2 MOMENTS OF INERTIA Objective Develop a method for determining the moment of inertia for an areaMoment of inertia is an important parameter to be determined when designing a structure or a mechanical part
3 Definition of Moments of Inertia for an Area To find the centroid of an area by the first moment of the area about an axis was determined ( x dA )Integral of the second moment is called moment of inertia ( x2 dA)Consider the area ( A )By definition, the moment of inertia of the differential area about the x and y axes are dIx and dIydIx = y2dA Ix = ( y2 dA)dIy = x2dA Iy = ( x2 dA)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 areaJo = r2 dA = Ix + Iy (since r2 = x2 + y2 )
4 Definition of Moments of Inertia for an Area dAyxOr
5 Parallel-Axis Theorem for an Area If moment of inertia of an area about an axis passing through its centroid is known, then it will be convenient to determine the moment of inertia about any parallel axis by the parallel-axis theorem.
6 How to derive the theorem? find the moment of inertia of the area about the axisIx = (y + dy)2 dA= y2 dA + 2dy y dA + d2y dAThe first integral is the moment of inertia about the centroid axis IxThe second integral is zero since the x axis passes through the area's centroid C y dA = y dA = 0 (since y = 0)Ix = Ix + A d2yIy = Iy + A d2xJo = Jc + A d2AdAdydxOyxdxyC
7 MOMENTS OF INERTIA FOR COMPOSITE AREAS Composite areas consist of simpler partsIf moment of inertia of each part is known or can be determined, then the moment of inertia of the composite area is the algebraic sum of moments of inertia of all parts
8 Procedure for Analysis Divide the area into simpler partsIndicate the perpendicular distance from the centroid of each part to the reference axisFind moment of inertia of each part about its centroidal axisUse parallel-axis theorem to find moments of inertia about the reference axis (I = ¯I + A d2 )Find moment of inertia of the entire area by summing moments of inertia of all parts algebraicallyIf a composite body has a hole, then its moment of inertia is found by subtracting the moment of inertia for the hole from the moment of inertia of the entire part including the hole.