"Momentum–depth relationship in a rectangular channel In classical physics, momentum is the product of mass and velocity and is a vector quantity, but in fluid mechanics it is treated as a longitudinal quantity (i.e. one dimension) evaluated in the direction of flow. Additionally, it is evaluated as momentum per unit time, corresponding to the product of mass flow rate and velocity, and therefore it has units of force. The momentum forces considered in open channel flow are dynamic force – dependent of depth and flow rate – and static force – dependent of depth – both affected by gravity. The"
"Figure 4. Stilling Basin On Oker River in the Harz-Mointains at Opened Scour Outlet (left) and Stilling Basin for Griggs Dam in Columbus, OH (right) Momentum is defined as the product of mass times velocity, and like velocity, it is a vector. French Scientist and Philosopher of the early 1600s René Descartes first discovered the concept of momentum but got stuck on the amount of motion (speed) which was not being conserved. Christian Huygens, a Dutch Scientist, pointed out that the ""quantity of motion"" did not need to be a positive value; a negative value meant that it was moving"
"being another term for mass. That force, applied to a mass, creates an acceleration, which when applied over time, changes the velocity of a mass. According to Newton's second law, the law of momentum -- changing the velocity of the mass changes its momentum, (mass multiplied by velocity). It is important to understand at this point that velocity is not simply speed. Velocity is the speed of a mass in a particular direction. In a very technical sense, speed is a scalar (mathematics), a magnitude, and velocity is a vector (physics), magnitude and direction. Newton's third law, known as conservation"
"a moment; commonly used quantities include forces, masses, and electric charge distributions. In its most simple and basic form, a moment is the product of the distance to some point, raised to some power, multiplied by some physical quantity such as the force, charge, etc. at that point: where formula_2 is the physical quantity such as a force applied at a point, or a point charge, or a point mass, etc. If the quantity is not concentrated solely at a single point, the moment is the integral of that quantity's density over space: where formula_4 is the distribution of the"
"Moment (physics) In physics, a moment is an expression involving the product of a distance and another physical quantity, and in this way it accounts for how the physical quantity is located or arranged. Moments are usually defined with respect to a fixed reference point; they deal with physical quantities as measured at some distance from that reference point. For example, the moment of force acting on an object, often called torque, is the product of the force and the distance to the object (i.e., the reference point). In principle, any physical quantity can be multiplied by distance to produce"
"to momentum. Just as space and time are, in that theory, different aspects of a more comprehensive entity called spacetime, energy and momentum are merely different aspects of a unified, four-dimensional quantity that physicists call four-momentum. In consequence, if energy is a source of gravity, momentum must be a source as well. The same is true for quantities that are directly related to energy and momentum, namely internal pressure and tension. Taken together, in general relativity it is mass, energy, momentum, pressure and tension that serve as sources of gravity: they are how matter tells spacetime how to curve. In"
"Moment (mathematics) In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment is the skewness, and the fourth"
"as well as the ordinary quark and lepton, and thus also anything made of mesons, which are unstable particles made up of a quark and an antiquark. In the context of relativity, mass is not an additive quantity, in the sense that one can not add the rest masses of particles in a system to get the total rest mass of the system. Thus, in relativity usually a more general view is that it is not the sum of rest masses, but the energy–momentum tensor that quantifies the amount of matter. This tensor gives the rest mass for the entire"
"Working mass Working mass, also referred to as reaction mass, is a mass against which a system operates in order to produce acceleration. In the case of a rocket, for example, the reaction mass is the fuel shot backwards to provide propulsion. All acceleration requires an exchange of momentum, which can be thought of as the ""unit of movement"". Momentum is related to mass and velocity, as given by the formula ""P = mv,"" where ""P"" is the momentum, ""m"" the mass, and ""v"" the velocity. The velocity of a body is easily changeable, but in most cases the mass"
"mass. This was clear at the beginning of the 20th century, before the advent of the theory of relativity. Mass, ""m"", denoted something like an amount of substance or quantity of matter. At the same time, mass was the quantitative measure of inertia of a body. The mass of a body determines the momentum, formula_1, of the body at given velocity, formula_2; it is a proportionality factor in the formula: The factor ""m"" is referred to as inertial mass. But mass, as related to the ""inertia"" of a body, can also be defined by the formula: where ""F"" is the"
"the mass of a moving particle."" When the relative velocity is zero, formula_33 is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light ""c"", the denominator of the right side approaches zero, and consequently formula_33 approaches infinity. In the formula for momentum the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum. While Newton's second law remains valid in the form the"
"the sun and planets. For a system of particles going off in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers, even those in relative motion. It is defined as the total energy (divided by ) in the center of mass frame (where by definition, the system total momentum is zero). A simple example of an object with moving parts but zero total momentum is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy"
"we now know it: a moment about a center of rotation. Moment (physics) In physics, a moment is an expression involving the product of a distance and another physical quantity, and in this way it accounts for how the physical quantity is located or arranged. Moments are usually defined with respect to a fixed reference point; they deal with physical quantities as measured at some distance from that reference point. For example, the moment of force acting on an object, often called torque, is the product of the force and the distance to the object (i.e., the reference point). In"
"what causes these particles to be termed ""massless."" However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference, The invariant mass is the ratio of four-momentum (the four-dimensional generalization of classical momentum) to four-velocity: and is also the ratio of four-acceleration to four-force when the rest mass is constant. The four-dimensional form of Newton's second law is: The relativistic expressions for ""E"" and ""p"" obey the relativistic energy–momentum relation: where the m is the rest mass, or the invariant mass for systems, and ""E"" is the total energy. The equation is"
"does momentum. However, for single particles the rest mass remains constant, and for systems of particles the invariant mass remain constant, because in both cases, the energy and momentum increases subtract from each other, and cancel. Thus, the invariant mass of systems of particles is a calculated constant for all observers, as is the rest mass of single particles. For systems of particles, the energy–momentum equation requires summing the momentum vectors of the particles: The inertial frame in which the momenta of all particles sums to zero is called the center of momentum frame. In this special frame, the relativistic"
"moment,rotational inertia, polar moment of inertia of mass, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbols I and J are usually used to refer to the moment of inertia or polar moment of inertia. While a simple scalar treatment"
"""x"" is finite for some . Moment (mathematics) In mathematics, a moment is a specific quantitative measure of the shape of a function. It is used in both mechanics and statistics. If the function represents physical density, then the zeroth moment is the total mass, the first moment divided by the total mass is the center of mass, and the second moment is the rotational inertia. If the function is a probability distribution, then the zeroth moment is the total probability (i.e. one), the first moment is the mean, the second central moment is the variance, the third standardized moment"
"respective masses and , and velocities and , the total momentum is The momenta of more than two particles can be added more generally with the following: A system of particles has a center of mass, a point determined by the weighted sum of their positions: If all the particles are moving, the center of mass will generally be moving as well (unless the system is in pure rotation around it). If the center of mass is moving at velocity , the momentum is: This is known as Euler's first law. If the net force applied to a particle is"
"the momentum has the classical form , then to first order in (i.e. retain the term for and neglect all terms for ) we have or where the second term is the classical kinetic energy, and the first is the rest mass of the particle. This approximation is not valid for massless particles, since the expansion required the division of momentum by mass. Incidentally, there are no massless particles in classical mechanics. In the case of many particles with relativistic momenta and energy , where (up to the total number of particles) simply labels the particles, as measured in a"
"separate in Newtonian mechanics, form a four-vector in relativity, and this relates the time component (the energy) to the space components (the momentum) in a non-trivial way. For an object at rest, the energy–momentum four-vector is : it has a time component which is the energy, and three space components which are zero. By changing frames with a Lorentz transformation in the x direction with a small value of the velocity v, the energy momentum four-vector becomes . The momentum is equal to the energy multiplied by the velocity divided by ""c"". As such, the Newtonian mass of an object,"
"Mass Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The basic SI unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often 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"
"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, fluids and deformable bodies, a momentum density can be defined, and 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"
"force the object exerts. Replacing momentum by mass times velocity, the law is also written more famously as since is a constant in Newtonian mechanics. Newton's second law applies to point-like particles, and to all points in a rigid body. They also apply to each point in a mass continua, like deformable solids or fluids, but the motion of the system must be accounted for, see material derivative. In the case the mass is not constant, it is not sufficient to use the product rule for the time derivative on the mass and velocity, and Newton's second law requires some"
"concept can also be thought of as a classical physics analogue of the quantum mechanical concept of quasiparticles. It is, however, not to be confused with relativistic mass increase. It is often erroneously stated that the added mass is determined by the momentum of the fluid. That this is not so becomes clear when considering the case of the fluid in a large box, where the fluid momentum is exactly zero at every moment of time. The added mass is actually determined by the quasi-momentum: the added mass times the body acceleration is equal to the time derivative of the"
"pendulum mass around the pivot, where formula_40 is the angular velocity of the mass about the pivot point. This angular momentum is given by using a similar derivation to the previous equation. Similarly, the kinetic energy of the pendulum mass is defined by the velocity of the pendulum around the pivot to yield This shows that the quantity formula_37 is how mass combines with the shape of a body to define rotational inertia. The moment of inertia of an arbitrarily shaped body is the sum of the values formula_1 for all of the elements of mass in the body. A"
"the ""momentum-associated mass"" (formula_4), instead of ""energy-associated mass"" (formula_5), and the photon's momentum and energy must constitute a Lorentz four-vector so that the global momentum-energy conservation law can be satisfied in the Einstein-box thought experiment within the principle-of-relativity frame. Sheppard and Kemp differently identified the difference between canonical (Minkowski) and kinetic (Abraham) momentums, explaining that the canonical momentum or wave momentum ""represents the combination of both field and material momentum values"", while the kinetic momentum ""represents the photon momentum void of material contributions"". This explanation is completely consistent with Alexander Feigel's finding that ""Abraham's expression is indeed the momentum of"
"for clarity. The energy and momentum of an object with invariant mass ""m"" are related by the formulas The first is referred to as the ""relativistic energy–momentum relation"". It can be derived by considering that formula_18 can be written as formula_19 where the denominator can be written as formula_20. Now, gamma can be replaced in the expression of energy. While the energy ""E"" and the momentum p depend on the frame of reference in which they are measured, the quantity ""E"" − (""pc"") is invariant, and arises as −""c"" times the squared magnitude of the 4-momentum vector which is −(""m""""c"")."
"vacuum, the momentum per unit volume is where is the vacuum permeability and is the speed of light. The momentum density is proportional to the Poynting vector which gives the directional rate of energy transfer per unit area: If momentum is to be conserved over the volume over a region , changes in the momentum of matter through the Lorentz force must be balanced by changes in the momentum of the electromagnetic field and outflow of momentum. If is the momentum of all the particles in , and the particles are treated as a continuum, then Newton's second law gives"
"also valid for photons, which have m = 0: and therefore A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always c. For an object at rest, the momentum p is zero, therefore The rest mass is only proportional to the total energy in the rest frame of the object. When the object is moving, the total energy is given by To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to formula_8, is the only"
"still the energy of the first particle in the frame of the second particle . In the rest frame of the particle the inner product of the momentum is Therefore, the rest mass (m) is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated. In many cases the rest mass is written as formula_30 to avoid confusion with the relativistic mass, which is formula_31 Note that The square of the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar. The"
"of space it occupies, despite the fact that matter is made of atoms, has voids, and is discrete. Therefore, when continuum mechanics refers to a point or particle in a continuous body it does not describe a point in the interatomic space or an atomic particle, rather an idealized part of the body occupying that point. Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces formula_28 and body forces formula_29. Thus, the total force formula_30 applied"
"including mass in Newtonian mechanics, see the article on mass. The term ""mass"" in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The ""invariant mass"" is another name for the ""rest mass"" of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the ""rest mass"" of a ""system"". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame (COM frame), as when any"
"with its thermal radiation. In developing special relativity, Einstein found that the kinetic energy of a moving body is with the velocity, the rest mass, and the Lorentz factor. He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the correspondence principle: Without this second term, there would be an additional contribution in the energy when the particle is not moving. Einstein found that the total momentum of a moving particle is: It is this quantity that is conserved in collisions. The ratio"