7.8.
Find the centroid of a solid region R as in Fig. 7-3.
Consider the volume element Ar, of the solid. The
mass of this volume element is
ΔΜ, = συ Δεν Ξ o, Ax₂ Ay, Az,
where is the density [mass per unit volume] and
ov
Ax,, Ay,, Az, are the dimensions of the volume element.
Then the centroid is given approximately by
Σr, o, Ax, Ay, Az,
Σo, Ax, Ay, Az,
Σr, AM, Σr, σ, ATV
ΣΔΜ,
Σου Δεν
-
where the summation is taken over all volume elements
of the solid.
Taking the limit as
ATV 0 or Ax, 0, Ay,
x
F
-
SSS
R
SSS
R
S
R
Fig. 7-3
the number of volume elements becomes infinite in such a way that
0, Az, → 0, we obtain for the centroid of the solid:
S₂
R
rdM
xo dx dy dz
o dx dy dz
dM
9
So
R
=
rodr
SR
o dr
where the integration is to be performed over R, as indicated.
Writing rxi+yj+zk, F = i +ÿj + zk, this can also be written in component form as
SSS
R
=
SSS
R
SSS
R
SSS
R
yo dx dy dz
ro dx dy dz
o dx dy dz
o dx dy dz
AT, Ax, Ay, Az,
Z
R
SSS
R
SSS
R
zo dx dy dz
Y
o dx dy dz
III