Centroids 1 rev

# Centroids 1 rev - Consider a body occupying the volume V...

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Unformatted text preview: Consider a body occupying the volume V and let the gravitational field act in the y direction Center of Gravity x y z V dV r d R (x,y,z) gravity If is the force/unit volume acting on the body from gravity, then the force of gravity acting on the small volume dV is and the total force, R , acting on V is where W is the total weight of the body O d dV = - R j V dV W = - = - R j j Gravity acting on dV produces a moment about the origin O given by so the total moment of the force of gravity about O is since M 0 is perpendicular to R we can replace the distributed force of gravity on the body by a single force acting at a point (xG , yG , zG ) called the center of gravity R r G (xG , yG , zG) ( 29 ( 29 d d x y z dV = = + + - M r R i j k j ( 29 ( 29 V V V x y z dV x dV z dV = + + - = - + M i j k j k i V dV W = - = - R j j To find the center of gravity we must satisfy which gives if gravity acted in a direction not parallel to the y-axis we would also find G G G V V G G V V x y z x dV z dV W z W x W x dV z dV = = - +-- = - + r R M i j k k i i k k i V V G V V V G V x dV x dV x W dV z dV z dV z W dV = = = = V V G V y dV y dV y W dV = = If is a constant we can factor it out of the integrals and obtain instead where the point (xc , yc , zc) is called the centroid of the volume V (a geometrical parameter) V V V V G C G C V V V V G C V xdV xdV zdV zdV x x z z V V dV dV ydV ydV y y V dV = = = = = = = = = In a similar manner we can define centroids of areas y dA or centroids of lines A x y dL L x A A C C xdA ydA x y A A = = L L C C xdL ydL x y L L = = If we use = mass/unit volume instead of = the weight/unit volume, we can define the coordinates of the center of mass as V V V V m m V V V V m V x dV x dV z dV z dV x z M M dV dV y dV y dV y M dV = = = = = = Centroids of Areas...
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## This note was uploaded on 10/17/2011 for the course EM 274 taught by Professor Boylan during the Fall '08 term at Iowa State.

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Centroids 1 rev - Consider a body occupying the volume V...

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