Sideway
output.to from Sideway
Draft for Information Only

Content

Centroid of 2D Plane Body
  Centroids of Areas
   Centroid by Single Integration
    Centroid of Area by Single 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.

image

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.

image

Centroids of Areas

The using of unit elemental areas of an object to determine the centroid of a 2D plane area can be expressed as

image

A double integation is needed to evaluate with respect to the two varables. Similar to finding the area of a 2D plane object, the centroid of an area can usually be determined by performing a single integration also.

Centroid by Single Integration

The unit elemental areas of an object used to determine the centroid of a 2D plane area can be rearranged into grouped elemental areas. Imply

image

After the grouping of unit elemental areas into one elemental area, the coordinates of the centroid of an area can also be determined by one single integration in a  similar way by considering the centroid of each elemental area strip. Imply

image

Centroid of Area by Single Integration

image

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

image

The unit element area of a region can be grouped into either a thin vertical rectangular strip or  a thin horizontal rectangular strip. And the elemental area ΔA becomes

image

Considering the thin rectangular strip as the elemental area, the centroid of the planar region can be determined by a single integration through sweeping the elemental centroid of the elemental area strip along either rectangular coordinate axis accordingly. Imply

By sweeping the centroid of horizontal strip along y axis vertically

Centroid of horizontal strip. Imply

image

Therefore, centroid of the bounded area is

image

By sweeping the centroid of vertical strip along x axis horizontally

Centroid of vertical strip. Imply

image

Therefore, centroid of the bounded area is

image

©sideway

ID: 120600004 Last Updated: 6/4/2012 Revision: 0 Ref:

close

References

  1. I.C. Jong; B.G. rogers, 1991, Engineering Mechanics: Statics and Dynamics
  2. F.P. Beer; E.R. Johnston,Jr.; E.R. Eisenberg, 2004, Vector Mechanics for Engineers: Statics
close

Latest Updated LinksValid XHTML 1.0 Transitional Valid CSS!Nu Html Checker Firefox53 Chromena IExplorerna
IMAGE

Home 5

Business

Management

HBR 3

Information

Recreation

Hobbies 8

Culture

Chinese 1097

English 339

Reference 79

Computer

Hardware 249

Software

Application 213

Digitization 32

Latex 52

Manim 205

KB 1

Numeric 19

Programming

Web 289

Unicode 504

HTML 66

CSS 65

SVG 46

ASP.NET 270

OS 429

DeskTop 7

Python 72

Knowledge

Mathematics

Formulas 8

Algebra 84

Number Theory 206

Trigonometry 31

Geometry 34

Coordinate Geometry 2

Calculus 67

Complex Analysis 21

Engineering

Tables 8

Mechanical

Mechanics 1

Rigid Bodies

Statics 92

Dynamics 37

Fluid 5

Fluid Kinematics 5

Control

Process Control 1

Acoustics 19

FiniteElement 2

Natural Sciences

Matter 1

Electric 27

Biology 1

Geography 1


Copyright © 2000-2024 Sideway . All rights reserved Disclaimers last modified on 06 September 2019