Page 203 - Dynamics of Mechanical Systems

0593_C06_fm Page 184 Monday, May 6, 2002 2:28 PM

184 Dynamics of Mechanical Systems

S
P 2
P 1
G r
i P i k

FIGURE 6.8.4 m g m g
2
1
Gravitational forces on the particles m g
i
of a body.
Equation (6.8.6) demonstrates the existence of G by providing an algorithm for its location.
We may think of a body as though it were composed of particles, just as a sandstone is
composed of particles of sand. Then, the sums in Eqs. (6.8.2) through (6.8.7) become very
large, and in the limit they may be replaced by integrals.
For homogeneous bodies, the mass is uniformly distributed throughout the region, or
geometric figure, occupied by the body. The mass center location is then solely determined
by the shape of the figure of the body. The mass center position is then said to be at the
centroid of the geometric figure of the body. The centroid location for common and simple
geometric figures may be determined by routine integration. The results of such integra-
tions are listed in figurative form in Appendix I. As the name implies, a centroid is at the
intuitive center or middle of a figure.
As an illustration of these concepts, consider the gravitational forces acting on a body
B with an arbitrary shape. Let B be composed of N particles P having masses m i
i
(i = 1,…, N), and let G be the mass center of B, as depicted in Figure 6.8.4.
Let the set of all the gravitational forces acting on B be replaced by a single force W
passing through G together with a couple with torque T. Then, from the definition of
equivalent force systems, W and T are:


N
N
W = ∑ mg k =  ∑ m g k = Mg k (6.8.8)

i
i
= i 1  i =1 
and
N  N 
T = r × mg k = ∑ m r × g k = 0 (6.8.9)

∑ i i ii
= i 1  i =1 
where k is a downward-directed unit vector as in Figure 6.8.4 and M is the total mass of
particles of B. The last equality of Eq. (6.8.9) follows from the definition of the mass center
in Eq. (6.8.3).
Equations (6.8.7) and (6.8.8) show how dramatic the reduction in forces can be through
the use of equivalent force systems.

6.9 Physical Forces: Inertia (Passive) Forces

Inertia forces arise due to acceleration of particles and their masses. Specifically, the inertia
force on a particle is proportional to both the mass of the particle and the acceleration of