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kinetic energy constraint, shells
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Expression 37: "D" left parenthesis, "v" Subscript, 10 , Baseline , "N" Subscript, 1 , Baseline "v" Subscript, 10 , Baseline , right parenthesis plus phi Subscript, 10 , Baseline "I" Subscript, 1 , Baseline phi Subscript, 10 , Baseline plus "D" left parenthesis, "v" Subscript, 20 , Baseline , "N" Subscript, 2 , Baseline "v" Subscript, 20 , Baseline , right parenthesis plus phi Subscript, 20 , Baseline "I" Subscript, 2 , Baseline phi Subscript, 20 , Baseline
D
v
1
0
,
N
1
v
1
0
+
ϕ
1
0
I
1
ϕ
1
0
+
D
v
2
0
,
N
2
v
2
0
+
ϕ
2
0
I
2
ϕ
2
0
equals
=
9.5 0 7 7 1 1 7 5
9
.
5
0
7
7
1
1
7
5
37
Expression 38: "D" left parenthesis, "v" Subscript, 1 , Baseline , "N" Subscript, 1 , Baseline "v" Subscript, 1 , Baseline , right parenthesis plus phi Subscript, 1 , Baseline "I" Subscript, 1 , Baseline phi Subscript, 1 , Baseline plus "D" left parenthesis, "v" Subscript, 2 , Baseline , "N" Subscript, 2 , Baseline "v" Subscript, 2 , Baseline , right parenthesis plus phi Subscript, 2 , Baseline "I" Subscript, 2 , Baseline phi Subscript, 2 , Baseline
D
v
1
,
N
1
v
1
+
ϕ
1
I
1
ϕ
1
+
D
v
2
,
N
2
v
2
+
ϕ
2
I
2
ϕ
2
equals
=
9.5 0 7 7 1 1 7 5
9
.
5
0
7
7
1
1
7
5
38
Expression 39: "D" left parenthesis, "v" Subscript, 12 , Baseline , "N" Subscript, 1 , Baseline "v" Subscript, 12 , Baseline , right parenthesis plus phi Subscript, 12 , Baseline "I" Subscript, 1 , Baseline phi Subscript, 12 , Baseline plus "D" left parenthesis, "v" Subscript, 22 , Baseline , "N" Subscript, 2 , Baseline "v" Subscript, 22 , Baseline , right parenthesis plus phi Subscript, 22 , Baseline "I" Subscript, 2 , Baseline phi Subscript, 22 , Baseline
D
v
1
2
,
N
1
v
1
2
+
ϕ
1
2
I
1
ϕ
1
2
+
D
v
2
2
,
N
2
v
2
2
+
ϕ
2
2
I
2
ϕ
2
2
equals
=
9.5 0 7 7 1 1 7 5
9
.
5
0
7
7
1
1
7
5
39
Expression 40: "D" left parenthesis, "v" Subscript, 12 , Baseline , "N" Subscript, 1 , Baseline "v" Subscript, 12 , Baseline , right parenthesis plus phi Subscript, 12 , Baseline "I" Subscript, 1 , Baseline phi Subscript, 12 , Baseline plus "D" left parenthesis, "v" Subscript, 22 , Baseline , "N" Subscript, 2 , Baseline "v" Subscript, 22 , Baseline , right parenthesis plus phi Subscript, 22 , Baseline "I" Subscript, 2 , Baseline phi Subscript, 22 , Baseline minus left parenthesis, "D" left parenthesis, "v" Subscript, 1 , Baseline , "N" Subscript, 1 , Baseline "v" Subscript, 1 , Baseline , right parenthesis plus phi Subscript, 1 , Baseline "I" Subscript, 1 , Baseline phi Subscript, 1 , Baseline plus "D" left parenthesis, "v" Subscript, 2 , Baseline , "N" Subscript, 2 , Baseline "v" Subscript, 2 , Baseline , right parenthesis plus phi Subscript, 2 , Baseline "I" Subscript, 2 , Baseline phi Subscript, 2 , Baseline , right parenthesis
D
v
1
2
,
N
1
v
1
2
+
ϕ
1
2
I
1
ϕ
1
2
+
D
v
2
2
,
N
2
v
2
2
+
ϕ
2
2
I
2
ϕ
2
2
−
D
v
1
,
N
1
v
1
+
ϕ
1
I
1
ϕ
1
+
D
v
2
,
N
2
v
2
+
ϕ
2
I
2
ϕ
2
equals
=
negative 1.7 7 6 3 5 6 8 3 9 4 times 10 to the negative 15th power
−
1
.
7
7
6
3
5
6
8
3
9
4
×
1
0
−
1
5
40
Hidden Label: "F" Subscript, 0 , Baseline equals left parenthesis, "C" Subscript, 1 , Baseline left parenthesis, "p" Subscript, 2 , Baseline left bracket, 1 , right bracket , phi Subscript, 2 , Baseline , right parenthesis plus "v" Subscript, 2 , Baseline minus "C" Subscript, 1 , Baseline left parenthesis, "p" Subscript, 1 , Baseline left bracket, 1 , right bracket , phi Subscript, 1 , Baseline , right parenthesis minus "v" Subscript, 1 , Baseline , right parenthesis "U"
F
0
=
C
1
p
2
1
,
ϕ
2
+
v
2
−
C
1
p
1
1
,
ϕ
1
−
v
1
U
Label
41
Hidden Label: "F" equals "F" Subscript, 0 , Baseline StartFraction, negative 2 left parenthesis, "D" left parenthesis, "v" Subscript, 1 , Baseline , "F" Subscript, 0 , Baseline , right parenthesis plus "D" left parenthesis, "v" Subscript, 2 , Baseline , negative "F" Subscript, 0 , Baseline , right parenthesis plus phi Subscript, 1 , Baseline "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 1 , Baseline left bracket, 1 , right bracket , right parenthesis plus phi Subscript, 2 , Baseline left parenthesis, negative "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 2 , Baseline left bracket, 1 , right bracket , right parenthesis , right parenthesis , right parenthesis Over left parenthesis, "D" left parenthesis, StartNestedFraction, "F" Subscript, 0 , Baseline NestedOver "N" Subscript, 1 , Baseline , EndNestedFraction , "F" Subscript, 0 , Baseline , right parenthesis plus "D" left parenthesis, StartNestedFraction, "F" Subscript, 0 , Baseline NestedOver "N" Subscript, 2 , Baseline , EndNestedFraction , "F" Subscript, 0 , Baseline , right parenthesis plus "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 1 , Baseline left bracket, 1 , right bracket , right parenthesis "I" to the negative 1st power "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 1 , Baseline left bracket, 1 , right bracket , right parenthesis plus "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 2 , Baseline left bracket, 1 , right bracket , right parenthesis "I" to the negative 1st power "C" Subscript, 0 , Baseline left parenthesis, "F" Subscript, 0 , Baseline , "p" Subscript, 2 , Baseline left bracket, 1 , right bracket , right parenthesis , right parenthesis , EndFraction "U"
F
=
F
0
−
2
D
v
1
,
F
0
+
D
v
2
,
−
F
0
+
ϕ
1
C
0
F
0
,
p
1
1
+
ϕ
2
−
C
0
F
0
,
p
2
1
D
F
0
N
1
,
F
0
+
D
F
0
N
2
,
F
0
+
C
0
F
0
,
p
1
1
I
−
1
1
C
0
F
0
,
p
1
1
+
C
0
F
0
,
p
2
1
I
−
1
2
C
0
F
0
,
p
2
1
U
Label
42
Expression 43: "U" equals 1
U
=
1
negative 10
−
1
0
10
1
0
43
Hidden Label: "v" Subscript, 12 , Baseline equals "v" Subscript, 1 , Baseline plus StartFraction, "F" Over "N" Subscript, 1 , Baseline , EndFraction
v
1
2
=
v
1
+
F
N
1
Label
44
Expression 45: phi Subscript, 12 , Baseline equals phi Subscript, 1 , Baseline plus "I" to the negative 1st power "C" Subscript, 0 , Baseline left parenthesis, "F" , "p" Subscript, 1 , Baseline left bracket, 1 , right bracket , right parenthesis
ϕ
1
2
=
ϕ
1
+
I
−
1
1
C
0
F
,
p
1
1
equals
=
negative 0.0 2 7 8 5 5 7 5 0 3 7 7 1
−
0
.
0
2
7
8
5
5
7
5
0
3
7
7
1
45
Hidden Label: "v" Subscript, 22 , Baseline equals "v" Subscript, 2 , Baseline minus StartFraction, "F" Over "N" Subscript, 2 , Baseline , EndFraction
v
2
2
=
v
2
−
F
N
2
Label
46
Expression 47: phi Subscript, 22 , Baseline equals phi Subscript, 2 , Baseline minus "I" to the negative 1st power "C" Subscript, 0 , Baseline left parenthesis, "F" , "p" Subscript, 2 , Baseline left bracket, 1 , right bracket , right parenthesis
ϕ
2
2
=
ϕ
2
−
I
−
1
2
C
0
F
,
p
2
1
equals
=
negative 0.4 2 3 2 0 6 4 9 2 6 6 7
−
0
.
4
2
3
2
0
6
4
9
2
6
6
7
47
48
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"x"
x
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y
"a" squared
a
2
"a" Superscript, "b" , Baseline
a
b
7
7
8
8
9
9
over
÷
functions
(
(
)
)
less than
<
greater than
>
4
4
5
5
6
6
times
×
| "a" |
|
a
|
,
,
less than or equal to
≤
greater than or equal to
≥
1
1
2
2
3
3
negative
−
A B C
StartRoot, , EndRoot
pi
π
0
0
.
.
equals
=
positive
+
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