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Centroid of Plane Body

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.

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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.

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Centroids of Lines

Curve Length of Quarter-Circular Arc by Integration

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For example, the curve segment bounded by curves in rectangular form , Imply

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An elemental curve fragment ΔL in rectangular form can be approximated by pythagorean theorem. Imply

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By taking the limit, the dimension of the curve element can be expressed as

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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

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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.

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Sweeping along y axis vertically

The x coordinate of centroid of the planar curve is.

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The y coordinate of centroid of the planar curve is.

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Centroid of Semi-Circular Arc by Integration

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For example, the curve segment bounded by curve in rectangular form , Imply

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By symmetry, the length of a semi-circular arc is

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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.

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Sweeping along x axis horizonatally

Coordinate x of centroid by symmetry.

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Coordinate x of centroid by integration.

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Centroid of Circular Arc by Integration

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For example, the curve segment bounded by curve in polar form , Imply

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The length of the curve segment

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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.

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By sweeping the centroid of circular sector slice along variable angle θ circularly

Coordinate x of centroid by integration. Imply

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Coordinate y of centroid by symmetry. Imply

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