Higher Engineering Mathematics, Sixth Edition

668 Higher Engineering Mathematics

Second moment of area and radius of gyration:

Shape Position of axis Second moment Radius of

of area,I gyration,k

Rectangle (1) Coinciding withb

bl^3

3

1

√

lengthl^3

(2) Coinciding withl

lb^3

3

b

√

3

breadthb

(3) Through centroid,

bl^3

12

1

√

parallel tob^12

(4) Through centroid, lb

3

12

b

√

parallel tol^12

Triangle (1) Coinciding withb

bh^3

12

h

√

Perpendicular^6

(2) Through centroid,

bh^3

36

h

√

18

heighth

baseb parallel to base

(3) Through vertex, bh

3

4

h

√

parallel to base^2

Circle (1) Through centre,

πr^4

2

r

√

radiusr perpendicular to plane^2

(i.e. polar axis)

(2) Coinciding with diameter

πr^4

4

r

2

(3) About a tangent

5 πr^4

4

√

5

2

r

Semicircle Coinciding with πr

4

8

r

2

radiusr diameter

Perpendicular axis theorem:

IfOXandOYlie in the plane of areaAin Fig. FA7,

thenAk^2 OZ=Ak^2 OX+Ak^2 OYork^2 OZ=k^2 OX+k^2 OY

Z

O

Y

X

Area A

Figure FA7

Numerical integration:

Trapezoidal rule

∫

ydx≈

(

width of

interval

)[

1

2

(

first+last

ordinates

)

+

(

sum of remaining

ordinates

)]

Mid-ordinate rule

∫

ydx≈

(

width of

interval

)(

sum of

mid-ordinates

)