Page 300 - Dynamics of Mechanical Systems

0593_C09_fm Page 281 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 281

FIGURE 9.3.1 FIGURE 9.3.2
A particle P with mass m and velocity v in A set S of moving particles.
reference frame R.

FIGURE 9.3.3
A rigid body B with mass center G moving in
a reference frame R.
Finally, consider a rigid body B with mass m and mass center G moving in a reference
frame R as in Figure 9.3.3. Consider B to be composed of N particles P (i = 1,…, N) with
i
B
masses m . Then, from Eq. (9.3.2), the linear momentum L of B in R is:
i
N
L = ∑ m i v P i (9.3.3)
B
i=1

Because P and G are both fixed in B, their velocities in R are related by the expression
i
(see Eq. (4.9.4)]:
v − v + ωω B × r (9.3.4)
G
P i
i
B
where ωω ωω is the angular velocity of B in R and where r locates P relative to G as in Figure
i
i
9.3.3. By substituting from Eq. (9.3.4) into (9.3.3) L becomes:
B
N
N
i ∑
L = ∑ m i( v + ωω B × r ) = N m i v + ∑ m ωω B × r i
G
B
G
i
i=1 i=1 i=1
(9.3.5)
 N  N
= ∑ m i v + ωω B × ∑ m ii M v G
r =
G
i  =1  i=1