ISM_T11_C15_C

ISM_T11_C15_C - 968 Chapter 15 Multiple Integrals 15.5...

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968 Chapter 15 Multiple Integrals 15.5 MASSES AND MOMENTS IN THREE DIMENSIONS 1. I y z dx dy dz a y z dy dz a yz dz x c 2 b 2 a 2 c 2 b 2 c 2 c 2 b 2 a 2 c 2 b 2 c 2 b 2 b 2 œ œ œ ' ' ' ' ' ' a b a b # # # # # y 3 a bz dz ab z ab b c b c ; œ œ œ œ œ ' c 2 c 2 c 2 c 2 Š Š a b a b b b z b c c abc M 12 12 3 12 12 1 12 # # # # # # R ; likewise R and R , by symmetry x y z œ œ œ É É É b c a c a b 12 12 12 2. The plane z is the top of the wedge I y z dz dy dx œ Ê œ 4 2y 3 # # x 3 2 4 3 3 4 4 2y 3 ' ' ' a b dy dx dx 208; I x z dz dy dx œ œ œ œ ' ' ' ' ' ' 3 2 3 3 2 4 3 3 4 3 3 4 4 2y 3 y a b 8y 2y 8(2 y) 3 3 81 81 3 64 104 # # dy dx 12x dx 280; œ œ œ ' ' ' 3 2 3 3 4 3 ˆ (4 2y) x (4 2y) 81 3 3 81 3 4x 64 32 # I x y dz dy dx x y dy dx 12 x 2 dx 360 z 3 2 4 3 3 2 3 3 4 4 2y 3 3 4 3 œ œ œ œ ' ' ' ' ' ' a b a b a b ˆ # # # # # 8 3 3 2y 3. I y z dz dy dx cy dy dx dx x 0 0 0 0 0 0 a b c a b a œ œ œ œ ' ' ' ' ' ' a b Š Š # # # c cb c b 3 3 3 3 abc b c a b b c where M abc; I a c and I a b , by symmetry œ œ œ œ M M M 3 3 3 a b a b a b # # # # # # y z 4. (a) M dz dy dx (1 x y) dy dx x dx ; œ œ œ œ ' ' ' ' ' ' 0 0 0 0 0 0 1 1 x 1 x y 1 1 x 1 Š x 6 # # " " M x dz dy dx x(1 x y) dy dx x 2x x dx yz 0 0 0 0 0 0 1 1 x 1 x y 1 1 x 1 œ œ œ œ ' ' ' ' ' ' " " # \$ # a b 24 x y z , by symmetry; I y z dz dy dx Ê œ œ œ œ " # # 4 x 0 0 0 1 1 x 1 x y ' ' ' a b y xy y dy dx (1 x) dx I I , by symmetry œ œ œ Ê œ œ ' ' ' 0 0 0 1 1 x 1 y x # # \$ % " " " (1 x y) 3 6 30 30 (b) R 0.4472; the distance from the centroid to the x-axis is 0 x œ œ œ ¸ œ œ É É É É I M 5 5 16 16 8 4 5 2 x " " " " # È È 0.3536 ¸ 5. M 4 dz dy dx 4 4 4y dy dx 16 dx ; M 4 z dz dy dx œ œ œ œ œ ' ' ' ' ' ' ' ' ' 0 0 4y 0 0 0 0 0 4y 1 1 4 1 1 1 1 1 4 xy a b # 2 32 3 3 2 16 16y dy dx dx z , and x y 0, by symmetry; œ œ œ Ê œ œ œ ' ' ' 0 0 0 1 1 1 a b % 128 128 12 5 5 5 I 4 y z dz dy dx 4 4y 4y dy dx 4 dx ; x 0 0 4y 0 0 0 1 1 4 1 1 1 œ œ œ œ ' ' ' ' ' ' a b ˆ Š # # # % 64 1976 7904 3 3 105 105 64y I 4 x z dz dy dx 4 4x 4x y dy dx 4 x dx y 0 0 4y 0 0 0 1 1 4 1 1 1 œ œ œ ' ' ' ' ' ' a b ˆ ˆ Š # # # # # # 64 8 128 3 3 3 7 64y ; I 4 x y dz dy dx 16 x x y y y dy dx œ œ œ 4832 63 z 0 0 4y 0 0 1 1 4 1 1 ' ' ' ' ' a b a b # # # # # # % 16 dx œ œ ' 0 1 Š 2x 2 256 3 15 45 6. (a) M dz dy dx (2 x) dy dx (2 x) 4 x dx 4 ; œ œ œ œ ' ' ' ' ' ' 2 4 x 2 0 2 4 x 2 2 2 4 x 2 2 x 2 4 x 2 2 Š È # 1 M x dz dy dx x(2 x) dy dx x(2 x) 4 x dx 2 ; yz 2 4 x 2 0 2 4 x 2 2 2 4 x 2 2 x 2 4 x 2 2 œ œ œ œ ' ' ' ' ' ' Š È # 1 M y dz dy dx y(2 x) dy dx xz 2 4 x 2 0 2 4 x 2 2 4 x 2 2 x 2 4 x 2 œ œ ' ' ' ' ' (2 x) dx 0 x and y 0 œ œ Ê œ œ " " # # ' 2 2 4 x 4 x 4 4

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Section 15.5 Masses and Moments in Three Dimensions 969 (b) M z dz dy dx (2 x) dy dx (2 x) 4 x dx xy 2 4 x 2 0 2 4 x 2 2 2 4 x 2 2 x 2 4 x 2 2 œ œ œ ' ' ' ' ' ' " " # # # # # Š È 5 z œ Ê œ 1 5 4 7. (a) M 4 dz dy dx 4 r dz dr d 4 4r r dr d 4 4 d 8 ; œ œ œ œ œ ' ' ' ' ' ' ' ' ' 0 0 x y 0 0 r 0 0 0 2 4 x 4 2 2 4 2 2 2 ) ) ) 1 a b \$ M zr dz dr d 16 r dr d d z , and x y 0, xy 0 0 r 0 0 0 2 2 4 2 2 2 œ œ œ œ Ê œ œ œ ' ' ' ' ' ' ) ) ) r 32 64 8 3 3 3 # % a b 1 by symmetry (b) M 8 4 r dz dr d cr r dr d d c 8 c 2 2, œ Ê œ œ œ œ Ê œ Ê œ 1 1 ) ) ) ' ' ' ' ' ' 0 0 r 0 0 0 2 c c 2 c 2 a b È \$ # # c c 4 1 since c 0 8. M 8; M z dz dy dx dy dx 0; M x dz dy dx œ œ œ œ œ xy yz 1 3 1 1 3 1 3 1 1 5 1 1 5 1 5 1 ' ' ' ' ' ' ' ' z 2 " " 2 x dy dx 4 x dx 0; M y dz dy dx 2 y dy dx 16 dx 32 œ œ œ œ œ œ œ ' ' ' ' ' ' ' ' ' 1 3 1 1 3 1 1 3 1 1 5 1 1 5 1 1 5 1 xz x 0, y 4, z 0; I y z dz dy dx 2y dy dx 100 dx ; Ê œ œ œ œ œ œ
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