Newton's first and third laws of motion were used extensively in statics
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 with the 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 the resultant 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 that particle.
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 and normal
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 and transverse components, with particular
emphasis on the motion of a particle under a central force. In Sec.
12.7, the angular momentumHO of a particle about a point O is defined
as the moment about O of the linear momentum of the particle:
HO = r X 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 of the 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.
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