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Z_-_Centroid___Moment_of_Inertia_for_Com

Z_-_Centroid___Moment_of_Inertia_for_Com - CENTROID MOMENTS...

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CENTROID & CENTROID & MOMENTS OF MOMENTS OF INERTIA INERTIA Centroid And Moment Of Inertia Centroid And Moment Of Inertia For Composite Areas For Composite Areas
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LEARNING OBJECTIVES LEARNING OBJECTIVES Be able to apply the parallel-axis Be able to apply the parallel-axis theorem theorem Be able to determine the moment of Be able to determine the moment of inertia for composite areas inertia for composite areas
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PRE-REQUISITE KNOWLEDGE PRE-REQUISITE KNOWLEDGE Unit of measurements Unit of measurements Concepts of center of gravity, center of Concepts of center of gravity, center of mass and centroid mass and centroid
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MOMENT OF INERTIA FOR COMPOSITE AREAS Many industrial objects can be considered as composite bodies made up of a series of connected “simpler” shaped parts or holes, like a rectangle, triangle, and semicircle. Knowing the location of the centroid or center of gravity of the simpler shaped part, we can easily determine the location of the centroid or center of gravity of the composite body.
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PARALLEL AXIS THEOREM PARALLEL AXIS THEOREM The moment of inertia of an area about an axis is The moment of inertia of an area about an axis is equal to the moment of inertia of the area about a equal to the moment of inertia of the area about a parallel axis passing through the area’s centroid plus parallel axis passing through the area’s centroid plus the product of the area and the square of the the product of the area and the square of the perpendicular distance between the axes perpendicular distance between the axes 2 2 ' 2 ' ; ; Ad J J Ad I I Ad I I C O x y y y x x + = + = + = Where I , I and J are the moments of inertia about an axis passing
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PARALLEL AXIS THEOREM PARALLEL AXIS THEOREM The parallel-axis theorem allows the calculation of the moments The parallel-axis theorem allows the calculation of the moments of inertia of segments, whose centroids are not located at the of inertia of segments, whose centroids are not located at the interested axes: interested axes: The calculation should be done in 3 steps: 1.Determine the location of the centroid 2.Calculate the moment of inertia about the axes passing through the centroid 3.Calculate the moment of inertia about the parallel interest axes
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EXAMPLE 1 EXAMPLE 1 Determine the moment of inertia of the above cross Determine the moment of inertia of the above cross section about the neutral axes (the axes passing through section about the neutral axes (the axes passing through the centroid) the centroid) x y
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SOLUTION 1 SOLUTION 1 Approach 1: Separate the cross section into three segments ( 29 in 2 48 96 ' = = = A A y y Determine the centroid of the cross section x x y 1 2 3 . . . X Segment Area (in 2 ) y’ (in) y’(A), (in 3 ) 1 12 3 36 2 24 1 24 3 12 3 36 Σ area = 48 Σy’(A) = 96 in 8 = x Let y’ = the distance from the X-axis to the centroid of each element, then
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x 1 2 3 x y .
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