Page 37 - Dynamics of Mechanical Systems

0593_C02_fm Page 18 Monday, May 6, 2002 1:46 PM

18 Dynamics of Mechanical Systems

B
C
Z Z
R
B
C A

Y Y
A
X X
FIGURE 2.3.6 FIGURE 2.3.7
Vectors A, B, and C to be added. Resultant of vectors A, B, and C of Figure 2.3.6.

Example 2.3.2: Resultant Magnitude in Three Dimensions
As an illustration, suppose C is perpendicular to the plane of vectors A and B of Example
2.3.1 and suppose the magnitude of C is 10 N. Then, from the results of Eq. (2.3.5), the
magnitude of R is:

[ ) +() 2 ] 12
/
2
R = (23 43 10 = 25 47N (2.3.8)
.
.
This procedure has several remarkable features. First, as before, the order of the compo-
nents in Eq. (2.3.7) is unimportant. That is,

+
=
+
+
=
+
+
+
AB C C AB B C A
(2.3.9)
+
=
+
+
=
+
+
= AC B B AC C B A = R
+
Second, to obtain the resultant R we may first add any two of the vectors, say A and B,
and then add the resultant of this sum to C. This means the summation in Eq. (2.3.7) is
associative. That is,
+
+
+
+
R = A BC = ( A B) + C = A +( BC) (2.3.10)
This feature may be used to obtain the magnitude of the resultant by repeated use of the
law of cosines as before.
Third, observe that configurations exist where the magnitude of the resultant is less
than the magnitude of the individual components. Indeed, the magnitude of the resultant
could be zero.
Finally, with three or more components, the procedure of finding the magnitude of the
resultant by repeated use of the law of cosines is cumbersome and tedious.
An attractive feature of vector addition is that the resultant magnitude may be obtained
by strictly analytical means — that is, without regard to triangle geometry. This is the
original reason for using vectors in analysis. We will explore this further in the next section.