This preview shows pages 1–7. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 yaxis 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...
View
Full
Document
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.
 Fall '08
 Boylan

Click to edit the document details