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Hence, we find, with the density = constant,
U = G 2 π
α =0
a
b
sdsdzdα
s 2 +( c − z ) 2
s =0
z =0
(3-5)
=2 πG a
s =0
b
sdsdz
s 2 +( c
z ) 2 .
z =0
The integration with respect to s yields
a
z ) 2
z ) 2 = s 2 +( c
sds
s 2 +( c
a
0
(3-6)
s =0
= a 2 +( c
z ) 2
c + z,
so that we have
U =2 πG b
0
z ) 2 dz .
c + z + a 2 +( c
(3-7)
The indefinite integral is 2 πG times
2 a 2 ln c
z ) 2 , (3-8)
z ) a 2 +( c
z + a 2 +( c
1
1
1
z ) 2
2 ( c
2 ( c
z ) 2
as may be verified by differentiation. Hence, U finally becomes
U e = πG ( c
b ) a 2 +( c
b ) 2 + c a 2 + c 2
b ) 2
c 2
( c
(3-9)
b ) 2 + a 2 ln c + a 2 + c 2 ,
b + a 2 +( c
a 2 ln c
where the subscript e denotes that P is external to the cylinder.
The vertical attraction A is the negative derivative of U with respect to
the height c [see Eq. (2-22)]:
∂U
∂c
A =
.
(3-10)
Differentiating (3-9), we obtain
A e =2 πG b + a 2 +( c
a 2 + c 2 .
b ) 2
(3-11)
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