Page 274 - Dynamics of Mechanical Systems

0593_C08_fm Page 255 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 255

FIGURE 8.8.4
Free body diagrams of the disk and
weight for the loading of Figure
8.8.2b.

By eliminating T from these last two expressions and solving for α , we obtain:
b
O (
α = Wr I + mr ) (8.8.8)
2
b
From Eq. (8.8.3), α is:
a
α = Wr I (8.8.9)
a O

By comparing Eqs. (8.8.8) and (8.8.9), we see the effect of the inertia of the weight in
reducing the angular acceleration of the disk.

8.9 The Rod Pendulum
For another illustration of the use of Eqs. (8.6.10), (8.6.11), and (8.6.12), consider a rod of
length supported by a frictionless hinge at one end and rotating in a vertical plane as
shown in Figure 8.9.1. Because the rod rotates in a vertical plane about a fixed horizontal
axis, the angular velocity and angular acceleration of the rod are simply:

˙
˙˙
ωω = θn and αα = θn (8.9.1)
z z
where θ is the inclination angle (see Figure 8.9.1), and n is a unit vector parallel to the
z
axis of rotation and perpendicular to the unit vectors n , n , n , and n as shown in
x y r θ
Figure 8.9.1.

FIGURE 8.9.1
The rod pendulum.