Chapter-3
Moment of Inertia and Centroid
3.7 Perpendicular axis theorem for an area If x, y & z are mutually perpendicular axes as shown, then I xx I yy
I zz J
Z-axis is perpendicular to the plane of x – y and vertical to this page as shown in figure.
x
To find the moment of inertia of the differential area about the pole (point of origin) or z-axis,
(r) is used. (r) is the perpendicular distance from the pole to dA for the entire area J = r2 dA = (x2 + y2 )dA = Ixx + Iyy (since r2 = x2 + y2 ) Where, J = polar moment of inertia
3.8 Moments of Inertia (area) of some common area (i) MOI of Rectangular area Moment of inertia about axis XX which passes through centroid. Take an element of width ‘dy’ at a distance y from XX axis.
? Area of the element (dA) = b u dy. and Moment of Inertia of the element about XX axis dA u y 2
b.y 2 .dy
?Total MOI about XX axis (Note it is area moment of Inertia) h
I xx
h
2 2
³ by dy
h
2
2
2 ³ by 2 dy 0
bh3 12
3
bh 12
I xx Similarly, we may find, I yy
hb3 12 3
3
12
12
?Polar moment of inertia (J) = Ixx + Iyy = bh hb
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