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0593_C10_fm Page 343 Monday, May 6, 2002 2:57 PM

Introduction to Energy Methods 343

O
G
O a
h = a 2 /2
G
Q
Q

FIGURE 10.12.3 FIGURE 10.12.4
Upward resting configuration of the plate. Plate mass center elevation change.
Recall that in Section 9.12 we discovered through the conservation of angular momen-
tum principle that the post-seizure rotation speed is related to the pre-seizure speed Ω
Ω
ˆ
by the expression:
/ )
ˆ
Ω = ( 28 Ω (10.12.1)

In view of the result of Eq. (10.12.1), a question that may be posed is what should the
pre-seizure speed be so that after seizure the plate rotates through exactly 180°, coming
to rest in the upward configuration shown in Figure 10.12.3?
We can answer this question using the work–energy principle. Because gravity is the
only force doing work on the plate, the work W is simply the plate weight multiplied by
the mass center elevation change (see Figure 10.12.4):

W =− mgh =− mga 22 (10.12.2)
/
where the negative sign occurs because the mass center elevation movement is opposite
to the direction of gravity.
The final kinetic energy K is zero because the plate comes to rest. The initial kinetic
f
energy K just after the plate edge is seized, is:
i
K = ( ) ( ) ] ( ) ( [ ) ma Ω 2 ] = ma Ω 2
[
2
2 ˆ
2 ˆ
m a 2 Ω
ˆ
i 1 2 + 1 2 1 12 6 (10.12.3)
The work–energy principle then produces:
2 ˆ
W = K − K or − mga 2 2 = − ma Ω 2 6 (10.12.4)
0
/
f i
ˆ
Ω
Solving for , we obtain:
[
ˆ
Ω= 32 ga ] / 12 (10.12.5)
Then, from Eq. (10.12.1), Ω, the pre-seizure angular speed, is:
Ω ) [ ] 12
/
ˆ
Ω = ( 8 2 = 96 2 ga (10.12.6)