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It is a three dimensional vector quantity, possessing a direction and a magnitude.If m is an objects mass and v is the velocity also a vector, then the momentum ispmv,displaystyle mathbf p mmathbf v ,In SI units, it is measured in kilogram meters per second kg ms.Newtons second law of motion states that a bodys rate of change in momentum is equal to the net force acting on it.An Introduction To Mechanics Kleppner Solutions Free Download' title='An Introduction To Mechanics Kleppner Solutions Free Download' />Momentum depends on the frame of reference, but in all frames, it is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.Momentum is also conserved in special relativity with a modified formula and, in a modified form, in electrodynamics, quantum mechanics, quantum field theory, and general relativity.It is an expression of one of the fundamental symmetries of space and time, that of translational symmetry.Advanced formulations of classical mechanics, Lagrangian and Hamiltonian mechanics, allow one to choose coordinate systems that incorporate symmetries and constraints.In these systems the conserved quantity is generalized momentum, and 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, 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 NavierStokes equations for fluids or the Cauchy momentum equation for deformable solids or fluids.NewtonianeditMomentum has both magnitude and direction.Since momentum has a direction, it can be used to predict the resulting direction of objects after they collide, and their speeds.Below, the basic properties of momentum are described in one dimension.The vector equations are almost identical to the scalar equations see multiple dimensions.Single particleeditThe momentum of a particle is traditionally represented by the letter p.An Introduction To Mechanics Kleppner Solutions Free Download' title='An Introduction To Mechanics Kleppner Solutions Free Download' />It is the product of two quantities, the mass represented by the letter m and velocity v 1pmv.The units of momentum are the product of the units of mass and velocity.In SI units, if the mass is in kilograms and the velocity in meters per second then the momentum is in kilogram meterssecond kg ms.In cgs units, if the mass is in grams and the velocity in centimeters per second, then the momentum is in gram centimeterssecond g cms.Being a vector, momentum has magnitude and direction.For example, a 1 kg model airplane, traveling due north at 1 ms in straight and level flight, has a momentum of 1 kg ms due north measured from the ground.Many particleseditThe momentum of a system of particles is the sum of their momenta.If two particles have masses m.The momenta of more than two particles can be added more generally with the following pimividisplaystyle psum imiviA system of particles has a center of mass, a point determined by the weighted sum of their positions rcmm.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.Online Documents Library Free download document and ebook.PDF file, ePUB. doc, and more.If the center of mass is moving at velocity vcm, the momentum is pmvcm.This is known as Eulers first law.Relation to forceeditIf a force F is applied to a particle for a time interval t, the momentum of the particle changes by an amountpFt.Delta pFDelta t. In differential form, this is Newtons second law the rate of change of the momentum of a particle is proportional to the force F acting on it,1Fdpdt.Ffrac dpdt. If the force depends on time, the change in momentum or impulse.J between times t.Jt. 1t. 2Ftdt. displaystyle Delta pJint t1t2Ft,dt.Impulse is measured in the derived units of the newton second 1 N s 1 kg ms or dyne second 1 dyne s 1 g msUnder the assumption of constant mass m, it is equivalent to write.Fmdvdtma,displaystyle Fmfrac dvdtma,so the force is equal to mass times acceleration.Example A model airplane of 1 kg accelerates from rest to a velocity of 6 ms due north in 2 s.The net force required to produce this acceleration is 3 newtons due north.The change in momentum is 6 kg ms.James-D.-Lee-9780470402559-500x500.jpg' alt='An Introduction To Mechanics Kleppner Solutions Free Download' title='An Introduction To Mechanics Kleppner Solutions Free Download' />The rate of change of momentum is 3 kg mss 3 N.ConservationeditIn a closed system one that does not exchange any matter with its surroundings and is not acted on by external forces the total momentum is constant.This fact, known as the law of conservation of momentum, is implied by Newtons laws of motion.Suppose, for example, that two particles interact.Because of the third law, the forces between them are equal and opposite.If the particles are numbered 1 and 2, the second law states that F1 dp.F2 dp. 2dt. Therefore,dp.Equivalently,ddtp.If the velocities of the particles are u.This law holds no matter how complicated the force is between particles.Similarly, if there are several particles, the momentum exchanged between each pair of particles adds up to zero, so the total change in momentum is zero.This conservation law applies to all interactions, including collisions and separations caused by explosive forces.It can also be generalized to situations where Newtons laws do not hold, for example in the theory of relativity and in electrodynamics.Dependence on reference frameedit.Newtons apple in Einsteins elevator.In person As frame of reference, the apple has non zero velocity and momentum.In the elevators and person Bs frames of reference, it has zero velocity and momentum.Momentum is a measurable quantity, and the measurement depends on the motion of the observer.For example if an apple is sitting in a glass elevator that is descending, an outside observer, looking into the elevator, sees the apple moving, so, to that observer, the apple has a non zero momentum.To someone inside the elevator, the apple does not move, so, it has zero momentum.The two observers each have a frame of reference, in which, they observe motions, and, if the elevator is descending steadily, they will see behavior that is consistent with those same physical laws.Suppose a particle has position x in a stationary frame of reference.From the point of view of another frame of reference, moving at a uniform speed u, the position represented by a primed coordinate changes with time asxxut.This is called a Galilean transformation.If the particle is moving at speed dxdt v in the first frame of reference, in the second, it is moving at speedvdxdtvu.Since u does not change, the accelerations are the same advdta.Thus, momentum is conserved in both reference frames.Moreover, as long as the force has the same form, in both frames, Newtons second law is unchanged.Forces such as Newtonian gravity, which depend only on the scalar distance between objects, satisfy this criterion.This independence of reference frame is called Newtonian relativity or Galilean invariance.A change of reference frame, can, often, simplify calculations of motion.For example, in a collision of two particles, a reference frame can be chosen, where, one particle begins at rest.Another, commonly used reference frame, is the center of mass frame one that is moving with the center of mass.In this frame, the total momentum is zero.Application to collisionseditBy itself, the law of conservation of momentum is not enough to determine the motion of particles after a collision.Another property of the motion, kinetic energy, must be known.This is not necessarily conserved.If it is conserved, the collision is called an elastic collision if not, it is an inelastic collision.Elastic collisionsedit.Elastic collision of equal masses.Elastic collision of unequal masses.An elastic collision is one in which no kinetic energy is lost. |
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