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Mechanics: Statics




2D Plane Body



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Centroid of 2D Plane Body
  Centroids of Areas
   Area by Integration
    Area by Double Integration

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 Areas

Area by Integration

Area by Double Integration

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For example, the signed area of the planar region R is bounded by curves in rectangular form , Imply

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An elemental area ΔA in rectangular form can be defined as Δx times Δy. Imply

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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

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Starting from vertical sweeping along y axis

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And for curves in polar form

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For example, the signed area of the planar region R is bounded by curves in polar form , For the curve profile, Imply

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And the curve profile in terms of θ, imply

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And other boundary curves are

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An elemental area ΔA in polar form can be approximated by Δr times rΔθ. Imply

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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

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Starting from circular sweeping along variable angle θ,

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References

  1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics, Saunders College Publishing, United States of America
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics, McGraw-Hill Companies, Inc., New York
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ID: 120500016 Last Updated: 2012/6/2 Revision: 1 Ref:

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