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3.8 At time t = 0, the inertial angular velocity of a rectangular plate in torque-free motion has a magnitude ft and is directed as shown in Fig. P3.8. Determine (a) the

maximum magnitude of the angular velocity subsequent to t = 0 and (b) the time at which the maximum angular velocity is first acquired.

3.9 Referring to Prob. 3.8, let n be a unit vector normal to the plate and suppose that at t = 0 the angular velocity to of the plate has a magnitude ft, is perpendicular to n, and makes a small angle with the side of length 3L. Determine approximately the largest value acquired by to • n for t > 0 and relate the result to considerations of stability of simple rotational motions of the plate.

Result 0.53ft

3.10 If /], I2,are the central principal moments of inertia of a rigid body B, and AT, and K2 are defined as

A h ~ h „ A h ~ h then every real body is represented by a point having coordinates x = K\, y = K2 in a rectangular Cartesian coordinate system; and all such points lie within a square bounded by the lines x - — 1, y = —l,x= 1 , v = 1. Draw this square and shade those of its portions corresponding to unstable simple rotational motions if B is torque-free and the inertial angular velocity of B is parallel to the 3-axis.

Result Figure P3.10

3.11 A uniform rectangular block B has the dimensions shown in Fig. P3.11(a), X|, X2, X} are the central principal axes of inertia of B. In Fig. P3.11(b), six regions of the K\-Kj space of Fig. 3.5.3 are labeled 1, . . . , 6.

Referring to Sec. 3.5, suppose that B remains at rest in A with each of X|, X2, X3 parallel to one of a,, a?, 83. To each such alignment there corresponds one of the numbers in Fig. P3,11(b). For example, the arrangement indicated in Fig. P3,11(c) corresponds to region 6. Draw sketches similar to Fig. P3.11(c) for regions 1 5.

Results Figure P3.11(d)

Figure P3.1 lib)

Figure P3.11(c)

Figure P3.1 lib)

Figure P3.11(c)

0 0

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