# 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) ( ) 2 2 xx I y z dm = + dm dxdy ρ = and m a 2 2 ρ = z ( ) 2 2 0 0 0 y a x a xx y x I y dx ρ = = = = = + dy 2a a x = 2 0 2 a a y dy ρ 2 4 2 3 3 xx ma I a ρ = = y ( ) 2 2 2 2 0 0 y a x a yy y x I x z dm x dxd ρ = = = = = + = y 3 0 8 3 a a dy ρ = 4 2 8 4 3 3 yy a m I ρ = = a From the perpendicular axis theorem: 2 5 3 zz xx yy ma I I I = + + 2 0 0 y a x a xy yx y x I I xydm xy dx ρ = = = = = = − = − dy 2 0 4 2 a a ydy ρ = − 2 4 2 xy yx ma I I a ρ = = − = − 0 xz zx yz zy I I xzdm I = = − = = = I (b) 2 5 α = cos , 1 cos 5 β = , cos 0 γ = From equation 9.1.10 … 2 2 2 4 4 1 2 1 2 3 5 3 5 2 5 5 ma ma ma I   = + +   2 2 15 ma = (c) ( ) ( ) ˆ ˆ ˆ cos cos 2 5 i j i ˆ j ω ω ω α β = + = G + From equation 9.1.29 … 1

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2 2 2 2 2 ˆ ˆ 3 2 2 5 5 5 5 ma ma ma ma L i j ω ω ω ω = + + + G 2 4 3 ( ) 2 ˆ ˆ 2 6 5 ma L i ω = + G j (d) From equation 9.1.32: ( ) 2 2 2 1 1 2 2 2 1 2 5 6 5 ma T L ma ω ω 5 ω ω = = + = G G 9.2 (a) ( ) ˆ ˆ ˆ 3 i j k ω ω = + + G ( ) 2 2 2 2 2 2 12 3 xx rod yy zz m a I I ma I I = = = = = a -a a -a -a a 0 xy I xydm = − = 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 ˆ ˆ 3 3 L ma i j k ω = + + G From equation 9.1.32, ( ) 2 2 2 2 2 1 2 1 1 1 2 3 3 3 3 ma ma T 2 ω ω ω = + + = (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 2 2 2 2 2 cos cos cos 3 3 ma 2 I ma α β 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 I I = + 2 zz xx I I = From Table 8.3.1, 2 6 zz ma I = 2 12 xx ma I = 2
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 θ = 2 45 22.5 θ θ = = D D 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 ( ) ( ) 2 2 2 1 13 2 3 12 12 m I a a m = + = a ( ) 2 2 2 2 10 3 12 12 m I a a m = + = a ( ) 2 2 2 3 5 2 12 12 m I a a m = + = a y ( ) 1 2 3 ˆ ˆ ˆ 2 3 14 e e e ω ω = + + G From equation 9.2.5, 2 2 2 2 2 1 13 10 4 5 9 2 12 14 12 14 12 14 T ma ma ma 2 ω ω ω = + + 2 2 7 24 ma ω = (b) From equation 9.2.4, 2 2 1 2 3 13 10 2 5 3 ˆ ˆ ˆ 12 12 12 14 14 14 L e ma e ma e ma 2 ω ω ω = + + G ( ) 2 1 2 ˆ ˆ ˆ 13 20 15 12 14 ma L e e ω = + + G 3 e ( )( ) ( )( ) ( )( ) ( ) ( ) 1 1 2 2 2 2 2 2 2 2 1 13 2 20 3 15 cos 1 2 3 13 20 15 L L ω θ ω + + = = + + + + G G ( ) 1 2 98 0.9295 11,116 = = 21.6 θ = D 9.5

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• Fall '04
• mokhtari

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