Higher Engineering Mathematics

Ess-For-H8152.tex 19/7/2006 18: 2 Page 714

714 ESSENTIAL FORMULAE

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

radiusr diameter^2

Theorem of Pappus

With reference to Fig. FA5, when the curve is rotated

one revolution about thex-axis between the limits

x=aandx=b, the volumeVgenerated is given by:

V= 2 πAy ̄.

Parallel axis theorem:

IfCis the centroid of areaAin Fig. FA6 then

Ak^2 BB=Ak^2 GG+Ad^2 ork^2 BB=k^2 GG+d^2

G B

C

Area A

d

G B

Figure FA6