Mechanical Engineering Principles

FIRST AND SECOND MOMENT OF AREAS 93

Problem 11. Determine the second moment

of area and radius of gyration for the

semicircle shown in Figure 7.15 about

axisXX.

G G

B B

XX

15.0 mm

10.0 mm

Figure 7.15

The centroid of a semicircle lies at 34 πr from its

diameter (see ‘Engineering Mathematics 3RDEdi-
tion’, page 471).

Using the parallel axis theorem:

IBB=IGG+AH^2 ,

where IBB=

πr^4

8

(from Table 7.1)

=

π( 10. 0 )^4

8

=3927 mm^4 ,

A=

πr^2

2

=

π( 10. 0 )^2

2

= 157 .1mm^2

and H=

4 r

3 π

=

4 ( 10. 0 )

3 π

= 4 .244 mm

Hence 3927=IGG+( 157. 1 )( 4. 244 )^2

i.e. 3927 =IGG+ 2830 ,from which,

IGG= 3927 − 2830 =1097 mm^4

Using the parallel axis theorem again:

IXX=IGG+A( 15. 0 + 4. 244 )^2

i.e. IXX= 1097 +( 157. 1 )( 19. 244 )^2

= 1097 + 58179 =59276 mm^4

or59280 mm^4 , correct to 4 significant figures.

Radius of gyration,

kXX=

√

IXX

area

=

√(

59276

157. 1

)

= 19 .42 mm

Problem 12. Determine the polar second

moment of area of an annulus about its

centre. The outer diameter of the annulus is

D 2 and its inner diameter isD 1

The polar second moment of area is denoted byJ.

Hence, for the annulus,

J=J of outer circle about its centre

−Jof inner circle about its centre

=

πD^42

32

−

πD^41

32

from Table 7.1

i.e. J=

π

32

(D 24 −D 14 )

Problem 13. Determine the polar second

moment of area of the propeller shaft

cross-section shown in Figure 7.16.

6.0 cm7.0 cm

Figure 7.16

The polar second moment of area of a circle,

J=

πd^4

32

The polar second moment of area of the shaded area

is given by the polar second moment of area of

the 7.0 cm diameter circle minus the polar second

moment of area of the 6.0 cm diameter circle.

Hence, from Problem 12, the polar second moment

of area of the cross-section shown

=

π

32

( 74 − 64 )=

π

32

( 1105 )

= 108 .5cm^4

Problem 14. Determine the second moment

of area and radius of gyration of a

rectangular lamina of length 40 mm and

width 15 mm about an axis through one

corner, perpendicular to the plane of the

lamina.