PROBLEM 9.185
Determine by direct integration the moments of inertia of the
shaded area with respect to the x and y axes.
SOLUTION
2
2
3
2
3
2
4
11
4
33
x
xx
yh
aa
xx
dI y dx h dx
aa
PROBLEM 9.186
Determine the moment of inertia and the radius of gyration of the shaded
area shown with respect to the y axis.
SOLUTION
22
22
2
2
22
1
1
2
2
y
xy
ab
x
yb a
dA ydx
dI x dA x ydx
PROBLEM 9.187
Determine the moment of inertia and the radius of gyration of the shaded
area shown with respect to the x axis.
SOLUTION
At 11 2
,:
x
ayyb
2
12
2
22
:or
:2 or
b
ybka k
a
b
ybbca c
a
Then
2
2
12
22
2
bx
yxyb
aa
PROBLEM 9.188
Determine the moments of inertia
x
I
and
y
I
of the area shown
with respect to centroidal axes respectively parallel and
perpendicular to side AB.
SOLUTION
Dimensions in mm
First locate centroid C of the area.
PROBLEM 9.188 (Continued)
Also 12
() ()
yy y
II I
y
PROBLEM 9.189
Determine the polar moment of inertia of the area shown with
respect to (a) Point O, (b) the centroid of the area.
SOLUTION
First locate centroid C of the area.
2
,mmA
,mmy
3
,mmyA
1 (54)(36) 3053.6
2
48 15.2789
46,656
PROBLEM 9.190
Two L4 4
1
2
-in. angles are welded to a steel plate as shown. Determine
the moments of inertia of the combined section with respect to centroidal
axes respectively parallel and perpendicular to the plate.
SOLUTION
PROBLEM 9.190 (Continued)
Entire section:
23 2 2
1
( ) (0.5)(10) (0.5)(10)(2.292) 2[5.52 (3.75)(1.528) ]
12
xx
IIAd
4
41.667 26.266 1604 17.511 96.48 in 4
96.5 in
x
I
PROBLEM 9.191
Using the parallel-axis theorem, determine the product of inertia
of the L5 3
1
2
-in. angle cross section shown with respect to the
centroidal x and y axes.
SOLUTION
We have
12
()()
xy xy xy
II I
For each rectangle:
PROBLEM 9.192
For the L5 3
1
2
-in.
angle cross section shown, use Mohr’s circle
to determine (a) the moments of inertia and the product of inertia
with respect to new centroidal axes obtained by rotating the x and y
axes 30 clockwise, (b) the orientation of the principal axes through
the centroid and the corresponding values of the moments of inertia.
SOLUTION
From Figure 9.13a:
44
9.43 in 2.55 in
xy
II
From the solution to Problem 9.191
PROBLEM 9.192 (Continued)
(a) We have
ave
cos 5.99 4.4433cos80.733
x
II R
or
4
5.27 in
x
I
(b) First observe that the principal axes are obtained by rotating the xy axes through
19.63° counterclockwise
about C.
Now
max, min ave 5.99 4.4433IIR
or
4
max
10.43 inI
4
min
1.547 inI
PROBLEM 9.193
A thin plate of mass m was cut in the shape of a parallelogram as
shown. Determine the mass moment of inertia of the plate with
respect to (a) the x axis, (b) the axis BB, which is perpendicular to
the plate.
SOLUTION
mass area area
mass
mtA
m
ItI I
A
(a) Consider parallelogram as made of horizontal strips and slide strips to form a square since distance
from each strip to x axis is unchanged.
PROBLEM 9.194
A thin plate of mass m was cut in the shape of a parallelogram as
shown. Determine the mass moment of inertia of the plate with
respect to (a) the y axis, (b) the axis AA, which is perpendicular to
the plate.
SOLUTION
See sketch of solution of Problem 9.117.
(a) From part b of solution of Problem 9.117:
PROBLEM 9.195
A 2-mm thick piece of sheet steel is cut and bent into the machine
component shown. Knowing that the density of steel is 7850 kg/m
3
,
determine the mass moment of inertia of the component with respect
to each of the coordinate axes.
SOLUTION
First compute the mass of each component. We have
ST ST
mV tA
Then
32
1
(7850 kg/m )(0.002 m)(0.76 0.48) m
5.72736 kg
m
PROBLEM 9.195 (Continued)
123
22
222 2
2
2 2
() () ()
10.760.48
(5.72736 kg)(0.76 0.48 ) m (5.72736 kg) m
12 2 2
10.761
(2.86368 kg)(0.76 m) (2.86368 kg) m (2.84101 kg)(0.48 m)
18 3 4
yy y y
II I I
PROBLEM 9.196
Determine the mass moment of inertia of the steel
machine element shown with respect to the z axis.
(The specific weight of steel is
3
490 lb/ft .)
SOLUTION
First compute the mass of each component. We have
ST
ST
mV V
g
Then
3
3
3
12
490 lb/ft 1 ft
(2.7 3.7 9) in 12 in.
32.2 ft/s
m
PROBLEM 9.196 (Continued)
Find: z
I
We have 1234
32 2 22
22 2
22
() () () ()
1(791.780 10 lb s /ft)[(2.7) (3.7) ] in
12
2.7 3.7 1 ft
(791.780 lb s /ft) in
22 12 in.
zz z z z
II I I I
32 2
2
116
(35.295 10 lb s /ft) (1.35 in.)
29