Some applications of integration 391
2.0 cm
5.0 cm
3.0 cm
(a)
(b)
(c)
15 cm
18 cm 10 cm
15 cm
5.0 cm
L
L
D ia 5 4 .0 c m
Figure 38.31
6. Calculate the radius of gyration of a rectangular door 2.0 m high by 1.5 m wide about a
vertical axis through its hinge.
[0.866 m]
7. A circular door of a boiler is hinged so that
it turns about a tangent. If its diameter is
1.0 m, determine its second moment of area
and radius of gyration about the hinge.
[0.245 m 4 , 0.559 m]
8. A circular cover, centre 0, has a radius of
12.0 cm. A hole of radius 4.0 cm and centre X ,
where OX = 6.0 cm, is cut in the cover. Determine the second moment of area and the radius
of gyration of the remainder about a diameter
through 0 perpendicular to OX .
[14280 cm 4 , 5.96 cm]
9. For the sections shown in Fig. 38.32, find
the second moment of area and the radius of
gyration about axis XX .
(a) 12190 mm 4 , 10.9 mm
(b) 549.5 cm 4 , 4.18 cm
18.0 mm
3.0 cm
2.5 cm
2.0 cm
2.0 cm
6.0 cm
12.0 mm
3.0 mm
X
X
X
X
4.0 mm
(a)
(b)
Figure 38.32
10. Determine the second moments of areas about
the given axes for the shapes shown in
Fig. 38.33. (In Fig. 38.33(b), the circular area
is removed.)
⎡
⎣
I AA = 4224 cm 4 ,
I BB = 6718 cm
4
,
I CC = 37300 cm 4
⎤
⎦
3.0 cm
16.0 cm
9.0 cm
10.0 cm
(a)
(b)
4.0 cm
15.0 cm
9.0 cm
4.5 cm
A
A
B
C
B
C
D ia 5 7 .0 c m
Figure 38.33
11. Find the second moment of area and radius
of gyration about the axis XX for the beam
section shown in Fig. 38.34.
1350 cm 4 ,
5.67 cm
2.0 cm
8.0 cm
2.0 cm
X
X
1.0 cm
10.0 cm
6.0 cm
Figure 38.34