Page 322 - Dynamics of Mechanical Systems

0593_C09_fm Page 303 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 303

ˆ ω
Equating the expressions of Eqs. (9.9.6) and (9.9.7), we find to be:
ˆ ω = [ { Imr (r h + 2 ω )} (9.9.8)
+
− )] ( I mr

9.10 Impact: Coefficient of Restitution
The principles of impulse and momentum are ideally suited to treat problems involving
impact, where large forces are exerted but only for a short time. Although a detailed
analysis of the contact forces, the stresses, and the deformations of colliding bodies requires
extensive analysis beyond our scope here, the principles of impulse and momentum can
be used to obtain a global description of the phenomenon. That is, by making a couple
of simplifying assumptions we can develop a direct and manageable procedure for study-
ing impact.
Our first assumption is that the time interval during which impact occurs is so short
that the positions and orientations of the colliding bodies do not change significantly
during the impact time interval, although the velocities may have significant incremental
changes. The second assumption is that after the bodies come together and collide they
then generally rebound and separate again. The speed of separation is assumed to be a
fraction of the speed of approach.
To develop these concepts, consider again the principle of linear impulse and momen-
tum. Specifically, consider a particle P of mass m initially at rest but free to move along
the X-axis as in Figure 9.10.1. Let P be subjected to a large force F acting for a short time
t*. Then, from the linear impulse–momentum principle (Eq. (9.5.3)), we have:

t *
I = ∫ Fdt = mv * (9.10.1)
0

*
*
where v is the speed of P at time t . If t is sufficiently small, the impulse I will be finite
*
even though F may be very large. Let the speed v of P be expressed as:
v = dx dt (9.10.2)

*
where x is the displacement of P along the X-axis of Figure 9.10.1. Then, at time t the
displacement is:

t *
x = ∫ vdt < v t (9.10.3)
* *
*
0
where the inequality follows from Eq. (9.10.1).

FIGURE 9.10.1
A large, short duration force applied
to a particle P.