Page 329 - Dynamics of Mechanical Systems

0593_C09_fm Page 310 Monday, May 6, 2002 2:50 PM

310 Dynamics of Mechanical Systems

FIGURE 9.12.1 FIGURE 9.12.2 FIGURE 9.12.3
Suspended spinning square Arrested plate spinning about an Plate geometry.
plate. edge.

The angular momentum of the plate about the seized edge after seizure is:

ˆ
ˆ
A = I ΩΩ n (9.12.3)
OQ OQ
where I is the moment of inertia of the plate about edge OQ. From the parallel axis
OQ
theorem (Eq. (7.6.6)), I is seen to be:
OQ
2
2
I =+ 2 = (1 12 ) ma +( ) 4 ma = ( ) 3 ma 2 (9.12.4)
1
1
I ma 4
OQ
(Note that due to symmetry the moment of inertia of a square plate about a diagonal is
ˆ
the same as the central moment of inertia about a line parallel to an edge.) Hence, A
OQ
becomes:
ˆ
ˆ
2
13
A = ( ) ma ΩΩ n (9.12.5)
OQ
ˆ
By equating A and A OQ we obtain the result:
OQ
ˆ
ˆ
ΩΩ = ( 28 ΩΩ ) or ΩΩΩΩ = 28 (9.12.6)

9.13 Closure

The principles of impulse and momentum are best applied with systems experiencing
impact and collision. The impact forces are usually large but the time of application is
short, thus producing a finite impulse. During this short impact time, the system config-
uration does not change significantly. The system velocities, however, may change sub-
stantially. Eqs. (9.5.14) and (9.6.37) are expressions typical of those which may be used for
calculating these velocity changes.
The impulse–momentum principles are especially useful if an impact is contained within
the system, as with a collision. With such events, the overall momentum of the system is