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Publicado: 6 de abril de 2012
CHAPTER
12
Kinetics of Particles: Newton’s Second Law
Caption to come
12.1. INTRODUCTION KINETICS OF PARTICLES: NEWTON’S SECOND LAW
12.1 12.2 12.3 Introduction Newton’s Second Law of Motion Linear Momentum of a Particle. Rate of Change of Linear Momentum Systems of Units Equations of Motion Dynamic Equilibrium Angular Momentum of a Particle. Rate of Change of Angular MomentumEquations of Motion in Terms of Radial and Transverse Components Motion under a Central Force. Conservation of Angular Momentum Newton’s Law of Gravitation Trajectory of a Particle under a Central Force Application to Space Mechanics Kepler’s Laws of Planetary Motion
12.4 12.5 12.6 12.7
12.8
12.9
12.10 12.11 12.12 12.13
Newton’s first and third laws of motion were used extensively instatics to study bodies at rest and the forces acting upon them. These two laws are also used in dynamics; in fact, they are sufficient for the study of the motion of bodies which have no acceleration. However, when bodies are accelerated, that is, when the magnitude or the direction of their velocity changes, it is necessary to use Newton’s second law of motion to relate the motion of the body withthe forces acting on it. In this chapter we will discuss Newton’s second law and apply it to the analysis of the motion of particles. As we state in Sec. 12.2, if the resultant of the forces acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force. Moreover, the ratio of the magnitudes of theresultant force and of the acceleration can be used to define the mass of the particle. In Sec. 12.3, the linear momentum of a particle is defined as the product L mv of the mass m and velocity v of the particle, and it is demonstrated that Newton’s second law can be expressed in an alternative form relating the rate of change of the linear momentum with the resultant of the forces acting on thatparticle. Section 12.4 stresses the need for consistent units in the solution of dynamics problems and provides a review of the International System of Units (SI units) and the system of U.S. customary units. In Secs. 12.5 and 12.6 and in the Sample Problems which follow, Newton’s second law is applied to the solution of engineering problems, using either rectangular components or tangential andnormal components of the forces and accelerations involved. We recall that an actual body—including bodies as large as a car, rocket, or airplane—can be considered as a particle for the purpose of analyzing its motion as long as the effect of a rotation of the body about its mass center can be ignored. The second part of the chapter is devoted to the solution of problems in terms of radial andtransverse components, with particular emphasis on the motion of a particle under a central force. In Sec. 12.7, the angular momentum HO of a particle about a point O is defined as the moment about O of the linear momentum of the particle: HO r mv. It then follows from Newton’s second law that the rate of change of the angular momentum HO of a particle is equal to the sum of the moments about O ofthe forces acting on that particle. Section 12.9 deals with the motion of a particle under a central force, that is, under a force directed toward or away from a fixed point O. Since such a force has zero moment about O, it follows that the angular momentum of the particle about O is conserved. This property greatly simplifies the analysis of the motion of a particle under a central force; in Sec.12.10 it is applied to the solution of problems involving the orbital motion of bodies under gravitational attraction. Sections 12.11 through 12.13 are optional. They present a more extensive discussion of orbital motion and contain a number of problems related to space mechanics.
692
12.2. NEWTON’S SECOND LAW OF MOTION
12.2. Newton’s Second Law of Motion
693
Newton’s second law...
12
Kinetics of Particles: Newton’s Second Law
Caption to come
12.1. INTRODUCTION KINETICS OF PARTICLES: NEWTON’S SECOND LAW
12.1 12.2 12.3 Introduction Newton’s Second Law of Motion Linear Momentum of a Particle. Rate of Change of Linear Momentum Systems of Units Equations of Motion Dynamic Equilibrium Angular Momentum of a Particle. Rate of Change of Angular MomentumEquations of Motion in Terms of Radial and Transverse Components Motion under a Central Force. Conservation of Angular Momentum Newton’s Law of Gravitation Trajectory of a Particle under a Central Force Application to Space Mechanics Kepler’s Laws of Planetary Motion
12.4 12.5 12.6 12.7
12.8
12.9
12.10 12.11 12.12 12.13
Newton’s first and third laws of motion were used extensively instatics to study bodies at rest and the forces acting upon them. These two laws are also used in dynamics; in fact, they are sufficient for the study of the motion of bodies which have no acceleration. However, when bodies are accelerated, that is, when the magnitude or the direction of their velocity changes, it is necessary to use Newton’s second law of motion to relate the motion of the body withthe forces acting on it. In this chapter we will discuss Newton’s second law and apply it to the analysis of the motion of particles. As we state in Sec. 12.2, if the resultant of the forces acting on a particle is not zero, the particle will have an acceleration proportional to the magnitude of the resultant and in the direction of this resultant force. Moreover, the ratio of the magnitudes of theresultant force and of the acceleration can be used to define the mass of the particle. In Sec. 12.3, the linear momentum of a particle is defined as the product L mv of the mass m and velocity v of the particle, and it is demonstrated that Newton’s second law can be expressed in an alternative form relating the rate of change of the linear momentum with the resultant of the forces acting on thatparticle. Section 12.4 stresses the need for consistent units in the solution of dynamics problems and provides a review of the International System of Units (SI units) and the system of U.S. customary units. In Secs. 12.5 and 12.6 and in the Sample Problems which follow, Newton’s second law is applied to the solution of engineering problems, using either rectangular components or tangential andnormal components of the forces and accelerations involved. We recall that an actual body—including bodies as large as a car, rocket, or airplane—can be considered as a particle for the purpose of analyzing its motion as long as the effect of a rotation of the body about its mass center can be ignored. The second part of the chapter is devoted to the solution of problems in terms of radial andtransverse components, with particular emphasis on the motion of a particle under a central force. In Sec. 12.7, the angular momentum HO of a particle about a point O is defined as the moment about O of the linear momentum of the particle: HO r mv. It then follows from Newton’s second law that the rate of change of the angular momentum HO of a particle is equal to the sum of the moments about O ofthe forces acting on that particle. Section 12.9 deals with the motion of a particle under a central force, that is, under a force directed toward or away from a fixed point O. Since such a force has zero moment about O, it follows that the angular momentum of the particle about O is conserved. This property greatly simplifies the analysis of the motion of a particle under a central force; in Sec.12.10 it is applied to the solution of problems involving the orbital motion of bodies under gravitational attraction. Sections 12.11 through 12.13 are optional. They present a more extensive discussion of orbital motion and contain a number of problems related to space mechanics.
692
12.2. NEWTON’S SECOND LAW OF MOTION
12.2. Newton’s Second Law of Motion
693
Newton’s second law...
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