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Unformatted text preview: MOMENT OF MOMENT OF INERTIA INERTIA Moment of Inertia by Integration Moment of Inertia by Integration LEARNING OBJECTIVES LEARNING OBJECTIVES Be able to define the moment of Be able to define the moment of inertia for an area inertia for an area Be able to determine the moment of Be able to determine the moment of inertia for an area by integration inertia for an area by integration Be able to determine the radii of Be able to determine the radii of gyration gyration PREREQUISITE KNOWLEDGE PREREQUISITE KNOWLEDGE Units of measurements Units of measurements Concepts of center of gravity, center Concepts of center of gravity, center of mass and centroid of mass and centroid Integration over lines, areas and Integration over lines, areas and volumes volumes MOMENTS OF INERTIA Area Moments of inertia are measures of how an area is distributed about particular axes. Area moments of inertia also called mass moments of inertia or the angular mass are measures of a body's resistance to changes in its rotation rate or angular acceleration. That is, it is the inertia of a rigid rotating body with respect to its rotation. The role of the moments of inertia to rotational dynamics is almost the same as that of mass to basic dynamics. They are used in determining the relationship between angular momentum and angular velocity, torque and angular acceleration, and other quantities. The moments of inertia are also used to relate the normal stress to the applied external moment. MOMENTS OF INERTIA MOMENTS OF INERTIA The concept of moments of inertia was introduced by Euler in his book in 1730. For an area, the moments of inertia are equal to the sum of the products of each element of an area multiplied by the square of its perpendicular distance from the reference axis (axis of rotation). For a mass, the moments of inertia are equal to the sum of the products of each point mass multiplied by the square of its perpendicular distance from the axis of rotation. ROTATION ROTATION MOMENTS OF INERTIA SCALAR NOTATION dm r I dm, r I dm, r I dA r I dA, r I dA, r I 2 y y 2 x x 2 A 2 y y A 2 x x A 2 = = = = = = mass a For area an For MOMENTS OF INERTIA dydx y y x y o = = 2 A 2 x dA I dydx x x x y = = 0 0 2 A 2 y dA I y x A 2 I I dA J + = = r Moment of inertia about xaxis : Moment of inertia about yaxis : Polar moment of inertia : EXAMPLE 1 EXAMPLE 1 Determine the moments of inertia of the above Determine the moments of inertia of the above cross section cross section SOLUTION 1 SOLUTION 1 ( 29 ( 29 4 7 2 7 3 2 2 4 2 3 2 1 4 2 in 571 . 1344 2 3 2 192 7 3 4 1 3 1 3 1 2 2 === = = = = x x dx x dx y dx dy y d y x x 2 x A x I A I Moments of inertia : SOLUTION 1 SOLUTION 1 4 2 5 3 2 2 2 2 1 4 2 in 067 . 1 20 3 4 1 2 ==  = = = x x dx x x dx dy x d x x 2 y A y I A I SOLUTION 1 SOLUTION 1 4 in 638 . 1 067 . 1 571 ....
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 Fall '10
 Baladi

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