Let o be the mass density of a (negligibly thick) distribution of mass over a surface S. That is,…

Let o be the mass density of a (negligibly thick) distribution of mass over a surface S. That is,…

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Let o be the mass density of
a (negligibly thick) distribution of mass over a surface S. That
is, is the mass per unit area at each point on S; it may vary
over S. Then the x, y, coordinates of the center of gravity
are defined as
where
Ic
31
Vc =
24
11
!!
M
1)
M
M =
xo dA,
yo dA,
zo dA.
= 11 od
dA
(1.1)
(11.2)
is the total mass. Evaluate ze in each case. (You need not
evaluate ye. e)
(a) S is the plane surface = 2 + 2y with vertices at (0,0,0).
(1.0, 1), (0.1.2);= constant = dg.
(b) S is the plane surface == x+y with vertices at (0,0,0).
(1,0, 1), (0.1, 1); o = 1 + y.
2 = 0 and 2 = 1.= constant = do
th) S is the spherical surface a2 + 1/²
(c) S is the plane surface z =
(0.1.2). (2.0.0). (2. 1.0):
2-z with vertices at (0, 0, 2).
4-*.
(d) S is the plane surface == 2-zwith vertices at (0,0, 2).
(0, 1, 2). (2,0,0);=i+r.
S is the cylindrical surface a² + y² = 4 between == 0 and
== h.o=1+r.
(1)S is the cylindrical surface ² + y² = 4 between = = 0 and
2=2+₁0= constant
dg.
(g) is the hemispherical surface a² + y² + 2 = 1 between

Expert Answer:

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a S is the plane surface z x 2y with vertices at 0 0 0 1 0 1 0 1 2 a constant 0 Calculate Total Mass M To calculate the total mass we need to integrate the mass density a over the surface S Since a is
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