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Fowles09 - CHAPTER 9 MOTION OF RIGID BODIES IN THREE...

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CHAPTER 9 MOTION OF RIGID BODIES IN THREE DIMENSIONS 9.1 (a) () 22 xx I yzd m =+ dm dxdy ρ = and ma 2 2 = z ( ) 2 2 00 0 ya x a xx yx I yd x == ∫∫ d y 2a a x = 2 0 2 a ayd y 2 4 2 33 xx ma Ia y ( ) 2 2 yy I xzd m xd x d = y 3 0 8 3 a a dy = 42 84 yy am I a From the perpendicular axis theorem: 2 5 3 zz xx yy ma III =++ 2 xy yx I Ix y d m x y d x = d y 2 0 4 2 a a ydy =− 2 4 2 xy yx ma II a − = 0 xz zx yz zy I z d m I = I (b) 2 5 α = cos , 1 cos 5 β = , cos 0 γ = From equation 9.1.10 … 2 44 1 2 1 2 35 3 5 2 55 ma ma ma I  =+ +   2 2 15 ma = (c) ˆˆ ˆ cos cos 2 5 ij i ˆ j ω ωω = G + From equation 9.1.29 … 1
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22 2 ˆˆ 32 2 55 5 5 ma ma ma ma Li j ωω ω   =⋅ +− + −+     G 2 4 3 () 2 2 65 ma =+ G j (d) From equation 9.1.32: 2 11 21 25 65 ma TL m a 5 = ⋅ += G G 9.2 ( a ) ( ) ˆ 3 ijk = ++ G 2 2 2 2 12 3 xx rod yy zz ma I Im a I I = == = = a -a a -a -a a 0 xy Ix y d m = −= since, for each rod, either x or or both are 0. The same is true for the other products of inertia. y From equation 9.1.29: ( ) 2 2 ˆ 3 3 Lm a i j k = ⋅+ + G From equation 9.1.32, 222 12 111 23 333 ma ma T 2 + = (b) From equation 9.1.10, with the moments of inertia equal to 2 2 3 ma and the products of inertia equal to 0: 2 cos cos cos 33 ma 2 I ma αβ γ + = (c) For the x-axis being any axis through the center of the lamina and in the plane of the lamina, and the y-axis also in the plane of the lamina … xx yy I I = due to the similar geometry of the mass distributions with respect to the x- and y-axes. From the perpendicular axis theorem: zz xx yy I II 2 zz xx I I = From Table 8.3.1, 2 6 zz ma I = 2 12 xx ma I = 2
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9.3 (a) From equation 9.2.13, 2 tan 2 xy xx yy I I I θ = From Prob. 9.1, 2 3 xx ma I = , 2 4 3 yy ma I = , 2 2 xy ma I =− tan 2 1 = 2 45 22.5 θθ == DD The 1-axis makes an angle of with the x-axis. 22.5 D (b) From symmetry, the principal axes are parallel to the sides of the lamina and perpendicular to the lamina, respectively. 9.4 (a) From symmetry, the coordinate axes are principal axes. From Table 8.3.1: ω G z x ()() 22 2 1 13 23 12 12 m I aa m  =+ =  a () 2 2 10 3 12 12 m I m =+= a 2 3 5 2 12 12 m I m a y 123 ˆˆˆ 14 eee + G From equation 9.2.5, 2 11 3 1 0 4 5 9 21 2 1 4 1 2 1 4 1 2 1 4 Tm a m a m a 2 ωω = +⋅  7 24 ma = (b) From equation 9.2.4, 13 10 2 5 3 12 12 12 14 14 14 L e ma e ma e ma 2  =⋅   G 2 12 13 20 15 12 14 ma Le e + G 3 e ( ) ( )( ) ( )( ) 222 2 2 2 3 0 31 5 cos 1 32 01 5 L L ++ + + G G 1 2 98 0.9295 11,116 21.6 = D 9.5 (a) Select coordinate axes such that the axis of the rod is the 3-axis, its center is at the 3
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origin, and ω G lies in the 1, 3 plane. G α 3 2 1 From Table 8.3.1, 2 12 12 ml == II , I 3 0 = ( ) 13 ˆˆ sin cos ee ωα =+ G α From equation 9.2.4, () 22 11 sin 0 0 sin 12 12 ml ml Le e αα + = G L G is perpendicular to the rod, and 2 sin 12 ml L = G (b) Since G is constant, from equations. 9.3.5 … 1 00 N 2 2 0c o s s i n 12 ml N    3 N 2 ˆ sin cos 12 ml Ne = G G N is perpendicular to the rod ( direction) and to 3 ˆ e L G ( e direction), and 1 ˆ sin cos 12 ml N = G 9.6 From Problem 9.4 … 123 ˆˆˆ 23 14 eee + G 2 1 13 12 I ma = , 2 2 10 12 I ma = , and 2 3 5 12 I ma = From eqns. 9.3.5: 2 1 02 35 1 0 14 12 28 ma ma N ωω = 2 5 2 2 03 3 5 14 12 28 ma ma N − = 2 4 ( ) 3
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Fowles09 - CHAPTER 9 MOTION OF RIGID BODIES IN THREE...

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