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0593_C06_fm Page 185 Monday, May 6, 2002 2:28 PM

Forces and Force Systems 185

the particle. However, the inertia force is directed opposite to the acceleration. Thus, if P
is a particle with mass m and with acceleration a in an inertial reference frame R, then the
*
inertia force F exerted on P is:
*
F =−m a (6.9.1)

An inertial reference frame is defined as a reference frame in which Newton’s laws of
motion are valid. From elementary physics, we recall that from Newton’s laws we have
the expression:

F = m a (6.9.2)

where F represents the resultant of all applied forces on a particle P having mass m and
acceleration a. By comparing Eqs. (6.9.1) and (6.9.2), we have:

+
*
FF = 0 (6.9.3)
Equation (6.9.3) is often referred to as d’Alembert’s principle. That is, the sum of the
applied and inertia forces on a particle is zero. Equation (6.9.3) thus also presents an
algorithm or procedure for the analysis of dynamic systems as though they were static
systems.
For rigid bodies, considered as sets of particles, the inertia force system is somewhat
more involved than for that of a single particle due to the large number of particles making
up a rigid body; however, we can accommodate the resulting large number of inertia
forces by using equivalent force systems, as discussed in Section 6.5. To do this, consider
the representation of a rigid body as a set of particles as depicted in Figure 6.9.1. As B
moves in an inertial frame R, the particles of B will be accelerated and thus experience
inertia forces; hence, the inertia force system exerted on B will be made up of the inertia
forces on the particles of B. The inertia force exerted on a typical particle P of B is:
i

F =− m a (6.9.4)
*
i i i
where m is the mass of P and A is the acceleration of P in R.
i
i
i
i
Using the procedures of Section 6.5, we can replace this system of many forces by a
single force F passing through an arbitrary point, together with a couple having a torque
*
B
P (m )
2 2
r
P (m ) i
1 1
P (m )
i i
P (m )
N N
FIGURE 6.9.1
Representation of a rigid body as a R
set of particles.