Centroid of 2D Plane Body

The 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 Lines

Curve 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 Integration

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 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