Page 309 - Dynamics of Mechanical Systems

0593_C09_fm Page 290 Monday, May 6, 2002 2:50 PM

290 Dynamics of Mechanical Systems

FIGURE 9.6.3 FIGURE 9.6.4
A set S of particles moving in an inertial reference Free-body diagram of typical particle P i.
frame R with reference point Q.

where v P i and a P i are the velocity and acceleration of P in R. Then, from d’Alembert’s
i
principle, we have:

F + F = or F = md v P i dt (9.6.10)
*
0
i i i i
By setting moments about Q equal to zero, we have:

QP × F + QP × F = or QP × F = QP × md v P i dt (9.6.11)
*
0
i i i i i i i i
Consider the final term in Eq. (9.6.11). By the product rule for differentiation, we have
(see Eq. (9.6.4)):

−
×
QP × m d v P i dt = d( QP × m i v ) dt d QP dt m i v P i
P i
i
i
i
i
(9.6.12)
= d A Q P i dt + v × m i v P i
Q
By substituting into Eq. (9.6.11) we have:
Q
M = d A dt + v × m v P i (9.6.13)
FQ P Q i
i
i
By adding the effects from all the particles, we have:
N
M = d A dt + v × ∑ m v P i (9.6.14)
Q
Q S Q i
i=1
If the velocity of Q is zero (that is, if Q is fixed in R), then we have:

M = d A dt (9.6.15)
Q S Q