Newton's laws of motion

Newton's laws of motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, its motion in response to those forces. More the first law defines the force qualitatively, the second law offers a quantitative measure of the force, the third asserts that a single isolated force doesn't exist; these three laws have been expressed in several ways, over nearly three centuries, can be summarised as follows: The three laws of motion were first compiled by Isaac Newton in his Philosophiæ Naturalis Principia Mathematica, first published in 1687. Newton used them to investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. A fourth law is also described in the bibliography, which states that forces add up like vectors, that is, that forces obey the principle of superposition.

Newton's laws are applied to objects which are idealised as single point masses, in the sense that the size and shape of the object's body are neglected to focus on its motion more easily. This can be done when the object is small compared to the distances involved in its analysis, or the deformation and rotation of the body are of no importance. In this way a planet can be idealised as a particle for analysis of its orbital motion around a star. In their original form, Newton's laws of motion are not adequate to characterise the motion of rigid bodies and deformable bodies. Leonhard Euler in 1750 introduced a generalisation of Newton's laws of motion for rigid bodies called Euler's laws of motion applied as well for deformable bodies assumed as a continuum. If a body is represented as an assemblage of discrete particles, each governed by Newton's laws of motion Euler's laws can be derived from Newton's laws. Euler's laws can, however, be taken as axioms describing the laws of motion for extended bodies, independently of any particle structure.

Newton's laws hold only with respect to a certain set of frames of reference called Newtonian or inertial reference frames. Some authors interpret the first law as defining. Other authors do treat the first law as a corollary of the second; the explicit concept of an inertial frame of reference was not developed until long after Newton's death. In the given interpretation mass, acceleration and force are assumed to be externally defined quantities; this is the most common, but not the only interpretation of the way one can consider the laws to be a definition of these quantities. Newtonian mechanics has been superseded by special relativity, but it is still useful as an approximation when the speeds involved are much slower than the speed of light; the first law states that if the net force is zero the velocity of the object is constant. Velocity is a vector quantity which expresses both the object's speed and the direction of its motion; the first law can be stated mathematically when the mass is a non-zero constant, as, ∑ F = 0 ⇔ d v d t = 0.

An object, at rest will stay at rest unless a force acts upon it. An object, in motion will not change its velocity unless a force acts upon it; this is known as uniform motion. An object continues to do. If it is at rest, it continues in a state of rest. If an object is moving, it continues to move without changing its speed; this is evident in space probes. Changes in motion must be imposed against the tendency of an object to retain its state of motion. In the absence of net forces, a moving object tends to move along a straight line path indefinitely. Newton placed the first law of motion to establish frames of reference for which the other laws are applicable; the first law of motion postulates the existence of at least one frame of reference called a Newtonian or inertial reference frame, relative to which the motion of a particle not subject to forces is a straight line at a constant speed. Newton's first law is referred to as the law of inertia. Thus, a condition necessary for the uniform motion of a particle relative to an inertial reference frame is that the total net force acting on it is zero.

In this sense, the first law can be restated as: In every material universe, the motion of a particle in a preferential reference frame Φ is determined by the action of forces whose total vanished for all times when and only when the velocity of the particle is constant in Φ. That is, a particle at rest or in uniform motion in the preferential frame Φ continues in that state unless compelled by forces to change it. Newton's first and second laws are valid only in an inertial reference frame. Any reference frame, in uniform motion with respect to an inertial frame is an in

Rigid body dynamics

Rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid, which means that they do not deform under the action of applied forces, simplifies the analysis by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body; this excludes bodies that display fluid elastic, plastic behavior. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form Lagrangian mechanics; the solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems. If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement.

In this case, Newton's laws for a rigid system of N particles, Pi, i=1... N, simplify. Determine the resultant force and torque at a reference point R, to obtain F = ∑ i = 1 N m i A i, T = ∑ i = 1 N ×, where ri denotes the planar trajectory of each particle; the kinematics of a rigid body yields the formula for the acceleration of the particle Pi in terms of the position R and acceleration A of the reference particle as well as the angular velocity vector ω and angular acceleration vector α of the rigid system of particles as, A i = α × + ω × + A. For systems that are constrained to planar movement, the angular velocity and angular acceleration vectors are directed along k perpendicular to the plane of movement, which simplifies this acceleration equation. In this case, the acceleration vectors can be simplified by introducing the unit vectors ei from the reference point R to a point ri and the unit vectors t i = k × e i, so A i = α − ω 2 + A; this yields the resultant force on the system as F = α ∑ i = 1 N m i − ω 2 ∑ i = 1 N m i + A, torque as T = ∑ i = 1 N × = α k → + × A, where e i × e i = 0 and e i × t i = k is the unit

Virtual work

Virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action; this displacement is therefore the displacement followed by the particle according to the principle of least action. The work of a force on a particle along a virtual displacement is known as the virtual work. Virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have been developed for the study of the mechanics of deformable bodies; the principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, Renaissance Italians as "the law of lever"; the idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Torricelli and Huygens, in varying degrees of generality, when solving problems in statics.

Working with Leibnizian concepts, Johann Bernoulli systematized the virtual work principle and made explicit the concept of infinitesimal displacement. He was able to solve problems for both rigid bodies as well as fluids. Bernoulli's version of virtual work law appeared in his letter to Pierre Varignon in 1715, published in Varignon's second volume of Nouvelle mécanique ou Statique in 1725; this formulation of the principle is today known as the principle of virtual velocities and is considered as the prototype of the contemporary virtual work principles. In 1743 D'Alembert published his Traité de Dynamique where he applied the principle of virtual work, based on Bernoulli's work, to solve various problems in dynamics, his idea was to convert a dynamical problem into static problem by introducing inertial force. In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved.

A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic D'Alembert's principle, was given in his Mécanique Analytique of 1788. Although Lagrange had presented his version of least action principle prior to this work, he recognized the virtual work principle to be more fundamental because it could be assumed alone as the foundation for all mechanics, unlike the modern understanding that least action does not account for non-conservative forces. If a force acts on a particle as it moves from point A to point B for each possible trajectory that the particle may take, it is possible to compute the total work done by the force along the path; the principle of virtual work, the form of the principle of least action applied to these systems, states that the path followed by the particle is the one for which the difference between the work along this path and other nearby paths is zero. The formal procedure for computing the difference of functions evaluated on nearby paths is a generalization of the derivative known from differential calculus, is termed the calculus of variations.

Consider a point particle that moves along a path, described by a function r from point A, where r, to point B, where r. It is possible that the particle moves from A to B along a nearby path described by r + δr, where δr is called the variation of r; the variation δr satisfies the requirement δr = δr = 0. The components of the variation, δr1, δr2 and δr3, are called virtual displacements; this can be generalized to an arbitrary mechanical system defined by the generalized coordinates qi, i = 1... n. In which case, the variation of the trajectory qi is defined by the virtual displacements δqi, i = 1... n. Virtual work is the total work done by the applied forces and the inertial forces of a mechanical system as it moves through a set of virtual displacements; when considering forces applied to a body in static equilibrium, the principle of least action requires the virtual work of these forces to be zero. Consider a particle P that moves from a point A to a point B along a trajectory r, while a force F is applied to it.

The work done by the force F is given by the integral W = ∫ r = A r = B F ⋅ d r = ∫ t 0 t 1 F ⋅ d r d t d t = ∫ t 0 t 1 F ⋅ v d t, where dr is the differential element along the curve, the trajectory of P, v is its velocity. It is important to notice. Now consider particle P that moves from point A to point B again, but this time it moves along the nearby trajectory that differs from r by the variation δr=εh, where ε is a scaling constant that can be made as small as desired and h(

Momentum

In Newtonian mechanics, linear momentum, translational momentum, or momentum is the product of the mass and velocity of an object. It is a vector quantity, possessing a direction in three-dimensional space. If m is an object's mass and v is the velocity the momentum is p = m v, In SI units, it is measured in kilogram meters per second. Newton's second law of motion states that a body's rate of change in momentum is equal to the net force acting on it. Momentum depends on the frame of reference, but in any inertial frame it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum does not change. Momentum is conserved in special relativity and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, general relativity, it is an expression of one of the fundamental symmetries of time: translational symmetry. Advanced formulations of classical mechanics and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.

In these systems the conserved quantity is generalized momentum, in general this is different from the kinetic momentum defined above. The concept of generalized momentum is carried over into quantum mechanics, where it becomes an operator on a wave function; the momentum and position operators are related by the Heisenberg uncertainty principle. In continuous systems such as electromagnetic fields and deformable bodies, a momentum density can be defined, a continuum version of the conservation of momentum leads to equations such as the Navier–Stokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids. Momentum is a vector quantity: it has both magnitude and direction. Since momentum has a direction, it can be used to predict the resulting direction and speed of motion of objects after they collide. Below, the basic properties of momentum are described in one dimension; the vector equations are identical to the scalar equations. The momentum of a particle is conventionally represented by the letter p.

It is the product of two quantities, the particle's mass and its velocity: p = m v. The unit of momentum is the product of the units of velocity. In SI units, if the mass is in kilograms and the velocity is in meters per second the momentum is in kilogram meters per second. In cgs units, if the mass is in grams and the velocity in centimeters per second the momentum is in gram centimeters per second. Being a vector, momentum has direction. For example, a 1 kg model airplane, traveling due north at 1 m/s in straight and level flight, has a momentum of 1 kg⋅m/s due north measured with reference to the ground; the momentum of a system of particles is the vector sum of their momenta. If two particles have respective masses m1 and m2, velocities v1 and v2, the total momentum is p = p 1 + p 2 = m 1 v 1 + m 2 v 2; the momenta of more than two particles can be added more with the following: p = ∑ i m i v i. A system of particles has a center of mass, a point determined by the weighted sum of their positions: r cm = m 1 r 1 + m 2 r 2 + ⋯ m 1 + m 2 + ⋯ = ∑ i m i r i ∑ i m i.

If all the particles are moving, the center of mass will be moving as well. If the center of mass is moving at velocity vcm, the momentum is: p = m v cm; this is known as Euler's first law. If the net force applied to a particle is a constant F, is applied for a time interval Δt, the momentum of the particle changes by an amount Δ p = F Δ t. In differential form, this is Newton's second law. If the net force experienced by a particle changes as a function of time, F, the change in momentum between times t1 and t2 is Δ p = J = ∫ t 1

Acceleration

In physics, acceleration is the rate of change of velocity of an object with respect to time. An object's acceleration is the net result of all forces acting on the object, as described by Newton's Second Law; the SI unit for acceleration is metre per second squared. Accelerations add according to the parallelogram law; the vector of the net force acting on a body has the same direction as the vector of the body's acceleration, its magnitude is proportional to the magnitude of the acceleration, with the object's mass as proportionality constant. For example, when a car starts from a standstill and travels in a straight line at increasing speeds, it is accelerating in the direction of travel. If the car turns, an acceleration occurs toward the new direction; the forward acceleration of the car is called a linear acceleration, the reaction to which passengers in the car experience as a force pushing them back into their seats. When changing direction, this is called radial acceleration, the reaction to which passengers experience as a sideways force.

If the speed of the car decreases, this is an acceleration in the opposite direction of the velocity of the vehicle, sometimes called deceleration or Retrograde burning in spacecraft. Passengers experience the reaction to deceleration as a force pushing them forwards. Both acceleration and deceleration are treated the same, they are both changes in velocity; each of these accelerations is felt by passengers until their velocity matches that of the uniformly moving car. An object's average acceleration over a period of time is its change in velocity divided by the duration of the period. Mathematically, a ¯ = Δ v Δ t. Instantaneous acceleration, meanwhile, is the limit of the average acceleration over an infinitesimal interval of time. In the terms of calculus, instantaneous acceleration is the derivative of the velocity vector with respect to time: a = lim Δ t → 0 Δ v Δ t = d v d t It can be seen that the integral of the acceleration function a is the velocity function v. V = ∫ a d t As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t: a = d v d t = d 2 x d t 2 Acceleration has the dimensions of velocity divided by time, i.e. L.

T−2. The SI unit of acceleration is the metre per second squared. An object moving in a circular motion—such as a satellite orbiting the Earth—is accelerating due to the change of direction of motion, although its speed may be constant. In this case it is said to be undergoing centripetal acceleration. Proper acceleration, the acceleration of a body relative to a free-fall condition, is measured by an instrument called an accelerometer. In classical mechanics, for a body with constant mass, the acceleration of the body's center of mass is proportional to the net force vector acting on it: F = m a → a = F / m where F is the net force acting on the body, m is the mass of the body, a is the center-of-mass acceleration; as speeds approach the speed of light, relativistic effects become large. The velocity of a particle moving on a curved path as a function of time can be written as: v = v v v = v u t, with v equal to the speed of travel along the path, u t = v v, {\displaystyle \mathbf _=

Classical mechanics

Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, astronomical objects, such as spacecraft, planets and galaxies. If the present state of an object is known it is possible to predict by the laws of classical mechanics how it will move in the future and how it has moved in the past; the earliest development of classical mechanics is referred to as Newtonian mechanics. It consists of the physical concepts employed by and the mathematical methods invented by Isaac Newton and Gottfried Wilhelm Leibniz and others in the 17th century to describe the motion of bodies under the influence of a system of forces. More abstract methods were developed, leading to the reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics; these advances, made predominantly in the 18th and 19th centuries, extend beyond Newton's work through their use of analytical mechanics. They are, with some modification used in all areas of modern physics.

Classical mechanics provides accurate results when studying large objects that are not massive and speeds not approaching the speed of light. When the objects being examined have about the size of an atom diameter, it becomes necessary to introduce the other major sub-field of mechanics: quantum mechanics. To describe velocities that are not small compared to the speed of light, special relativity is needed. In case that objects become massive, general relativity becomes applicable. However, a number of modern sources do include relativistic mechanics into classical physics, which in their view represents classical mechanics in its most developed and accurate form; the following introduces the basic concepts of classical mechanics. For simplicity, it models real-world objects as point particles; the motion of a point particle is characterized by a small number of parameters: its position and the forces applied to it. Each of these parameters is discussed in turn. In reality, the kind of objects that classical mechanics can describe always have a non-zero size.

Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom, e.g. a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made of a large number of collectively acting point particles; the center of mass of a composite object behaves like a point particle. Classical mechanics uses common-sense notions of how matter and forces interact, it assumes that matter and energy have definite, knowable attributes such as location in space and speed. Non-relativistic mechanics assumes that forces act instantaneously; the position of a point particle is defined in relation to a coordinate system centered on an arbitrary fixed reference point in space called the origin O. A simple coordinate system might describe the position of a particle P with a vector notated by an arrow labeled r that points from the origin O to point P. In general, the point particle does not need to be stationary relative to O.

In cases where P is moving relative to O, r is defined as a function of time. In pre-Einstein relativity, time is considered an absolute, i.e. the time interval, observed to elapse between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space; the velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time: v = d r d t. In classical mechanics, velocities are directly subtractive. For example, if one car travels east at 60 km/h and passes another car traveling in the same direction at 50 km/h, the slower car perceives the faster car as traveling east at 60 − 50 = 10 km/h. However, from the perspective of the faster car, the slower car is moving 10 km/h to the west denoted as -10 km/h where the sign implies opposite direction. Velocities are directly additive as vector quantities. Mathematically, if the velocity of the first object in the previous discussion is denoted by the vector u = ud and the velocity of the second object by the vector v = ve, where u is the speed of the first object, v is the speed of the second object, d and e are unit vectors in the directions of motion of each object then the velocity of the first object as seen by the second object is u ′ = u − v. Similarly, the first object sees the velocity of the second object as v ′ = v − u.

When both objects are moving in the same direction, this equation can be simplified to u ′ = d. Or, by ignoring direction, the difference can be given in terms of speed only: u ′ = u − v; the acceleration, or rate of change of velocity, is th

Mass

Mass is both a property of a physical body and a measure of its resistance to acceleration when a net force is applied. The object's mass determines the strength of its gravitational attraction to other bodies; the basic SI unit of mass is the kilogram. In physics, mass is not the same as weight though mass is determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass; this is because weight is a force, while mass is the property that determines the strength of this force. There are several distinct phenomena. Although some theorists have speculated that some of these phenomena could be independent of each other, current experiments have found no difference in results regardless of how it is measured: Inertial mass measures an object's resistance to being accelerated by a force. Active gravitational mass measures the gravitational force exerted by an object.

Passive gravitational mass measures the gravitational force exerted on an object in a known gravitational field. The mass of an object determines its acceleration in the presence of an applied force; the inertia and the inertial mass describe the same properties of physical bodies at the qualitative and quantitative level by other words, the mass quantitatively describes the inertia. According to Newton's second law of motion, if a body of fixed mass m is subjected to a single force F, its acceleration a is given by F/m. A body's mass determines the degree to which it generates or is affected by a gravitational field. If a first body of mass mA is placed at a distance r from a second body of mass mB, each body is subject to an attractive force Fg = GmAmB/r2, where G = 6.67×10−11 N kg−2 m2 is the "universal gravitational constant". This is sometimes referred to as gravitational mass. Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical.

The standard International System of Units unit of mass is the kilogram. The kilogram is 1000 grams, first defined in 1795 as one cubic decimeter of water at the melting point of ice. However, because precise measurement of a decimeter of water at the proper temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of the international prototype kilogram of cast iron, thus became independent of the meter and the properties of water. However, the mass of the international prototype and its identical national copies have been found to be drifting over time, it is expected that the re-definition of the kilogram and several other units will occur on May 20, 2019, following a final vote by the CGPM in November 2018. The new definition will use only invariant quantities of nature: the speed of light, the caesium hyperfine frequency, the Planck constant. Other units are accepted for use in SI: the tonne is equal to 1000 kg. the electronvolt is a unit of energy, but because of the mass–energy equivalence it can be converted to a unit of mass, is used like one.

In this context, the mass has units of eV/c2. The electronvolt and its multiples, such as the MeV, are used in particle physics; the atomic mass unit is 1/12 of the mass of a carbon-12 atom 1.66×10−27 kg. The atomic mass unit is convenient for expressing the masses of molecules. Outside the SI system, other units of mass include: the slug is an Imperial unit of mass; the pound is a unit of both mass and force, used in the United States. In scientific contexts where pound and pound need to be distinguished, SI units are used instead; the Planck mass is the maximum mass of point particles. It is used in particle physics; the solar mass is defined as the mass of the Sun. It is used in astronomy to compare large masses such as stars or galaxies; the mass of a small particle may be identified by its inverse Compton wavelength. The mass of a large star or black hole may be identified with its Schwarzschild radius. In physical science, one may distinguish conceptually between at least seven different aspects of mass, or seven physical notions that involve the concept of mass.

Every experiment to date has shown these seven values to be proportional, in some cases equal, this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or operationally defined: Inertial mass is a measure of an object's resistance to acceleration when a force is applied, it is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says. Active gravitational mass is a measure of the strength of an object's gravitational flux. Gravitational field can be measured by allowing a small "test object" to fall and measuring its free-fall acceleration. For example, an object in free fall near the Moon is subject to a smaller gravitational field, hence