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 with b
bl 3
3
1
√
3
length l
(2) Coinciding with l
lb 3
3
b
√
3
breadth b
(3) Through centroid,
bl 3
12
1
√
12
parallel to b
(4) Through centroid,
lb 3
12
b
√
12
parallel to l
Triangle
(1) Coinciding with b
bh 3
12
h
√
6
Perpendicular
(2) Through centroid,
bh 3
36
h
√
18
height h
parallel to base
base b
(3) Through vertex,
bh 3
4
h
√
2
parallel to base
Circle
(1) Through centre,
πr 4
2
r
√
2
radius r
perpendicular to plane
(i.e. polar axis)
(2) Coinciding with diameter
πr 4
4
r
2
(3) About a tangent
5πr 4
4
√
5
2
r
Semicircle
Coinciding with
πr 4
8
r
2
radius r
diameter
Perpendicular axis theorem:
If OX and OY lie in the plane of area A in Fig. FA7,
then Ak
2
O Z = Ak
2
O X + Ak
2
OY or k
2
O Z = k
2
O X + 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