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:

Answer rating: 100% (QA)

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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