Page 320 - Dynamics of Mechanical Systems

0593_C09_fm Page 301 Monday, May 6, 2002 2:50 PM

Principles of Impulse and Momentum 301

9.9 Additional Examples: Conservation of Momentum
The most useful impulse–momentum principles are those related to the conservation of
momentum, where external impulse to a system is absent. In this section we will consider
a few simple examples to illustrate these principles.

Example 9.9.1: Colliding Collinear Blocks
First, consider two blocks A and B with masses m and m moving along the same line
A B
as in Figure 9.9.1. Let the blocks be moving to the right as shown, and let the speed of A
be greater than the speed of B (v > v ) so that collision occurs. Let the collision be “plastic”
A B
so that the blocks move together as a unit after collision. Given the masses and speeds of
the blocks before collision, the objective is to determine their common speed after collision.
If we consider the two blocks as a system, we observe that there are no external impulses
on the system. This means that the overall momentum of the system is unchanged by the
collision. Thus, we have:

mv + m v = ( m + ) (9.9.1)
m v
AA B B A B
where v represents the common speed of the blocks after collision. Solving for v we obtain:

v = ( m v + m v ) ( m + ) (9.9.2)
m
AA B B A B
Observe in Eq. (9.9.2) that if the masses are equal, v is the average of the speeds of the
blocks prior to collision.

Example 9.9.2: Engaging Coaxial Disks
As a second example of momentum conservation, consider two disks A and B rotating
freely on a common shaft as depicted in Figure 9.9.2. Let the disks have axial moments
of inertia I and I and angular speeds ω and ω as shown. Let the disks be brought
A B A B
together by sliding them toward each other along their common shaft. Let the disks engage
one another and then rotate together with the common angular velocity ω. The objective
is to determine ω.

FIGURE 9.9.1
Colliding collinear blocks (plastic collision).