In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a closed system remains constant. In three dimensions, the angular momentum for a point particle is a pseudovector r × p, the cross product of the particle's position vector r and its momentum vector p = mv; this definition can be applied to each point in physical fields. Unlike momentum, angular momentum does depend on where the origin is chosen, since the particle's position is measured from it. Just like for angular velocity, there are two special types of angular momentum: the spin angular momentum and the orbital angular momentum; the spin angular momentum of an object is defined as the angular momentum about its centre of mass coordinate. The orbital angular momentum of an object about a chosen origin is defined as the angular momentum of the centre of mass about the origin; the total angular momentum of an object is the sum of orbital angular momenta.
The orbital angular momentum vector of a particle is always parallel and directly proportional to the orbital angular velocity vector ω of the particle, where the constant of proportionality depends on both the mass of the particle and its distance from origin. However, the spin angular momentum of the object is proportional but not always parallel to the spin angular velocity Ω, making the constant of proportionality a second-rank tensor rather than a scalar. Angular momentum is additive. For a continuous rigid body, the total angular momentum is the volume integral of angular momentum density over the entire body. Torque can be defined as the rate of change of angular momentum, analogous to force; the net external torque on any system is always equal to the total torque on the system. Therefore, for a closed system, the total torque on the system must be 0, which means that the total angular momentum of the system is constant; the conservation of angular momentum helps explain many observed phenomena, for example the increase in rotational speed of a spinning figure skater as the skater's arms are contracted, the high rotational rates of neutron stars, the Coriolis effect, the precession of gyroscopes.
In general, conservation does limit the possible motion of a system, but does not uniquely determine what the exact motion is. In quantum mechanics, angular momentum is an operator with quantized eigenvalues. Angular momentum is subject to the Heisenberg uncertainty principle, meaning that at any time, only one component can be measured with definite precision; because of this, it turns out that the notion of an elementary particle "spinning" about an axis does not exist. For technical reasons, elementary particles still possess a spin angular momentum, but this angular momentum does not correspond to spinning motion in the ordinary sense. Angular momentum is a vector quantity that represents the product of a body's rotational inertia and rotational velocity about a particular axis. However, if the particle's trajectory lies in a single plane, it is sufficient to discard the vector nature of angular momentum, treat it as a scalar. Angular momentum can be considered a rotational analog of linear momentum.
Thus, where linear momentum p is proportional to mass m and linear speed v, p = m v, angular momentum L is proportional to moment of inertia I and angular speed ω, L = I ω. Unlike mass, which depends only on amount of matter, moment of inertia is dependent on the position of the axis of rotation and the shape of the matter. Unlike linear speed, which does not depend upon the choice of origin, angular velocity is always measured with respect to a fixed origin; therefore speaking, L should be referred to as the angular momentum relative to that center. Because I = r 2 m for a single particle and ω = v r for circular motion, angular momentum can be expanded, L = r 2 m ⋅ v r, reduced to, L = r m v, the product of the radius of rotation r and the linear momentum of the particle p = m v, where v in this case is the equivalent linear speed at the radius; this simple analysis can apply to non-circular motion if only the component of the motion, perpendicular to the radius vector is considered. In that case, L
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(
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
Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788. In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations using Lagrange multipliers. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, time, contains the information about the dynamics of the system. No new physics are introduced in applying Lagrangian mechanics compared to Newtonian mechanics, it is, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange equations.
Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem. Lagrangian mechanics is important not just for its broad applications, but for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton developed Hamilton's principle that can be used to derive the Lagrange equation and was recognized to be applicable to much of fundamental theoretical physics as well quantum mechanics and the theory of relativity, it can be applied to other systems by analogy, for instance to coupled electric circuits with inductances and capacitances. Lagrangian mechanics is used to solve mechanical problems in physics and when Newton's formulation of classical mechanics is not convenient.
Lagrangian mechanics applies to the dynamics of particles, while fields are described using a Lagrangian density. Lagrange's equations are used in optimization problems of dynamic systems. In mechanics, Lagrange's equations of the second kind are used much more than those of the first kind. Suppose we have a bead sliding around on a wire, or a swinging simple pendulum, etc. If one tracks each of the massive objects as a particle, calculation of the motion of the particle using Newtonian mechanics would require solving for the time-varying constraint force required to keep the particle in the constrained motion. For the same problem using Lagrangian mechanics, one looks at the path the particle can take and chooses a convenient set of independent generalized coordinates that characterize the possible motion of the particle; this choice eliminates the need for the constraint force to enter into the resultant system of equations. There are fewer equations since one is not directly calculating the influence of the constraint on the particle at a given moment.
For a wide variety of physical systems, if the size and shape of a massive object are negligible, it is a useful simplification to treat it as a point particle. For a system of N point particles with masses m1, m2... mN, each particle has a position vector, denoted r1, r2... rN. Cartesian coordinates are sufficient, so r1 =, r2 = and so on. In three dimensional space, each position vector requires three coordinates to uniquely define the location of a point, so there are 3N coordinates to uniquely define the configuration of the system; these are all specific points in space to locate the particles. The velocity of each particle is how fast the particle moves along its path of motion, is the time derivative of its position, thusIn Newtonian mechanics, the equations of motion are given by Newton's laws; the second law "net force equals mass times acceleration",applies to each particle. For an N particle system in 3 dimensions, there are 3N second order ordinary differential equations in the positions of the particles to solve for.
Instead of forces, Lagrangian mechanics uses the energies in the system. The central quantity of Lagrangian mechanics is the Lagrangian, a function which summarizes the dynamics of the entire system. Overall, the Lagrangian has no single expression for all physical systems. Any function which generates the correct equations of motion, in agreement with physical laws, can be taken as a Lagrangian, it is possible to construct general expressions for large classes of applications. The non-relativistic Lagrangian for a system of particles can be defined by L = T − V where T = 1 2 ∑ k = 1 N m k v k 2 is the total kinetic energy of the system, equalling the sum Σ of the kinetic energies of the particles, V is the potential energy of the system. Kinetic energy is the energy of the system's motion, vk2 = vk
In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes; the same amount of work is done by the body when decelerating from its current speed to a state of rest. In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is 1 2 m v 2. In relativistic mechanics, this is a good approximation only when v is much less than the speed of light; the standard unit of kinetic energy is the joule. The imperial unit of kinetic energy is the foot-pound; the adjective kinetic has its roots in the Greek word κίνησις kinesis, meaning "motion". The dichotomy between kinetic energy and potential energy can be traced back to Aristotle's concepts of actuality and potentiality; the principle in classical mechanics that E ∝ mv2 was first developed by Gottfried Leibniz and Johann Bernoulli, who described kinetic energy as the living force, vis viva.
Willem's Gravesande of the Netherlands provided experimental evidence of this relationship. By dropping weights from different heights into a block of clay, Willem's Gravesande determined that their penetration depth was proportional to the square of their impact speed. Émilie du Châtelet published an explanation. The terms kinetic energy and work in their present scientific meanings date back to the mid-19th century. Early understandings of these ideas can be attributed to Gaspard-Gustave Coriolis, who in 1829 published the paper titled Du Calcul de l'Effet des Machines outlining the mathematics of kinetic energy. William Thomson Lord Kelvin, is given the credit for coining the term "kinetic energy" c. 1849–51. Energy occurs in many forms, including chemical energy, thermal energy, electromagnetic radiation, gravitational energy, electric energy, elastic energy, nuclear energy, rest energy; these can be categorized in two main classes: kinetic energy. Kinetic energy is the movement energy of an object.
Kinetic energy can be transformed into other kinds of energy. Kinetic energy may be best understood by examples that demonstrate how it is transformed to and from other forms of energy. For example, a cyclist uses chemical energy provided by food to accelerate a bicycle to a chosen speed. On a level surface, this speed can be maintained without further work, except to overcome air resistance and friction; the chemical energy has been converted into kinetic energy, the energy of motion, but the process is not efficient and produces heat within the cyclist. The kinetic energy in the moving cyclist and the bicycle can be converted to other forms. For example, the cyclist could encounter a hill just high enough to coast up, so that the bicycle comes to a complete halt at the top; the kinetic energy has now been converted to gravitational potential energy that can be released by freewheeling down the other side of the hill. Since the bicycle lost some of its energy to friction, it never regains all of its speed without additional pedaling.
The energy is not destroyed. Alternatively, the cyclist could connect a dynamo to one of the wheels and generate some electrical energy on the descent; the bicycle would be traveling slower at the bottom of the hill than without the generator because some of the energy has been diverted into electrical energy. Another possibility would be for the cyclist to apply the brakes, in which case the kinetic energy would be dissipated through friction as heat. Like any physical quantity, a function of velocity, the kinetic energy of an object depends on the relationship between the object and the observer's frame of reference. Thus, the kinetic energy of an object is not invariant. Spacecraft use chemical energy to launch and gain considerable kinetic energy to reach orbital velocity. In an circular orbit, this kinetic energy remains constant because there is no friction in near-earth space. However, it becomes apparent at re-entry. If the orbit is elliptical or hyperbolic throughout the orbit kinetic and potential energy are exchanged.
Without loss or gain, the sum of the kinetic and potential energy remains constant. Kinetic energy can be passed from one object to another. In the game of billiards, the player imposes kinetic energy on the cue ball by striking it with the cue stick. If the cue ball collides with another ball, it slows down and the ball it hit accelerates its speed as the kinetic energy is passed on to it. Collisions in billiards are elastic collisions, in which kinetic energy is preserved. In inelastic collisions, kinetic energy is dissipated in various forms of energy, such as heat, binding energy. Flywheels have been developed as a method of energy storage; this illustrates that kinetic energy is stored in rotational motion. Several mathematical descriptions of kinetic energy exist that describe it in the appropriate physical situation. For objects and processes in common human experience, the formula ½mv² given by Newtonian mechanics is suitable. However, if the speed of the object is comparabl
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