Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

Answers 451
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27i 2 h cosec \f
The resulting attraction is equal to \ \ dq> d\|? \ Q sin \f cos \|> dr.
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2272. Solution. We introduce cylindrical coordinates (Q, cp, z) with origin
at the centre of the sphere and with the z-axis passing through a material
point whose mass we assume equal to m. We denote by %
the distance of
this point from the centre of the sphere. Let r= >^Q 2
-h(| z) 2 be the distance
from the element of volume dv to the mass m. The attractive force of
the element of volume dv of the sphere and the material point m is directed
along r and is numerically equal to kym 2 , where
y= -^
la
is the
density of the sphere and dv qd^dQdz is the element of volume. The projection
of this force on the z-axis is
._, kmydv /\
H z ...
f dF =
c s (rz) = kmy *^~ Q d