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0593_C08_fm Page 253 Monday, May 6, 2002 2:45 PM

Principles of Dynamics: Newton’s Laws and d’Alembert’s Principle 253

where d is the distance from the origin to where G returns again to the horizontal plane,
or to the X-axis (see Figure 8.7.3). For a given V , Eq. (8.7.14) shows that d has a maximum
O
value when θ is 45°.
Next, consider Eq. (8.7.5). Suppose that the unit vectors n (i = 1, 2, 3) are not only parallel
i
to principal inertia axes but are also fixed in B. Then, from Eq. (4.6.6), we have:
ω
ω
ω
α = ˙ , α = ˙ , α = ˙ (8.7.15)
1 1 2 2 3 3
Equation (8.7.5) then takes the form (see Eqs. (8.6.10) to (8.6.12)):
˙ ω = ω ω (I − I (8.7.16)
1 2 3 22 33) I 11

˙ ω = ω ω (I − I (8.7.17)
2 3 1 33 11) I 22

˙ ω = ω ω (I − I (8.7.18)
3 1 2 11 22) I 33

These equations form a system of nonlinear differential equations. A simple solution of
the equations is seen to be:

ω = ω , ω = ω = 0 (8.7.19)
1 0 2 3
That is, a projectile can rotate with constant speed about a central principal inertia axis.
(We will examine the stability of such rotation in Chapter 13.)
Finally, observe that if a projectile B is rotating about a central principal axis and a point
Q of B is not on the central principal axis, then Q will move on a circle whose center
moves on a parabola. Moreover, a projectile always rotates about its mass center, which
in turn has planar motion on a parabola.

8.8 A Rotating Circular Disk

For another illustration of the effects of inertia forces and inertia torques, consider the
circular disk D with radius r rotating in a vertical plane as depicted in Figure 8.8.1. Let D
be supported by frictionless bearings at its center O.

FIGURE 8.8.1
A rotating circular disk.