Page 313 - Dynamics of Mechanical Systems

0593_C09_fm Page 294 Monday, May 6, 2002 2:50 PM

294 Dynamics of Mechanical Systems

By integrating Eqs. (9.6.27) and (9.6.33) over the time interval when the forces are
applied, we obtain:

t 2 t 2
∫ M dt = ( R d A BO ) A () − A () (9.6.34)
∫
dt dt =
t
t
1
O
2
BO
BO
t 1 t 1
and
t 2 t 2
∫
∫ M dt = ( R d A BG ) A () − A () (9.6.35)
dt dt =
t
t
BG
G
1
BG
2
t 1 t 1
or
J =∆ A (9.6.36)
O B O
and
J =∆ A (9.6.37)
G B G
where J and J are the angular impulses of the applied forces about O and G.
G
O
Eqs. (9.6.36) and (9.6.37) are, of course, analogous to Eqs. (9.6.8) and (9.6.16) for a single
particle and for a set of particles. They all state that the angular impulse is equal to the
change in angular momentum when the object point is fixed in an inertial reference frame.
For a rigid body, the object point may also be the mass center. Other object points do not
produce the simple relation between angular impulse and angular momentum as in Eqs.
(9.6.8), (9.6.16), (9.6.36), and (9.6.37).

9.7 Conservation of Momentum Principles

Before looking at examples illustrating the impulse momentum principles, it is helpful to
consider also the conservation of momentum principles. Simply stated, these principles
assert that if the impulse is zero, the momentum is unchanged — that is, the momentum
is conserved.
Referring to Eq. (9.5.3), if the impulse I on a particle is zero we have:

t
∆L = 0 or L() = L() (9.7.1)
t
2 1
Specifically, the linear momentum is the same at the beginning and end of the impulse
time interval. Moreover, because the impulse is zero the time interval is arbitrary. Hence,
t and t are arbitrary, and the linear momentum is unchanged throughout all time inter-
1 2
vals. That is, the linear momentum is constant, or conserved. Expressions analogous to
Eq. (9.7.1) may also be obtained for sets of particles and for a rigid body using Eqs. (9.5.9)
and (9.5.14).