"changing mass, such as rocket exhaust) and is included in the quantity F. Then, by substituting the definition of acceleration, the equation becomes F = ""m""a. The third law states that all forces between two objects exist in equal magnitude and opposite direction: if one object ""A"" exerts a force F on a second object ""B"", then ""B"" simultaneously exerts a force F on ""A"", and the two forces are equal in magnitude and opposite in direction: F = −F. The third law means that all forces are ""interactions"" between different bodies, or different regions within one body, and thus"
"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"
"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"
"with the corresponding components for other objects and fields. In special relativity, Newton's second law does not hold in the form F = ""m""a, but it does if it is expressed as where p = γ(v)""m""v is the momentum as defined above and ""m"" is the invariant mass. Thus, the force is given by Consequently, in some old texts, γ(v)""m"" is referred to as the ""longitudinal mass"", and γ(v)""m"" is referred to as the ""transverse mass"", which is numerically the same as the relativistic mass. See mass in special relativity. If one inverts this to calculate acceleration from force, one"
"Momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction in three-dimensional space. If is an object's mass and is the velocity (also a vector), then the momentum is In SI units, it is measured in kilogram meters per second (kg⋅m/s). 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"
"the following consequences: For some problems, it is convenient to use rotating coordinates (reference frames). Thereby one can either keep a mapping to a convenient inertial frame, or introduce additionally a fictitious centrifugal force and Coriolis force. Newton was the first to mathematically express the relationship between force and momentum. Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as ""Newton's Second Law"": The quantity ""m""v is called the (canonical) momentum. The net force"
"Newton's second law, The final equation derives from Newton's third law; the force of the second body on the first body (F) is equal and opposite to the force of the first body on the second (F). Thus, the equation of motion for r can be written in the form where formula_12 is the reduced mass As a special case, the problem of two bodies interacting by a central force can be reduced to a central-force problem of one body. The motion of a particle under a central force F always remains in the plane defined by its initial position"
"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"
"second law states that where F is the force on mass 1 due to its interactions with mass 2, and F is the force on mass 2 due to its interactions with mass 1. The two dots on top of the x position vectors denote their second derivative with respect to time, or their acceleration vectors. Adding and subtracting these two equations decouples them into two one-body problems, which can be solved independently. ""Adding"" equations (1) and (2) results in an equation describing the center of mass (barycenter) motion. By contrast, ""subtracting"" equation (2) from equation (1) results in an"
"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"
"is satisfied: Any free motion has a constant magnitude and direction. Newton's second law for a particle takes the form: with F the net force (a vector), ""m"" the mass of a particle and a the acceleration of the particle (also a vector) which would be measured by an observer at rest in the frame. The force F is the vector sum of all ""real"" forces on the particle, such as electromagnetic, gravitational, nuclear and so forth. In contrast, Newton's second law in a rotating frame of reference, rotating at angular rate ""Ω"" about an axis, takes the form: which"
"v is also constant (conservation of momentum). Hence, the position R (""t"") of the center of mass can be determined at all times from the initial positions and velocities. Dividing both force equations by the respective masses, subtracting the second equation from the first and rearranging gives the equation where we have again used Newton's third law F = −F and where r is the displacement vector from mass 2 to mass 1, as defined above. The force between the two objects, which originates in the two objects, should only be a function of their separation r and not of"
"If a body is in equilibrium, there is zero ""net"" force by definition (balanced forces may be present nevertheless). In contrast, the second law states that if there is an ""unbalanced"" force acting on an object it will result in the object's momentum changing over time. By the definition of momentum, where ""m"" is the mass and formula_5 is the velocity. If Newton's second law is applied to a system of constant mass, ""m"" may be moved outside the derivative operator. The equation then becomes By substituting the definition of acceleration, the algebraic version of Newton's Second Law is derived:"
"relativity can be turned into their general relativity counterparts by replacing the Minkowski metric (""η"") with the relevant metric of spacetime (""g"") and by replacing any partial derivatives with covariant derivatives. In the discussions that follow, the change of metrics is implied. Inertial motion is motion free of all forces. In Newtonian mechanics, the force ""F"" acting on a particle with mass ""m"" is given by Newton's second law, formula_1, where the acceleration is given by the second derivative of position ""r"" with respect to time ""t"" . Zero force means that inertial motion is just motion with zero acceleration:"
"mass ""m"" varies with time by either ejecting or accreting mass is obtained by applying the second law to the entire, constant-mass system consisting of the body and its ejected/accreted mass; the result is where u is the velocity of the escaping or incoming mass relative to the body. From this equation one can derive the equation of motion for a varying mass system, for example, the Tsiolkovsky rocket equation. Under some conventions, the quantity u d""m""/d""t"" on the left-hand side, which represents the advection of momentum, is defined as a force (the force exerted on the body by the"
"motion, and is the time derivative of its position, thusformula_1In Newtonian mechanics, the equations of motion are given by Newton's laws. The second law ""net force equals mass times acceleration"",formula_2applies to each particle. For an ""N"" particle system in 3 dimensions, there are 3""N"" 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 units of energy, but no single expression for"
"results in force contributions by added mass and the Basset force. The BBO equation states: This is Newton's second law, in which the left-hand side is the rate of change of the particle's linear momentum, and the right-hand side is the summation of forces acting on the particle. The terms on the right-hand side are, respectively, the: The particle Reynolds number formula_14 has to be less than unity, formula_16, for the BBO equation to give an adequate representation of the forces on the particle. Also suggest to estimate the pressure gradient from the Navier–Stokes equations: with formula_18 the material derivative"
"of change of the momentum. Any mass that is gained or lost by the system will cause a change in momentum that is not the result of an external force. A different equation is necessary for variable-mass systems (see below). Newton's second law is an approximation that is increasingly worse at high speeds because of relativistic effects. An impulse J occurs when a force F acts over an interval of time Δ""t"", and it is given by Since force is the time derivative of momentum, it follows that This relation between impulse and momentum is closer to Newton's wording of"
"more complicated than for test particles, although it remains true that spacetime tells matter how to move. In Newton's description of gravity, the gravitational force is caused by matter. More precisely, it is caused by a specific property of material objects: their mass. In Einstein's theory and related theories of gravitation, curvature at every point in spacetime is also caused by whatever matter is present. Here, too, mass is a key property in determining the gravitational influence of matter. But in a relativistic theory of gravity, mass cannot be the only source of gravity. Relativity links mass with energy, and"
"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 then are fixed. The use of Newton's Second Law as a ""definition"" of force has been disparaged in some of the more rigorous textbooks, because it is essentially a mathematical truism. Notable physicists, philosophers and mathematicians who have sought a more explicit definition of the concept of force include Ernst Mach and Walter Noll. Newton's Second Law can be used to measure the strength of forces. For instance, knowledge of the masses of planets along"
"more than the simple product of the mass and velocity of the exiting projectile, for this reason. A consideration of the energy released during the firing of a gun leads to an additional equation useful in recoil analysis. Using Newton's second law, force equals mass times acceleration, the energy within a moving body due to its velocity change caused by that acceleration can be stated mathematically from the translational kinetic energy as: where: This equation is known as the ""classic statement"" and yields a measurement of energy in joules (or foot-pound force in non-SI units). formula_37 is the amount of"
"namely where ""F"" is the force acting on the mass ""m"" caused by the gravitational attraction mass ""m"" has for ""m"", ""G"" is the universal gravitational constant, and ""r"" is the distance between the two masses centers. From Newton's Second Law, the summation of the forces acting on ""m"" related to that bodies acceleration: where ""A"" is the acceleration of ""m"" caused by the force of gravitational attraction ""F"" of ""m"" acting on ""m"". Combining Eq 1 and 2: Solving for the acceleration, ""A"": where formula_5 is the standard gravitational parameter, in this case formula_6. It is understood that the"
"the two particles are related by the equation ""θ""(""t"") = ""k θ""(""t""). Newton showed that the motion of the second particle can be produced by adding an inverse-cube central force to whatever force ""F""(""r"") acts on the first particle where ""L"" is the magnitude of the first particle's angular momentum, which is a constant of motion (conserved) for central forces. If ""k"" is greater than one, ""F"" − ""F"" is a negative number; thus, the added inverse-cube force is ""attractive"", as observed in the green planet of Figures 1–4 and 9. By contrast, if ""k"" is less than one, ""F""−""F"""
"force, ""m"" is the inertial mass, and ""a"" is the acceleration. By this formula, the greater its mass, the less a body accelerates under a given force. The masses defined by the above formulas are equal because the latter formula is a consequence of the former, if mass does not depend on time and velocity. Thus, ""mass is the quantitative or numerical measure of a body’s inertia, that is of its resistance to being accelerated"". This meaning of a ""body's inertia"" therefore is altered from the popular meaning as ""a tendency to maintain momentum"" to a description of the measure"
"the center of mass accelerate in proportion to the magnitude of the external force divided by the mass of the system. Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of a system is conserved. In a system of two particles, if formula_14 is the momentum of object 1 and formula_15 the momentum of object 2, then Using similar arguments, this can be generalized to a system with an arbitrary number of particles. In general, as long as all forces are due to the interaction of objects with mass, it is possible to define"
"use of calculus. An example of the use of calculus in mechanics is Newton's second law of motion: historically stated it expressly uses the term ""change of motion"" which implies the derivative saying ""The"" change ""of momentum of a body is equal to the resultant force acting on the body and is in the same direction."" Commonly expressed today as Force = Mass × acceleration, it implies differential calculus because acceleration is the time derivative of velocity or second time derivative of trajectory or spatial position. Starting from knowing how an object is accelerating, we use calculus to derive its"
"is an infinitesimal volume element, ""ρ"" is the mass density, and g is the ""external body force density field"" acting on the system. Like any other force, a body force will cause an object to accelerate. For a non-rigid object, Newton's second law applied to a small volume element is where ""ρ""(r) is the mass density of the substance, ƒ the force density, and a(r) all at point r. In the case of gravity on a planet surface, g(r) is simply the approximately constant and uniform gravitational field, like on Earth where: Body force A body force is a force"
"equals the change in the kinetic energy of the particle."" That is, the work ""W"" done by the resultant force on a particle equals the change in the particle's kinetic energy formula_42, where formula_44 and formula_45 are the speeds of the particle before and after the work is done, and ""m"" is its mass. The derivation of the ""work–energy principle"" begins with Newton’s second law of motion and the resultant force on a particle. Computation of the scalar product of the forces with the velocity of the particle evaluates the instantaneous power added to the system. Constraints define the direction"
"consider a spherically symmetrical Earth and subtract from the mass of the Earth that of the shell outside the point of observation, because that does not cause gravity inside. This gives the same result. From the law of universal gravitation, the force on a body acted upon by Earth's gravity is given by where ""r"" is the distance between the centre of the Earth and the body (see below), and here we take ""m"" to be the mass of the Earth and ""m"" to be the mass of the body. Additionally, Newton's second law, ""F"" = ""ma"", where ""m"" is"
"embraces Galileo's, Newton's, and Einstein's conceptualization. The equivalence principle does not deny the existence of measurable effects caused by a ""rotating"" gravitating mass (frame dragging), or bear on the measurements of light deflection and gravitational time delay made by non-local observers. By definition of active and passive gravitational mass, the force on formula_2 due to the gravitational field of formula_3 is: Likewise the force on a second object of arbitrary mass due to the gravitational field of mass is: By definition of inertial mass: If formula_7 and formula_8 are the same distance formula_9 from formula_10 then, by the weak equivalence"
"constant factor ""k"" The areal velocity of the second particle equals that of the first particle multiplied by the same factor ""k"" Since ""k"" is a constant, the second particle also sweeps out equal areas in equal times. Therefore, by Proposition 2, the second particle is also acted upon by a central force ""F""(""r""). This is the conclusion of Proposition 43. To find the magnitude of ""F""(""r"") from the original central force ""F""(""r""), Newton calculated their difference using geometry and the definition of centripetal acceleration. In Proposition 44 of his ""Principia"", he showed that the difference is proportional to the"
"law, with mass fixed, is not invariant under a Lorentz transformation. However, it can be made invariant by making the ""inertial mass"" of an object a function of velocity: The modified momentum, obeys Newton's second law: Within the domain of classical mechanics, relativistic momentum closely approximates Newtonian momentum: at low velocity, is approximately equal to , the Newtonian expression for momentum. In the theory of special relativity, physical quantities are expressed in terms of four-vectors that include time as a fourth coordinate along with the three space coordinates. These vectors are generally represented by capital letters, for example for position."
"a system such that net momentum is never lost nor gained. In the special theory of relativity, mass and energy are equivalent (as can be seen by calculating the work required to accelerate an object). 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 remains valid because it is a mathematical definition. But for relativistic momentum to be conserved, it must be redefined as: where formula_19 is the rest mass and formula_20 the"