Link:http://output.to/sideway/default.asp?qno=120500016 Centroid of Plane Body Centroid of 2D Plane BodyThe centroid of an plate is determined by the first moment of a two dimensional plane body with the method of the first moment of area. ![]() And the centroid of a wire is determined by the first moment of a two dimensional plane body with the method of the first moment of line. ![]() Centroids of AreasArea by IntegrationArea by Double Integration![]() For example, the signed area of the planar region R is bounded by curves in rectangular form , Imply ![]() An elemental area ΔA in rectangular form can be defined as Δx times Δy. Imply ![]() In general, the area of a region can be determined by multiple integration through sweeping the signed elemental area starting from along either rectangular coordinate axis. Imply Starting from horizontal sweeping along x axis ![]() Starting from vertical sweeping along y axis ![]() And for curves in polar form ![]() For example, the signed area of the planar region R is bounded by curves in polar form , For the curve profile, Imply ![]() And the curve profile in terms of θ, imply ![]() And other boundary curves are ![]() An elemental area ΔA in polar form can be approximated by Δr times rΔθ. Imply ![]() Unlike rectangular form, the polar form of an elemental area ΔA is not a constant but a function of r and in turn a function of θ also. In general, the area of a region can be determined by multiple integration through sweeping the signed elemental area starting from along either polar variables. Imply Starting from radical sweeping along variable radius r ![]() Starting from circular sweeping along variable angle θ, ![]() |
Sideway BICK Blog 30/05 |