Mass and center of mass Let S be a surface that represents a thin shell with density ρ. The moments about the coordinate planes ( see Section 13.6 ) are M y z = ∬ S x ρ ( x , y , z ) d S , M x z = ∬ S y ρ ( x , y , z ) d S , and M x y = ∬ S z ρ ( x , y , z ) d S . The coordinates of the center of mass of the shell are x ¯ = M y z m , y ¯ = M x z m , z ¯ = M x y m , where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible . 66. The constant-density hemispherical shall x 2 + y 2 + z 2 = a 2 , z ≥ 0
Mass and center of mass Let S be a surface that represents a thin shell with density ρ. The moments about the coordinate planes ( see Section 13.6 ) are M y z = ∬ S x ρ ( x , y , z ) d S , M x z = ∬ S y ρ ( x , y , z ) d S , and M x y = ∬ S z ρ ( x , y , z ) d S . The coordinates of the center of mass of the shell are x ¯ = M y z m , y ¯ = M x z m , z ¯ = M x y m , where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible . 66. The constant-density hemispherical shall x 2 + y 2 + z 2 = a 2 , z ≥ 0
Mass and center of massLet S be a surface that represents a thin shell with density ρ. The moments about the coordinate planes (see Section 13.6) are
M
y
z
=
∬
S
x
ρ
(
x
,
y
,
z
)
d
S
,
M
x
z
=
∬
S
y
ρ
(
x
,
y
,
z
)
d
S
, and
M
x
y
=
∬
S
z
ρ
(
x
,
y
,
z
)
d
S
. The coordinates of the center of mass of the shell are
x
¯
=
M
y
z
m
,
y
¯
=
M
x
z
m
,
z
¯
=
M
x
y
m
, where m is the mass of the shell. Find the mass and center of mass of the following shells. Use symmetry whenever possible.
66. The constant-density hemispherical shall
x
2
+
y
2
+
z
2
=
a
2
,
z
≥
0
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