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PROBLEM 9.24 (Continued)
32 93
r
PROBLEM 9.25
(a) Determine by direct integration the polar moment of inertia of the annular
area shown with respect to Point O. (b) Using the result of Part a, determine
the moment of inertia of the given area with respect to the x axis.
PROBLEM 9.26
(a) Show that the polar radius of gyration k
O
of the annular area shown is
approximately equal to the mean radius 12
()2
m
RRR for small values of
the thickness
21
.tR R (b) Determine the percentage error introduced by
using R
m
in place of k
O
for the following values of t/R
m
: 1,
1
2
, and
1
10
.
PROBLEM 9.26 (Continued)
PROBLEM 9.27
Determine the polar moment of inertia and the polar radius of gyration of
the shaded area shown with respect to Point O.
PROBLEM 9.28
Determine the polar moment of inertia and the polar radius of gyration of
the isosceles triangle shown with respect to Point O.
PROBLEM 9.28 (Continued)
PROBLEM 9.29*
Using the polar moment of inertia of the isosceles triangle of Problem
9.28, show that the centroidal polar moment of inertia of a circular area of
radius r is
4
/2.r
(Hint: As a circular area is divided into an increasing
number of equal circular sectors, what is the approximate shape of each
circular sector?)
PROBLEM 9.28
Determine the polar moment of inertia and the polar
radius of gyration of the isosceles triangle shown with respect to Point O.
PROBLEM 9.30*
Prove that the centroidal polar moment of inertia of a given area A cannot be smaller than
2
/2 .A
(Hint:
Compare the moment of inertia of the given area with the moment of inertia of a circle that has the same
area and the same centroid.)
PROBLEM 9.30* (Continued)
