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0593_C04*_fm Page 90 Monday, May 6, 2002 2:06 PM

90 Dynamics of Mechanical Systems

b. For the launch at the Equator, the position vector OR is (r + h)i. In this case, Eq.
(4.6.7) becomes:

dr h i )
V = ( ×( r h i )
A
A R E + dt+ ω E +
˙
= hi +( r h)ω j
+
×
= o V i + ( [ 3960 )(5280 ) + h]( .7 27 10 −5 j )
or

h )
R
A V = o V i +(1520 + ω jft sec (4.6.11)
Observe that h is small (at least, initially) compared with r. Thus, a reasonable
A
approximation to V is:
R
V = V i + 1520 jft sec (4.6.12)
A R
O
Observe also the differences in the results of Eqs. (4.6.10) and (4.6.11).

4.7 Addition Theorem for Angular Velocity
Equation (4.6.6) is useful for establishing the addition theorem for angular velocity — one
of the most important equations of rigid body kinematics. Consider a body B moving in
ˆ
R
a reference frame , which in turn is moving in a reference frame R as depicted in Figure
4.7.1. Let V be an arbitrary vector fixed in B. Using Eq. (4.6.6), the derivative of V in R is:
R R ˆ R R ˆ
dV dt = dV dt+ ωω × V (4.7.1)
ˆ
Because V is fixed in B its derivatives in R and may be expressed in the forms (see
R
Eq. (4.5.2)):

R ˆ
R dV dt = ωω R × V and R ˆ dV dt = ωω B × V (4.7.2)
R
B

V

ˆ
R

FIGURE 4.7.1 R
Body B moving in reference frame
ˆ
R and R.