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

Sec. II] Moments. Centres of Gravity. Guldin's Theorems 169
In particular: 1) for the curve * = *(s); y=y(s) t whare the parameter s
is the arc length, we have
L L
(ds = V(dx)* + (dy)* is the differential of the arc);
y (s) ds\ M Y =. x (s) ds (2)
Fig. 57 Fig. 58
2) for a plane figure bounded by th3 curve y = y(x), ihz Jt-axis and two
vertical lines x = a and y b t we obtain
b b
a a
x\y\dx. (3)
Example 1. Find the static moments about the x- and /-axes of a triangle
bounded by the straight lines: ~-f-^ = l, x = 0, // = (Fig. 57)
a b
and
Solution. Here, y = b II ~ )
. Applying formula (3), we obtain
2. Moment of inertia. The moment of inertia, about an /-axis, of a imfe-
rial point of mass m at a distance d from the /-axis, is the number l t -=-tnd 2 .
The moment of inertia, about an /-axis, of a system of n material points
with masses m lt m 2t ..., m n is the sum
ab*