Page 201 - Dynamics of Mechanical Systems

0593_C06_fm Page 182 Monday, May 6, 2002 2:28 PM

182 Dynamics of Mechanical Systems

By eliminating S and S between these three equations, we have:
B
A
ˆ
ˆ
S + S = 0 (6.7.13)
A B
Now, suppose that S ˆ A is represented by a single force, say F , passing through some
A
common point C of A and B (or A and B extended) together with a couple with torque
T . Similarly, let S ˆ B be represented by a single force F passing through C together with
A
B
a couple with torque T . Then, because S ˆ A and S ˆ B taken together form a zero system (Eq.
B
(6.7.3)), the resultant of S ˆ A and S ˆ B and the moment of S ˆ A and S ˆ B about C must be zero.
That is,
F + F = 0 or F = − F (6.7.14)
A B A B
and

T + T = 0 or T = − T (6.7.15)
A B A B

Equations (6.7.13), (6.7.14), and (6.7.15), or the equivalent wording, represent the law of
action and reaction.

6.8 First Moments
Consider a particle P with mass m (or, alternatively, a point P with associated mass m) as
depicted in Figure 6.8.1. Let O be an arbitrary reference point, and let p be a position
vector locating P relative to O. The first moment of P relative to O, φ P/O , is defined as:

φ PO D mp (6.8.1)
=
Consider next a set S of N particles P (i = 1,…, N) having masses as in Figure 6.8.2,
i
where O is an arbitrary reference point. The first moment of S for O, φ S/O , is defined as
the sum of the first moments of the individual particles of S for O. That is,

N
=
φ SO D + m p +K + m p N ∑ m p (6.8.2)
= mp
11 2 2 N i i
i=1
Observe that, in general, φ S/O is not zero. However, if a point G can be found such that the
S/G
first moment of S relative to G, φ , is zero, then G is defined as the mass center of S.
Using this definition, the existence and location of G can be determined from Eq. (6.8.2).
Specifically, if G is the mass center and if r locates p relative to G, as in Figure 6.8.3, then
i i
the first moment of S relative to G may be expressed as:

N
∑
φ = m r = 0 (6.8.3)
SG
ii
i=1