Ahsanullah University of Science And Technology

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Prepared By Course no: CE 416

:Objectives Introduction of Moment of Inertia Polar moment of inertia Radious of Gyration Moment of inertia of composite bodies Parallel axis theorm Perpendicular axis theorm

MOMENT OF INERTIA  

The product of the elemental area and square of the The product of the elemental area and square of the perpendicular distance between the perpendicular distance between the centroids centroids of area of area and the axis of reference is the “Moment of Inertia” and the axis of reference is the “Moment of Inertia” about the reference axis. about the reference axis.   

IIxxxx = ∫dA. y = ∫dA. y22

IIyyyy = ∫dA. x = ∫dA. x22

x

dAY

x

y

Fig: Relation between Moment of inertia (I) and angular velocity (w)

Moment of Inertia of a Disk and a Ring : A wooden disk and a metal ring with the same diameter and equal mass roll with

different accelerations down an inclined plane. Now, if both are rolled along the lecture bench by being given an equal impulse, the metal ring will continue to roll longer than the disk because of its greater moment of inertia. However, when the ring and disk are released simultaneously from the same height on an inclined plane, the wooden disk reaches the bottom first due to its lesser moment of inertia.

Polar Moment of Inertia

• The polar moment of inertia is an important parameter in problems involving torsion of cylindrical shafts and rotations of slabs.

dArJ 20

• The polar moment of inertia is related to the rectangular moments of inertia,

xy II

dAydAxdAyxdArJ

222220

Radius of Gyration

Frequently tabulated data related to moments of inertia will be presented in terms of radius of gyration.

m

IkormkI 2

m = Mass of the body

K= Radious of Gyration

Where

Moment of inertia of ’ :composite area s

Parallel Axis Theorem

• The moment of inertia about any axis parallel to and at distance d away from the axis that passes through the centre of mass is:

• Whereo IG= moment of inertia for mass centre G

o m = mass of the bodyo d = perpendicular distance between the parallel axes.

2mdII GO

Perpendicular Axis Theorem:

yxz III

For flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane.

M

Ix = (1/12) Mb2

Iy= (1/12) Ma2

a

b

Iz = (1/12) M(a2 + b2)