of one positive Neutrinos are Elementary particles that travel close to the Speed of light, lack an Electric charge, are able to pass through ordinary matter almost The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature While the Feynman diagram for this interaction has similar features to a repulsive interaction, the decay is more complicated than a simple "repulsive force". [21]

### Special relativity

In the special theory of relativity mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός When an object's velocity increases so does its energy and hence its mass equivalent (inertia). It thus requires more force to accelerate it the same amount than it did at a lower velocity. Newton's second law $\vec{F} = \mathrm{d}\vec{p}/\mathrm{d}t$ remains formally valid. [22] But in order to be conserved, momentum must be redefined as:

$\vec{p} = \frac{m\vec{v}}{\sqrt{1 - v^2/c^2}}$

where

v is the velocity and
c is the speed of light.

The relativistic expression relating force and acceleration for a particle with non-zero rest mass $m\,$ moving in the $x\,$ direction is:

$F_x = \gamma^3 m a_x \,$
$F_y = \gamma m a_y \,$
$F_z = \gamma m a_z \,$

where the Lorentz factor

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$[23]

Here a constant force does not produce a constant acceleration, but an ever decreasing acceleration as the object approaches the speed of light. The Lorentz factor or Lorentz term appears in several equations in Special relativity, including Time dilation, Length contraction, and the Note that γ is undefined for an object with a non zero rest mass at the speed of light, and the theory yields no prediction at that speed. In

One can however restore the form of

$F^\mu = mA^\mu \,$

for use in relativity through the use of four-vectors. In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space. This relation is correct in relativity when Fμ is the four-force, m is the invariant mass, and Aμ is the four-acceleration. In the Special theory of relativity four-force is a Four-vector that replaces the classical Force; the four-force is the four-vector defined as the change In Special relativity, four-acceleration is a Four-vector and is defined as the change in Four-velocity over the particle's Proper time: [24]

## Fundamental models

All the forces in the universe are based on four fundamental forces. The strong and weak forces act only at very short distances, and are responsible for holding certain nucleons and compound nuclei together. In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom The electromagnetic force acts between electric charges and the gravitational force acts between masses. Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object All other forces are based on the existence of the four fundamental interactions. For example, friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces, and the Pauli Exclusion Principle,[25] which does not allow atoms to pass through each other. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. The forces in springs, modeled by Hooke's law, are also the result of electromagnetic forces and the Exclusion Principle acting together to return the object to its equilibrium position. A spring is a flexible elastic object used to store mechanical Energy. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating frames of reference. See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to [3]

The development of fundamental theories for forces proceeded along the lines of unification of disparate ideas. In Physics, a unified field theory is a type of Field theory that allows all of the Fundamental forces between Elementary particles to be written For example, Isaac Newton unified the force responsible for objects falling at the surface of the Earth with the force responsible for the orbits of celestial mechanics in his universal theory of gravitation. Michael Faraday and James Clerk Maxwell demonstrated that electric and magnetic forces were unified through one consistent theory of electromagnetism. Michael Faraday, FRS ( September 22 1791 – August 25 1867) was an English James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. In the twentieth century, the development of quantum mechanics led to a modern understanding that the first three fundamental forces (all except gravity) are manifestations of matter (fermions) interacting by exchanging virtual particles called gauge bosons. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. In Physics, a virtual particle is a particle that exists for a limited time and space introducing uncertainty in their energy and momentum due to the Heisenberg Uncertainty In Particle physics, gauge bosons are Bosonic particles that act as carriers of the fundamental forces of nature [26] This standard model of particle physics posits a similarity between the forces and led scientists to predict the unification of the weak and electromagnetic forces in electroweak theory subsequently confirmed by observation. The Standard Model of Particle physics is a theory that describes three of the four known Fundamental interactions together with the Elementary particles Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the The complete formulation of the standard model predicts an as yet unobserved Higgs mechanism, but observations such as neutrino oscillations indicate that the standard model is incomplete. The Higgs mechanism is Spontaneous symmetry breaking in a Gauge theory. Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvo whereby a Neutrino created with a specific Lepton A grand unified theory allowing for the combination of the electroweak interaction with the strong force is held out as a possibility with candidate theories such as supersymmetry proposed to accommodate some of the outstanding unsolved problems in physics. Grand Unification, grand unified theory, or GUT refers to any of several very similar unified field theories or models in Physics that In Particle physics, supersymmetry (often abbreviated SUSY) is a Symmetry that relates elementary particles of one spin to another particle that This is a list of some of the major unsolved problems in Physics. Physicists are still attempting to develop self-consistent unification models that would combine all four fundamental interactions into a theory of everything. A theory of everything ( TOE) is a putative Theory of Theoretical physics that fully explains and links together all known physical phenomena Einstein tried and failed at this endeavor, but currently the most popular approach to answering this question is string theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings [6]

### Gravity

Main article: Gravity
An initially stationary object which is allowed to fall freely under gravity drops a distance which is proportional to the square of the elapsed time. Gravitation is a natural Phenomenon by which objects with Mass attract one another An image was taken 20 flashes per second. During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12 mm); by 2/20ths it has dropped a total of 4 units; by 3/20ths, 9 units and so on.

What we now call gravity was not identified as a universal force until the work of Isaac Newton. Before Newton, the tendency for objects to fall towards the Earth was not understood to be related to the motions of celestial objects. Galileo was instrumental in describing the characteristics of falling objects by determining that the acceleration of every object in free-fall was constant and independent of the mass of the object. Free fall is motion with no Acceleration other than that provided by Gravity. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object Today, this acceleration due to gravity towards the surface of the Earth is usually designated as $\vec{g}$ and has a magnitude of about 9. 81 meters per second squared (this measurement is taken from sea level and may vary depending on location), and points toward the center of the Earth. The metre or meter is a unit of Length. It is the basic unit of Length in the Metric system and in the International The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units [27] This observation means that the force of gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an object that has a mass of m will experience a force:

$\vec{F} = m\vec{g}$

In free-fall, this force is unopposed and therefore the net force on the object is the force of gravity. For objects not in free-fall, the force of gravity is opposed by the weight of the object. In the Physical sciences weight is a Measurement of the gravitational Force acting on an object For example, a person standing on the ground experiences zero net force, since the force of gravity is balanced by the weight of the person that is manifested by a normal force exerted on the person by the ground. In Physics, the normal force F_n\ (or in some books N) is the component perpendicular to the surface of contact of the Contact force [3]

Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which Aristotle had assumed were in a natural state of constant motion, with falling motion observed on the Earth. He proposed a law of gravity that could account for the celestial motions that had been described earlier using Kepler's Laws of Planetary Motion. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. [28]

Newton came to realize that the effects of gravity might be observed in different ways at larger distances. In particular, Newton determined that the acceleration of the moon around the Earth could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an inverse square law. In Physics, an inverse-square law is any Physical law stating that some physical Quantity or strength is inversely proportional Further, Newton realized that the mass of the gravitating object directly affected the acceleration due to gravity. [28] Combining these ideas gives a formula that relates the mass of the Earth ($M_\oplus$), the radius of the Earth ($R_\oplus$) to the acceleration due to gravity:

$\vec{g}=-\frac{GM_\oplus}{{R_\oplus}^2} \hat{r}$

where the vector direction is given by $\hat{r}$ which is the unit vector directed outward from the center of the Earth. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length [10]

In this equation, a dimensional constant G is used to describe the relative strength of gravity. This constant has come to be known as Newton's Universal Gravitation Constant,[29] though it was of an unknown value in Newton's lifetime. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass Not until 1798 was Henry Cavendish able to make the first measurement of G using a torsion balance; this was widely reported in the press as a measurement of the mass of the Earth since knowing the G could allow one to solve for the Earth's mass given the above equation. Henry Cavendish, FRS (10 October 1731 - 24 February 1810 was a British Scientist noted for his discovery of Hydrogen or what he called "inflammable A torsion spring is a spring that works by torsion or twisting that is a flexible elastic object that stores Mechanical energy when it is twisted Newton, however, realized that since all celestial bodies followed the same laws of motion, his law of gravity had to be universal. In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Succinctly stated, Newton's Law of Gravitation states that the force on an object of mass m1 due to the gravitational pull of mass m2 is

$\vec{F}=-\frac{Gm_{1}m_{2}}{r^2} \hat{r}$

where r is the distance between the two objects' centers of mass and $\hat{r}$ is the unit vector pointed in the direction away from the center of the first object toward the center of the second object. Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass [10]

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion within the solar system until the twentieth century. During that time, sophisticated methods of perturbation analysis[30] were invented to calculate the deviations of orbits due to the influence of multiple bodies on a planet, moon, comet, or asteroid. This article describes perturbation theory as a general mathematical method In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star A planet, as defined by the International Astronomical Union (IAU is a celestial body Orbiting a Star or stellar remnant that is A comet is a small Solar System body that orbits the Sun and when close enough to the Sun exhibits a visible coma (atmosphere or a tail — Asteroids, sometimes called Minor planets or planetoids', are bodies—primarily of the inner Solar System —that are smaller than planets but These techniques are so powerful that they can be used to predict precisely the motion of celestial bodies to an arbitrary precision at any length of time in the future. The formalism was exact enough to allow mathematicians to predict the existence of the planet Neptune before it was observed. Neptune ( English|AmE] ] is the eighth and farthest Planet from the Sun in the Solar System. [31]

It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would explain the discrepancies; however, despite some early indications, no such planet could be found. Vulcan was a small Planet proposed to exist in an Orbit between Mercury and the Sun in a 19th-century hypothesis When Albert Einstein finally formulated his theory of general relativity (GR) he turned his attention to the problem of Mercury's orbit and found that his theory added a correction which could account for the discrepancy. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 At its introduction in 1915 the general theory of relativity did not have a solid empirical foundation This was the first time that Newton's Theory of Gravity had been shown to be less correct than an alternative. [32]

Since then, and so far, general relativity has been acknowledged as the theory which best explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in gravitational fields travel under their own inertia in straight lines through curved space-time – defined as the shortest space-time path between two space-time events. In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 From the perspective of the object, all motion occurs as if there were no gravitation whatsoever. It is only when observing the motion in a global sense that the curvature of space-time can be observed and the force is inferred from the object's curved path. Thus, the straight line path in space-time is seen as a curved line in space, and it is called the ballistic trajectory of the object. External ballistics is the part of the science of Ballistics that deals with the behaviour of a non-powered projectile in flight Trajectory is the path a moving object follows through space The object might be a Projectile or a Satellite, for example For example, a basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field. Basketball is a team Sport in which two teams of five active players each try to score points against one another by propelling a ball through a 10 feet (3 m In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly curved (with the radius of curvature of the order of few light-years). Radius of curvature is a term characterizing the measure of how curved or bent a given Curve or Surface is A light-year or light year (symbol ly) is a unit of Length, equal to just under ten trillion Kilometres As defined by The time derivative of the changing momentum of the object is what we label as "gravitational force". [3]

### Electromagnetic forces

Main article: Electromagnetic force

The electrostatic force was first described in 1784 by Coulomb as a force which existed intrinsically between two charges. In Physics, the electromagnetic force is the force that the Electromagnetic field exerts on electrically charged particles ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction. [33] The properties of the electrostatic force were that it varied as an inverse square law directed in the radial direction, was both attractive and repulsive (there was intrinsic polarity), was independent of the mass of the charged objects, and followed the law of superposition. In Physics, an inverse-square law is any Physical law stating that some physical Quantity or strength is inversely proportional In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by The law of superposition (or the principle of superposition) is a key axiom based on observations of Natural history that is a foundational principle of sedimentary Coulomb's Law unifies all these observations into one succinct statement. ---- Bold text Coulomb's law', developed in the 1780s by French physicist Charles Augustin de Coulomb, may be stated in scalar form [34]

Subsequent mathematicians and physicists found the construct of the electric field to be useful for determining the electrostatic force on an electric charge at any point in space. In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can The electric field was based on using a hypothetical "test charge" anywhere in space and then using Coulomb's Law to determine the electrostatic force. In physical theories, a test particle is an idealized model of an object whose physical properties (usually Mass, Charge, or size) are assumed [35] Thus the electric field anywhere in space is defined as

$\vec{E} = {\vec{F} \over{q}}$

where q is the magnitude of the hypothetical test charge.

Meanwhile, the Lorentz force of magnetism was discovered to exist between two electric currents. In Physics, the Lorentz force is the Force on a Point charge due to Electromagnetic fields It is given by the following equation In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. Electric current is the flow (movement of Electric charge. The SI unit of electric current is the Ampere. It has the same mathematical character as Coulomb's Law with the proviso that like currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be used to determine the magnetic force on an electric current at any point in space. In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges In this case, the magnitude of the magnetic field was determined to be

$B = {F \over{I \ell}}$

where I is the magnitude of the hypothetical test current and $\ell$ is the length of hypothetical wire through which the test current flows. The magnetic field exerts a force on all magnets including, for example, those used in compasses. A magnet (from Greek grc μαγνήτης λίθος " Magnesian stone" is a material or object that produces a Magnetic field. A compass, magnetic compass or mariner's compass is a navigational instrument for determining direction relative to the earth's Magnetic poles It consists The fact that the Earth's magnetic field is aligned closely with the orientation of the Earth's axis causes compass magnets to become oriented because of the magnetic force pulling on the needle. Earth 's magnetic field (and the surface magnetic field) is approximately a Magnetic dipole, with one pole near the North pole (see A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation

Through combining the definition of electric current as the time rate of change of electric charge, a rule of vector multiplication called Lorentz's Law describes the force on a charge moving in an magnetic field. [35] The connection between electricity and magnetism allows for the description of a unified electromagnetic force that acts on a charge. This force can be written as a sum of the electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field). Fully stated, this is the law:

$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

where $\vec{F}$ is the electromagnetic force, q is the magnitude of the charge of the particle, $\vec{E}$ is the electric field, $\vec{v}$ is the velocity of the particle which is crossed with the magnetic field ($\vec{B}$). In Physics, velocity is defined as the rate of change of Position. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which

The origin of electric and magnetic fields would not be fully explained until 1864 when James Clerk Maxwell unified a number of earlier theories into a succinct set of four equations. James Clerk Maxwell (13 June 1831 &ndash 5 November 1879 was a Scottish mathematician and theoretical physicist. These "Maxwell Equations" fully described the sources of the fields as being stationary and moving charges, and the interactions of the fields themselves. In Classical electromagnetism, Maxwell's equations are a set of four Partial differential equations that describe the properties of the electric This led Maxwell to discover that electric and magnetic fields could be "self-generating" through a wave that traveled at a speed which he calculated to be the speed of light. In Physics, especially Quantum mechanics, the Schrödinger equation is an equation that describes how the Quantum state of a Physical system This insight united the nascent fields of electromagnetic theory with optics and led directly to a complete description of the electromagnetic spectrum. The electromagnetic (EM spectrum is the range of all possible Electromagnetic radiation frequencies [36]

However, attempting to reconcile electromagnetic theory with two observations, the photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome. Introduction When a Metallic surface is exposed to Electromagnetic radiation above a certain threshold Frequency, the light is absorbed and Electrons The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe was a prediction of early 20th century Classical physics that an ideal Black body at Through the work of leading theoretical physicists, a new theory of electromagnetism was developed using quantum mechanics. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This final modification to electromagnetic theory ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic phenomena as being mediated by wave particles known as photons. Quantum electrodynamics ( QED) is a relativistic Quantum field theory of Electrodynamics. In Physics, the photon is the Elementary particle responsible for electromagnetic phenomena In QED, photons are the fundamental exchange particle which described all interactions relating to electromagnetism including the electromagnetic force. [37]

It is a common misconception to ascribe the stiffness and rigidity of solid matter to the repulsion of like charges under the influence of the electromagnetic force. Solid-state physics, the largest branch of Condensed matter physics, is the study of rigid Matter, or Solids The bulk of solid-state physics theory and However, these characteristics actually result from the Pauli Exclusion Principle. The Pauli exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925 Since electrons are fermions, they cannot occupy the same quantum mechanical state as other electrons. The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J In Particle physics, fermions are particles which obey Fermi-Dirac statistics; they are named after Enrico Fermi. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system When the electrons in a material are densely packed together, there are not enough lower energy quantum mechanical states for them all, so some of them must be in higher energy states. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός This means that it takes energy to pack them together. While this effect is manifested macroscopically as a structural "force", it is technically only the result of the existence of a finite set of electron states.

### Nuclear forces

Main article: Nuclear force

There are two "nuclear forces" which today are usually described as interactions that take place in quantum theories of particle physics. The nuclear force (or nucleon-nucleon interaction or residual strong force) is the force between two or more Nucleons It is responsible for In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them The strong nuclear force[38] is the force responsible for the structural integrity of atomic nuclei while the weak nuclear force[39] is responsible for the decay of certain nucleons into leptons and other types of hadrons. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom The weak interaction (often called the weak force or sometimes the weak nuclear force) is one of the four Fundamental interactions of nature In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. Leptons are a family of fundamental Subatomic particles comprising the Electron, the Muon, and the Tauon (or tau particle as well as their In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( [3]

The strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). Interaction is a kind of action that occurs as two or more objects have an Effect upon one another In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the Quantum chromodynamics (abbreviated as QCD is a theory of the Strong interaction ( color force a Fundamental force describing the interactions of the [40] The strong force is the fundamental force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the In Physics, a quark (kwɔrk kwɑːk or kwɑːrk is a type of Subatomic particle. to most kinds of particles, there is an associated antiparticle with the same Mass and opposite Electric charge. Gluons ( Glue and the suffix -on) are Elementary particles that cause Quarks to interact and are indirectly responsible for the The strong interaction is the most powerful of the four fundamental forces.

The strong force only acts directly upon elementary particles. However, a residual of the force is observed between hadrons (the best known example being the force that acts between nucleons in atomic nuclei) as the nuclear force. In Particle physics, a hadron ( from the ἁδρός hadrós, " stout, thick " ( In Physics a nucleon is a collective name for two Baryons the Neutron and the Proton. The nuclear force (or nucleon-nucleon interaction or residual strong force) is the force between two or more Nucleons It is responsible for Here the strong force acts indirectly, transmitted as gluons which form part of the virtual pi and rho mesons which classically transmit the nuclear force (see this topic for more). In Particle physics, a meson is a strongly interacting Boson &mdashthat is a Hadron with integer spin. The failure of many searches for free quarks has shown that the elementary particles affected are not directly observable. This phenomenon is called colour confinement. Color confinement, often called just confinement, is the Physics phenomenon that Color charged particles (such as Quarks cannot be isolated singularly

The weak force is due to the exchange of the heavy W and Z bosons. The W and Z bosons are the Elementary particles that mediate the Weak force. Its most familiar effect is beta decay (of neutrons in atomic nuclei) and the associated radioactivity. In Nuclear physics, beta decay is a type of Radioactive decay in which a Beta particle (an Electron or a Positron) is emitted This article is a discussion of neutrons in general For the specific case of a neutron found outside the nucleus see Free neutron. The nucleus of an Atom is the very dense region consisting of Nucleons ( Protons and Neutrons, at the center of an atom Radioactive decay is the process in which an unstable Atomic nucleus loses energy by emitting ionizing particles and Radiation. The word "weak" derives from the fact that the field strength is some 1013 times less than that of the strong force. In particle physics the strong interaction, or strong force, or color force, holds Quarks and Gluons together to form Protons and Still, it is stronger than gravity over short distances. A consistent electroweak theory has also been developed which shows that electromagnetic forces and the weak force are indistinguishable at a temperatures in excess of approximately 1015 Kelvin. In Particle physics, the electroweak interaction is the unified description of two of the four Fundamental interactions of nature Electromagnetism and the The kelvin (symbol K) is a unit increment of Temperature and is one of the seven SI base units The Kelvin scale is a thermodynamic Such temperatures have been probed in modern particle accelerators and show the conditions of the universe in the early moments of the Big Bang. The Universe is defined as everything that Physically Exists: the entirety of Space and Time, all forms of Matter, Energy The Big Bang is the cosmological model of the Universe that is best supported by all lines of scientific evidence and Observation.

## Non-fundamental models

Some forces can be modeled by making simplifying assumptions about the physical conditions. In such situations, idealized models can be utilized to gain physical insight.

### Normal force

Fn represents the normal force exerted on the object. In Physics, the normal force F_n\ (or in some books N) is the component perpendicular to the surface of contact of the Contact force
Main article: Normal force

The normal force is the surface force which acts normal to the surface interface between two objects. In Physics, the normal force F_n\ (or in some books N) is the component perpendicular to the surface of contact of the Contact force [41] The normal force, for example, is responsible for the structural integrity of tables and floors as well as being the force that responds whenever an external force pushes on a solid object. An example of the normal force in action is the impact force of an object crashing into an immobile surface. This force is proportional to the square of that object's velocity due to the conservation of energy and the work energy theorem when applied to completely inelastic collisions. In Physics, the law of conservation of energy states that the total amount of Energy in an isolated system remains constant and cannot be created although it may In Physics, mechanical work is the amount of Energy transferred by a Force. An inelastic collision is a Collision in which kinetic energy is not conserved (see elastic collision) [3]

### Friction

Main article: Friction

Friction is a surface force that opposes motion. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e The frictional force is directly related to the normal force which acts to keep two solid objects separated at the point of contact. In Physics, the normal force F_n\ (or in some books N) is the component perpendicular to the surface of contact of the Contact force There are two broad classifications of frictional forces: static friction and kinetic friction. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e

The static friction force (Fsf) will exactly oppose forces applied to an object parallel to a surface contact up to the limit specified by the coefficient of static friction (μsf) multiplied by the normal force (FN). The coefficient of friction is a Dimensionless quantity symbolized by the Greek letter μ and is used to approximate the Force of Friction. In other words the magnitude of the static friction force satisfies the inequality:

$0 \le F_{sf} \le \mu_{sf} F_N$.

The kinetic friction force (Fkf) is independent of both the forces applied and the movement of the object. Thus, the magnitude of the force is equal to

Fkf = μkfFN,

where μkf is the coefficient of kinetic friction. The coefficient of friction is a Dimensionless quantity symbolized by the Greek letter μ and is used to approximate the Force of Friction. For most surface interfaces, the coefficient of kinetic friction is less than the coefficient of static friction. [3]

### Continuum mechanics

When the drag force (Fd) associated with air resistance becomes equal in magnitude to the force of gravity on a falling object (Fg), the object reaches a state of dynamical equilibrium at terminal velocity. A free falling object achieves its terminal velocity when the downward force of gravity ( Fg)equals the upward force of drag ( Fd)

Newton's laws and Newtonian mechanics in general were first developed to describe how forces affect idealized point particles rather than three-dimensional objects. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a Stress is a measure of the average amount of Force exerted per unit Area. A point particle (or point-like, often spelled pointlike) is an idealized object heavily used in Physics. However, in real life, matter has extended structure and forces that act on one part of an object might affect other parts of an object. For situations where lattice holding together the atoms in an object is able to flow, contract, expand, or otherwise change shape, the theories of continuum mechanics describe the way forces affect the material. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e For example, in extended fluids, differences in pressure result in forces being directed along the pressure gradients as follows:

$\frac{\vec{F}}{V} = - \vec{\nabla} P$

where V is the volume of the object in the fluid and P is the scalar function that describes the pressure at all locations in space. Fluid mechanics is the study of how Fluids move and the Forces on them Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point Pressure gradients and differentials result in the buoyant force for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated with aerodynamics and flight. In Physics, buoyancy ( BrE IPA: /ˈbɔɪənsi/ is the upward Force on an object produced by the surrounding liquid or gas in which it is Wind is the flow of Air or other Gases that compose an Atmosphere (including but not limited to the Earth's) Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes the effects other systems have on the atmosphere and the effects of the atmosphere In the context of a Fluid flow relative to a body the lift force is the component of the Aerodynamic force that is Perpendicular to the flow Flight is the process by which an object achieves sustained movement either through the Air (or movement beyond Earth's atmosphere, in the case of [3]

A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a body force that resists the motion of an object through a fluid due to viscosity. In Fluid dynamics dynamic pressure (indicated with q, or Q, and sometimes called velocity pressure) is the quantity defined by Viscosity is a measure of the resistance of a Fluid which is being deformed by either Shear stress or Extensional stress. For so-called "Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:

$\vec{F}_d = - b \vec{v} \,$

where:

b is a constant that depends on the properties of the fluid and the dimensions of the object (usually the cross-sectional area), and
$\vec{v}$ is the velocity of the object. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a [3]

More formally, forces in continuum mechanics are fully described by a stress tensor with terms that are roughly defined as

$\sigma = \frac{F}{A}$

where A is the relevant cross-sectional area for the volume for which the stress-tensor is being calculated. Continuum mechanics is a branch of Mechanics that deals with the analysis of the Kinematics and mechanical behavior of materials modeled as a continuum e Stress is a measure of the average amount of Force exerted per unit Area. History The word tensor was introduced in 1846 by William Rowan Hamilton to describe the norm operation in a certain type of algebraic system (eventually This formalism includes pressure terms associated with forces that act normal to the cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with forces that act parallel to the cross-sectional area (the off-diagonal elements). In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero A shear stress, denoted \tau\ ( Tau) is defined as a stress which is applied Parallel or tangential to a face of a material The stress tensor accounts for forces that cause all deformations including also tensile stresses and compressions. Stress is a measure of the average amount of Force exerted per unit Area.

### Tension

Main article: Tension (physics)

Tension forces can be modeled using ideal strings which are massless, frictionless, unbreakable, and unstretchable. In Physics String Tension is the magnitude of the pulling force exerted by a string cable chain or similar object on another object They can be combined with ideal pulleys which allow ideal strings to switch physical direction. A pulley (also called a sheave or block) is a Wheel with a groove between two Flanges around its Circumference Ideal strings transmit tension forces instantaneously in action-reaction pairs so that if two objects are connected by an ideal string, any force directed along the string by the first object is accompanied by a force directed along the string in the opposite direction by the second object. [42] By connecting the same string multiple times to the same object through the use of a set-up that uses movable pulleys, the tension force on a load can be multiplied. For every string that acts on a load, another factor of the tension force in the string acts on the load. However, even though such machines allow for an increase in force, there is a corresponding increase in the length of string that must be displaced in order to move the load. In Physics, especially Mechanics, a simple machine is a mechanical device that changes the direction or magnitude of a Force. In Physics and Engineering, mechanical advantage (MA is the factor by which a mechanism multiplies the force put into it These tandem effects result ultimately in the conservation of mechanical energy since the work done on the load is the same no matter how complicated the machine. Energy conservation is the practice of decreasing the quantity of energy used [43][3]

### Elastic force

Fk is the force that responds to the load on the spring. A material is said to be elastic if it deforms under stress (e In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the

An elastic force acts to return a spring to its natural length. A spring is a flexible elastic object used to store mechanical Energy. An ideal spring is taken to be massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push when contracted, or pull when extended, in proportion to the displacement of the spring from its equilibrium position. [44] This linear relationship was described by Robert Hooke in 1676, for whom Hooke's law is named. Robert Hooke, FRS (18 July 1635 – 3 March 1703 was an English Natural philosopher and Polymath who played an important role in the In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the If Δx is the displacement, the force exerted by an ideal spring is equal to:

$\vec{F}=-k \Delta \vec{x}$

where k is the spring constant (or force constant), which is particular to the spring. The minus sign accounts for the tendency of the elastic force to act in opposition to the applied load. [3]

### Centripetal force

Main article: Centripetal force

For an object accelerating in circular motion, the unbalanced force acting on the object is equal to[45]

$\vec{F} = - \frac{mv^2 \hat{r}}{r}$

where m is the mass of the object, v is the velocity of the object and r is the distance to the center of the circular path and $\hat{r}$ is the unit vector pointing in the radial direction outwards from the center. The centripetal force is the external force required to make a body follow a curved path In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length This means that the unbalanced centripetal force felt by any object is always directed toward the center of the curving path. Such forces act perpendicular to the velocity vector associated with the motion of an object, and therefore do not change the speed of the object (magnitude of the velocity), but only the direction of the velocity vector. Speed is the rate of motion, or equivalently the rate of change in position often expressed as Distance d traveled per unit of The unbalanced force that accelerates an object can be resolved into a component that is perpendicular to the path, and one that is tangential to the path. This yields both the tangential force which accelerates the object by either slowing it down or speeding it up and the radial (centripetal) force which changes its direction. [3]

### Fictitious forces

Main article: Fictitious forces

There are forces which are frame dependent, meaning that they appear due to the adoption of non-Newtonian (that is, non-inertial) reference frames. A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent Force that acts on all masses in a non-inertial See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to See also Inertial frame A frame of reference in Physics, may refer to a Coordinate system or set of axes within which to Such forces include the centrifugal force and the Coriolis force. In physics the Coriolis effect is an apparent deflection of moving objects when they are viewed from a Rotating frame of reference. [46] These forces are considered fictitious because they do not exist in frames of reference that are not accelerating. [3] In general relativity, gravity becomes a fictitious force that arises in situations where spacetime deviates from a flat geometry. General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916 As an extension, Kaluza-Klein theory and string theory ascribe electromagnetism and the other fundamental forces respectively to the curvature of differently scaled dimensions, which would ultimately imply that all forces are fictitious. In Physics, Kaluza–Klein theory (or KK theory, for short is a model that seeks to unify the two fundamental forces of Gravitation and String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

## Rotations and torque

Relationship between force (F), torque (τ), and momentum vectors (p and L) in a rotating system. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position
Main article: Torque

Forces that cause extended objects to rotate are associated with torques. A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation A torque (τ in Physics, also called a moment (of force is a pseudo- vector that measures the tendency of a force to rotate an object about Mathematically, the torque on a particle is defined as the cross-product:

$\vec{\tau} = \vec{r} \times \vec{F}$

where

$\vec{r}$ is the particle's position vector relative to a pivot
$\vec{F}$ is the force acting on the particle. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which A position, location or radius vector is a vector which represents the position of an object in space in relation to an arbitrary reference

Torque is the rotation equivalent of force in the same way that angle is the rotational equivalent for position, angular velocity for velocity, and angular momentum for momentum. In Geometry and Trigonometry, an angle (in full plane angle) is the figure formed by two rays sharing a common Endpoint, called Do not confuse with Angular frequency The unit for angular velocity is rad/s In Physics, velocity is defined as the rate of change of Position. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product All the formal treatments of Newton's Laws that applied to forces equivalently apply to torques. Thus, as a consequence of Newton's First Law of Motion, there exists rotational inertia that ensures that all bodies maintain their angular momentum unless acted upon by an unbalanced torque. This article is about the moment of inertia of a rotating object. Likewise, Newton's Second Law of Motion can be used to derive an alternative definition of torque:

$\vec{\tau} = I\vec{\alpha}$

where

I is the moment of inertia of the particle
$\vec{\alpha}$ is the angular acceleration of the particle. This article is about the moment of inertia of a rotating object.

This provides a definition for the moment of inertia which is the rotational equivalent for mass. Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object In more advanced treatments of mechanics, the moment of inertia acts as a tensor that, when properly analyzed, fully determines the characteristics of rotations including precession and nutation. This article is about the moment of inertia of a rotating object. Precession refers to a change in the direction of the axis of a rotating object Nutation is a slight irregular motion (etymologically a "nodding" in the Axis of rotation of a largely axially symmetric object such as a Gyroscope

Equivalently, the differential form of Newton's Second Law provides an alternative definition of torque:

$\vec{\tau} = \frac{d\vec{L}}{dt}$[47]

where $\vec{L}$ is the angular momentum of the particle.

Newton's Third Law of Motion requires that all objects exerting torques themselves experience equal and opposite torques,[48] and therefore also directly implies the conservation of angular momentum for closed systems that experience rotations and revolutions through the action of internal torques. In Physics, the angular momentum of a particle about an origin is a vector quantity equal to the mass of the particle multiplied by the Cross product of the position A revolution (from the Latin revolutio, "a turnaround" is a fundamental change in power or organizational structures that takes place in a relatively

## Kinematic integrals

Forces can be used to define a number of physical concepts by integrating with respect to kinematic variables. The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Kinematics ( Greek κινειν, kinein, to move is a branch of Classical mechanics which describes the motion of objects without For example, integrating with respect to time gives the definition of impulse:

$\vec{I}=\int{\vec{F} dt}$

which, by Newton's Second Law, must be equivalent to the change in momentum (yielding the Impulse momentum theorem). In Classical mechanics, an impulse is defined as the Integral of a Force with respect to Time: \mathbf{I} = \int \mathbf{F}\ In Classical mechanics, an impulse is defined as the Integral of a Force with respect to Time: \mathbf{I} = \int \mathbf{F}\

Similarly, integrating with respect to position gives a definition for the work done by a force:

$W=\int{\vec{F} \cdot{d\vec{x}}}$

which, in a system where all the forces are conservative (see below) is equivalent to changes in kinetic and potential energy (yielding the Work energy theorem). In Physics, mechanical work is the amount of Energy transferred by a Force. The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, mechanical work is the amount of Energy transferred by a Force. The time derivative of the definition of work gives a definition for power in term of force and the velocity ($\vec{v}$):

$P=\frac{dW}{dt}=\int{\vec{F} \cdot{d\vec{v}}}$

## Potential energy

Main article: Potential energy

Instead of a force, often the mathematically related concept of a potential energy field can be used for convenience. In Physics, power (symbol P) is the rate at which work is performed or energy is transmitted or the amount of energy required or expended for Potential energy can be thought of as Energy stored within a physical system Potential energy can be thought of as Energy stored within a physical system For instance, the gravitational force acting upon an object can be seen as the action of the gravitational field that is present at the object's location. A gravitational field is a model used within Physics to explain how gravity exists in the universe Restating mathematically the definition of energy (via the definition of work), a potential scalar field $U(\vec{r})$ is defined as that field whose gradient is equal and opposite to the force produced at every point:

$\vec{F}=-\vec{\nabla} U.$

Forces can be classified as conservative or nonconservative. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Physics, mechanical work is the amount of Energy transferred by a Force. In Mathematics and Physics, a scalar field associates a scalar value which can be either mathematical in definition or physical, to every point In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar A conservative force is defined as a Force with the following property when an object moves from one location to another the force changes the Potential energy of Conservative forces are equivalent to the gradient of a potential while non-conservative forces are not. In Vector calculus, the gradient of a Scalar field is a Vector field which points in the direction of the greatest rate of increase of the scalar The Mathematical study of potentials is known as Potential theory; it is the study of Harmonic functions on Manifolds This mathematical [3]

### Conservative forces

Main article: Conservative force

A conservative force that acts on a closed system has an associated mechanical work that allows energy to convert only between kinetic or potential forms. A conservative force is defined as a Force with the following property when an object moves from one location to another the force changes the Potential energy of A Closed system is a System in the state of being isolated from the environment In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The kinetic energy of an object is the extra Energy which it possesses due to its motion Potential energy can be thought of as Energy stored within a physical system This means that for a closed system, the net mechanical energy is conserved whenever a conservative force acts on the system. In Physics, mechanical energy describes the Potential energy and Kinetic energy present in the components of a mechanical system. The force, therefore, is related directly to the difference in potential energy between two different locations in space,[49] and can be considered to be an artifact of the potential field in the same way that the direction and amount of a flow of water can be considered to be an artifact of the contour map of the elevation of an area. A contour line (also Level set, isopleth, isoline, isogram or isarithm) of a function of two [3]

Conservative forces include gravity, the electromagnetic force, and the spring force. Gravitation is a natural Phenomenon by which objects with Mass attract one another Electromagnetism is the Physics of the Electromagnetic field: a field which exerts a Force on particles that possess the property of In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the Each of these forces has models which are dependent on a position often given as a radial vector $\vec{r}$ emanating from spherically symmetric potentials. Remote Authentication Dial In User Service ( RADIUS) is a networking protocol that provides centralized access authorization and accounting management for people or computers This article is about rotations in three-dimensional Euclidean space [50] Examples of this follow:

For gravity:

$\vec{F} = - \frac{G m_1 m_2 \vec{r}}{r^3}$

where G is the gravitational constant, and mn is the mass of object n. The gravitational constant, denoted G, is a Physical constant involved in the calculation of the gravitational attraction between objects with mass

For electrostatic forces:

$\vec{F} = \frac{q_{1} q_{2} \vec{r}}{4 \pi \epsilon_{0} r^3}$

where ε0 is electric permittivity of free space, and qn is the electric charge of object n. Permittivity is a Physical quantity that describes how an Electric field affects and is affected by a Dielectric medium and is determined by the ability Electric charge is a fundamental conserved property of some Subatomic particles which determines their Electromagnetic interaction.

For spring forces:

$\vec{F} = - k \vec{r}$

where k is the spring constant. In Mechanics, and Physics, Hooke's law of elasticity is an approximation that states that the amount by which a material body is deformed (the [3]

### Nonconservative forces

For certain physical scenarios, it is impossible to model forces as being due to gradient of potentials. This is often due to macrophysical considerations which yield forces as arising from a macroscopic statistical average of microstates. In Statistical mechanics, a microstate describes a specific detailed microscopic configuration of a system that the system visits in the course of its thermal fluctuations For example, friction is caused by the gradients of numerous electrostatic potentials between the atoms, but manifests as a force model which is independent of any macroscale position vector. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny Nonconservative forces other than friction include other contact forces, tension, compression, and drag. Friction is the Force resisting the relative motion of two Surfaces in contact or a surface in contact with a fluid (e In Physics, a contact force is a force between two objects (or an object and a surface that are in contact with each other In Physics String Tension is the magnitude of the pulling force exerted by a string cable chain or similar object on another object Physical compression is the result of the subjection of a material to Compressive stress, resulting in reduction of Volume. In Fluid dynamics, drag (sometimes called fluid resistance) is the force that resists the movement of a Solid object through a Fluid (a However, for any sufficiently detailed description, all these forces are the results of conservative ones since each of these macroscopic forces are the net results of the gradients of microscopic potentials. [3]

The connection between macroscopic non-conservative forces and microscopic conservative forces is described by detailed treatment with statistical mechanics. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In macroscopic closed systems, nonconservative forces act to change the internal energies of the system, and are often associated with the transfer of heat. In Thermodynamics, the internal energy of a Thermodynamic system, or a body with well-defined boundaries, denoted by  U, or sometimes  In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature According to the Second Law of Thermodynamics, nonconservative forces necessarily result in energy transformations within closed systems from ordered to more random conditions as entropy increases. The second law of Thermodynamics is an expression of the universal law of increasing Entropy, stating that the entropy of an Isolated system which In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy [3]

## Units of measurement

The SI unit of force is the newton (symbol N), which is the force required to accelerate a one kilogram mass at a rate of one meter per second squared, or kg•m•s−2. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical [51] The corresponding CGS unit is the dyne, the force required to accelerate a one gram mass by one centimeter per second squared, or g•cm•s−2. The centimetre-gram-second system ( CGS) is a system of physical units. 1 newton is thus equal to 100,000 dyne.

The foot-pound-second Imperial unit of force is the pound-force (lbf), defined as the force exerted by gravity on a pound-mass in the standard gravitational field of 9. This article deals with the unit of force For the unit of mass see Pound (mass. Imperial units or the Imperial system is a collection of units first defined in the British Weights and Measures Act of 1824 This article deals with the unit of force For the unit of mass see Pound (mass. The pound or pound-mass (abbreviation lb, lbm, or sometimes in the United States #) is a unit of Mass Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level 80665 m•s−2. [51] The pound-force provides an alternate unit of mass: one slug is the mass that will accelerate by one foot per second squared when acted on by one pound-force. The slug is an English unit of Mass. It is a mass that accelerates by 1 ft/s² when a force of one Pound-force (lbf is exerted on it [51] An alternate unit of force in the same system is the poundal, defined as the force required to accelerate a one pound mass at a rate of one foot per second squared. The poundal is a non- SI unit of Force. It is a part of the Foot-pound-second system of units a coherent subsystem of English units introduced [51] The units of slug and poundal are designed to avoid a constant of proportionality in Newton's Second Law. The slug is an English unit of Mass. It is a mass that accelerates by 1 ft/s² when a force of one Pound-force (lbf is exerted on it The poundal is a non- SI unit of Force. It is a part of the Foot-pound-second system of units a coherent subsystem of English units introduced Newton's laws of motion are three Physical laws which provide relationships between the Forces acting on a body and the motion of the

The pound-force has a metric counterpart, less commonly used than the newton: the kilogram-force (kgf) (sometimes kilopond), is the force exerted by standard gravity on one kilogram of mass. The unit kilogram-force ( kgf, often incorrectly just kg) or kilopond ( kp) is defined as the Force exerted by Earth's gravity The unit kilogram-force ( kgf, often incorrectly just kg) or kilopond ( kp) is defined as the Force exerted by Earth's gravity [51] The kilogram-force leads to an alternate, but rarely used unit of mass: the metric slug (sometimes mug or hyl) is that mass which accelerates at 1 m•s−2 when subjected to a force of 1 kgf. In the gravitational metric system(s the base unit of force is not normalised to one mass unit ( Gram or Kilogram) times one length unit ( Metre or centimetre A mug is a sturdily built type of cup often used for drinking hot beverages such as Coffee, Tea, or Hot chocolate. In the gravitational metric system(s the base unit of force is not normalised to one mass unit ( Gram or Kilogram) times one length unit ( Metre or centimetre The kilogram-force is not a part of the modern SI system, and is generally deprecated; however it still sees use for some purposes as expressing jet thrust, bicycle spoke tension, torque wrench settings and engine output torque. Other arcane units of force include the sthène which is equivalent to 1000 N and the kip which is equivalent to 1000 lbf. The sthène is the unit of force in the Metre-tonne-second system of units (mts invented in France and used in the Soviet Union 1933 - 1955. In the United States a kip is a unit of Force that equals 1000 pounds-force, used primarily by architects and engineers of mayapur to measure engineering loads

Units of force
newton
(SI unit)
dynekilogram-force,
kilopond
pound-forcepoundal
1 N≡ 1 kg·m/s²= 105 dyn≈ 0. The newton (symbol N) is the SI derived unit of Force, named after Isaac Newton in recognition of his work on Classical The unit kilogram-force ( kgf, often incorrectly just kg) or kilopond ( kp) is defined as the Force exerted by Earth's gravity This article deals with the unit of force For the unit of mass see Pound (mass. The poundal is a non- SI unit of Force. It is a part of the Foot-pound-second system of units a coherent subsystem of English units introduced 10197 kp≈ 0. 22481 lbf≈ 7. 2330 pdl
1 dyn= 10−5 N≡ 1 g·cm/s²≈ 1. 0197×10−6 kp≈ 2. 2481×10−6 lbf≈ 7. 2330×10−5 pdl
1 kp= 9. 80665 N= 980665 dyngn·(1 kg)≈ 2. 2046 lbf≈ 70. 932 pdl
1 lbf≈ 4. 448222 N≈ 444822 dyn≈ 0. 45359 kpgn·(1 lb)≈ 32. The pound or pound-mass (abbreviation lb, lbm, or sometimes in the United States #) is a unit of Mass 174 pdl
1 pdl≈ 0. 138255 N≈ 13825 dyn≈ 0. 014098 kp≈ 0. 031081 lbf≡ 1 lb·ft/s²
The value of gn as used in the official definition of the kilogram-force is used here for all gravitational units. A foot (plural feet or foot; symbol or abbreviation ft or sometimes &prime – the prime symbol) is a non-SI unit Standard gravity, usually denoted by g 0 or g n is the nominal acceleration due to gravity at the Earth's surface at sea level

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