Higher Engineering Mathematics, Sixth Edition

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

Dia^5

4.0 cm

Figure 38.31

6. Calculate the radius of gyration of a rectan-
   gular door 2.0m high by 1.5m wide about a
   vertical axis through its hinge.
   [0.866m]


7. A circular door of a boiler is hinged so that
   it turns about a tangent. If its diameter is
   1.0m, determine its second moment of area
   and radius of gyration about the hinge.
   [0.245m^4 , 0.559m]


8. A circular cover, centre 0, has a radius of
   12.0cm. A hole of radius 4.0cm and centreX,
   whereOX= 6 .0cm, is cut in thecover. Deter-
   mine the second moment of area and the radius
   of gyration of the remainder about a diameter
   through 0 perpendicular toOX.
   [14280cm^4 ,5.96cm]


9. For the sections shown in Fig. 38.32, find
   the second moment of area and the radius of
   gyration about axis[XX.
   (a)12190mm^4 , 10 .9mm
   (b) 549 .5cm^4 , 4 .18cm

]

18.0 mm

2.5 cm 3.0 cm

2.0 cm

2.0 cm

6.0 cm

12.0 mm

3.0 mm

XX

XX

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.) ⎡

⎣

IAA=4224cm^4 ,

IBB=6718cm^4 ,

ICC=37300cm^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

AA

B

C

B

C

Dia^5

7.0 cm

Figure 38.33

11. Find the second moment of area and radius
    of gyration about the axisXXfor the beam
    section shown in Fig. 38.34. [
    1350cm^4 ,
    5 .67cm

]

2.0 cm

8.0 cm

2.0 cm

XX

1.0 cm

10.0 cm

6.0 cm

Figure 38.34