Link:http://output.to/sideway/default.asp?qno=120600006 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 LinesCurve Length of Quarter-Circular Arc by Integration![]() For example, the curve segment bounded by curves in rectangular form , Imply ![]() An elemental curve fragment ΔL in rectangular form can be approximated by pythagorean theorem. Imply ![]() By taking the limit, the dimension of the curve element can be expressed as ![]() In general, the length of the curce in a region can be determined by integration through sweeping the curve element along either rectangular coordinate axis. Imply Sweeping along x axis horizontally ![]() The length of the curce in a region can also be determined by integration through sweeping the curve element along y axis in similar way. Centroid of Quarter-Circular Arc by IntegrationThe centroid of the planar curve can be determined by a single integration through sweeping the elemental centroid of the curve fragment along either rectangular coordinate axis. ![]() Sweeping along y axis vertically The x coordinate of centroid of the planar curve is. ![]() The y coordinate of centroid of the planar curve is. ![]() Centroid of Semi-Circular Arc by Integration![]() For example, the curve segment bounded by curve in rectangular form , Imply ![]() By symmetry, the length of a semi-circular arc is ![]() The centroid of the planar curve can be determined by a single integration through sweeping the elemental centroid of the curve fragment along either rectangular coordinate axis. ![]() Sweeping along x axis horizonatally Coordinate x of centroid by symmetry. ![]() Coordinate x of centroid by integration. ![]() Centroid of Circular Arc by Integration![]() For example, the curve segment bounded by curve in polar form , Imply ![]() The length of the curve segment ![]() The centroid of the planar curve can be determined by a single integration through sweeping the elemental centroid of the curve fragment along variable angle c circularly. ![]() By sweeping the centroid of circular sector slice along variable angle θ circularly Coordinate x of centroid by integration. Imply ![]() Coordinate y of centroid by symmetry. Imply ![]() |
Sideway BICK Blog 07/06 |