Thomas Calculus 13th [Solutions]

Section 15.6 Moments and Centers of Mass 1123
(c)
1 4 (2 )/2 2 2 1 4
2
I y 1 1
x
1 2 1 ( x 1) y z dz dy dx
1 2
( x 1) 2 y y 3 2 3 1 y dy dx
2
45;
3 3
1 4 (2 )/2 2 2 1 4
2
I y 1 1
y
1 2 1 ( x 1) x z dz dy dx
1 2
( x 1) 2 x x y 3 2 3 1 y dy dx
2
15;
2 3
1 4 (2 y)/2 2 2 1 4
2 2
1 2 1 ( 1) 1 2
( 1) 2 y
Iz
x x y dz dy dx x x y dy dx
2
42
33.
M 1 1 z z 1 1 z
1
(2 5) 5 2 5 (1 )
1 z 1 0 y dy dx dz 0 z 1 z z dx dz 0
z z z dz
2 1 1/2 3/2 2 3/2 2 5/2 3
1
10 1 1
9 3
0 5 z z 5 z z dz 2 z z
3 2 2 z z
3 2 0 3 2
3
34.
2 2 2
2
2 4 x 16 2 x y
2 2 2 4 x 2 2 2 2
M 2 x y dz dy dx
2 2
2 x y 16 4 x y dy dx
2 4 x 2 x y
2 4 x
2 2 2 2 2
4 r 4 r r dr d 4 4r
r d 4 64 d 512
0 0 0 3 5
0 0 15 15
3 5 2
M yz
35. (a) x
M
0 x ( x, y, z) dx dy dz 0 M yz 0
R
2 2
(b) IL
v hi dm ( x h) i yj
dm
2
x 2xh 2
h
2
y dm
D D D
2
x
2
y dm 2h x dm
2
h dm Ix
0
2
h m Ic.m.
2
h m
D D D
36.
2 2 2 2 7 2
IL
Ic.m. mh ma ma ma
5 5
37. (a)
(b)
2
2 2
abc a
( x, y, z ) a , b , c I a b
2 2 2 z Ic.m.
abc I
4 4 c.m. Iz
4
2 2 2 2 2 2
abc a b abc a b abc a b I 2 2
c.m.
; R
a b
3 4 12 c.m. M 12
2
2 2 2 2 2 2 2 2
2
2 abc a b abc a 9b abc 4a 28b abc a 7b
I a b
L Ic.m. abc 2 b
;
4 2 12 4 12 3
2 2
b
IL
R a 7b
L M 3
2 2
38.
3 4 (4 2 y)/3 3 4
2
3
2 y
2
M dz dy dx (4 y) dy dx 4y dx 12 dx 72; x y z 0
3 2 4/3 3 2 3 3 3 2
3
2
2 2
2 2
from Exercise 22 I 16 160
x Ic.m. 72 0 0 Ic.m. IL
Ic.m.
72 16 208 72 1488
9 9
2 4
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