Basics of Centroid Mechanical Engineering Notes | EduRev

Engineering Mechanics - Notes, Videos, MCQs & PPTs

Mechanical Engineering : Basics of Centroid Mechanical Engineering Notes | EduRev

The document Basics of Centroid Mechanical Engineering Notes | EduRev is a part of the Mechanical Engineering Course Engineering Mechanics - Notes, Videos, MCQs & PPTs.
All you need of Mechanical Engineering at this link: Mechanical Engineering

Centroid and Center Of Mass of the Composite Bodies

What is center of gravity?

The center of gravity G is a point which locates the resultant weight of a system of particles.

What is composite body?

A composite body consists of a series of connected simpler-shaped bodies, which may be rectangular, triangular, semicircular, etc. 

What is centroid?

Basics of Centroid Mechanical Engineering Notes | EduRev

The line of action of the resultant force passes through the geometric center or centroid of the volume under the distributed–loading diagram.

Basics of Centroid Mechanical Engineering Notes | EduRevBasics of Centroid Mechanical Engineering Notes | EduRev

Find the  Centroid of A Rectangular Area by integration

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

Find the centroid of a quarter circle by double integration in rectangular coordinates

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

Procedure for locatingcenter of gravity of a body or the centroid of a composite geometrical object

  • Using a sketch, divide the body or object into a finite number of composite parts that have simpler shapes.
  • If a composite part has a hole, or a geometric region having no material, then consider the composite part without the hole and consider the hole as an additional composite part having negative. 
  • Establish the coordinate axes on the sketch and determine the coordinates Basics of Centroid Mechanical Engineering Notes | EduRev of the center of gravity or centroid of each part.
  • Determine Basics of Centroid Mechanical Engineering Notes | EduRev  by applying the center of gravity equations
  • If an object is symmetrical about an axis, the centroid of the object lies on this axis. 

First Moments of Areas and Lines 

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

  • An area is symmetric with respect to an axis BB’ if for every point P there exists a point P’ such that PP’ is perpendicular to BB’ and is divided into two equal parts by BB’.

 

Basics of Centroid Mechanical Engineering Notes | EduRev
  • The first moment of an area with respect to a line of symmetry is zero.
  • If an area possesses a line of symmetry, its centroid lies on that axis
  • If an area possesses two lines of symmetry, its centroid lies at their intersection.

 

Basics of Centroid Mechanical Engineering Notes | EduRev

 

  • An area is symmetric with respect to a center O if for every element dA at (x,y) there exists an area dA’ of equal area at (-x,-y).  
  • The centroid of the area coincides with the center of symmetry.

 

Center Of Mass – locates the point in a system where the resultant mass can be concentrated so that the moment of the concentrated mass with respect to any axis equals the moment of the distributed mass with respect to the same axis.

Basics of Centroid Mechanical Engineering Notes | EduRev

Center Of Gravity– locates where the resultant, concentrated weight acts on a body. 

Basics of Centroid Mechanical Engineering Notes | EduRev

Centre of gravity and centroid:

The center of gravity (G) is a point that locates the resultant weight of a system of particles.

Particles with weights W1, W2, …, Wn can be replaced by a resultant force of weight W located at the center of gravity G.

Basics of Centroid Mechanical Engineering Notes | EduRev

To find the location of the center of gravity G(x,y,z):

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

(We can obtain z by imagining the coordinate system, with the particles fixed in it, as being rotated 90 degrees about the x (or the y) axis).

Knowing W=mg, if we assume that the acceleration due to gravity (g) for every particle is constant (g will be cancelled out)

Basics of Centroid Mechanical Engineering Notes | EduRev

A rigid body is composed of an infinite number of particles, hence it is necessary to use integration instead of summation.

Basics of Centroid Mechanical Engineering Notes | EduRev

But dm=ρdV, with ρ being the density and dv the volume of each particle. Therefore, the centre of mass has the coordinates of

Basics of Centroid Mechanical Engineering Notes | EduRev

Centroid:

The centroid (C) is a point which defines the geometric center of an object.  If the material composing a body is uniform or homogeneous, the density of material is constant (ρ = constant). Hence, the resulting formulas that define the centroid of a body depend only on the geometry of the body {Volume (V), Area (A), or Length (L)}.

Basics of Centroid Mechanical Engineering Notes | EduRevBasics of Centroid Mechanical Engineering Notes | EduRevBasics of Centroid Mechanical Engineering Notes | EduRev

Centroid (volume):

Basics of Centroid Mechanical Engineering Notes | EduRev

Centroid (area):

Basics of Centroid Mechanical Engineering Notes | EduRev

Centroid (line):

Basics of Centroid Mechanical Engineering Notes | EduRev

The centroids of some shapes may be partially or completely specified by using conditions of symmetry.  In cases where the shape has an axis of symmetry, the centroid of the shape will lie along that axis.

Basics of Centroid Mechanical Engineering Notes | EduRev

Basics of Centroid Mechanical Engineering Notes | EduRev

Offer running on EduRev: Apply code STAYHOME200 to get INR 200 off on our premium plan EduRev Infinity!

Dynamic Test

Content Category

Related Searches

practice quizzes

,

pdf

,

Free

,

Extra Questions

,

Summary

,

past year papers

,

Important questions

,

ppt

,

Basics of Centroid Mechanical Engineering Notes | EduRev

,

study material

,

video lectures

,

MCQs

,

Semester Notes

,

Objective type Questions

,

Sample Paper

,

Previous Year Questions with Solutions

,

Basics of Centroid Mechanical Engineering Notes | EduRev

,

Basics of Centroid Mechanical Engineering Notes | EduRev

,

Exam

,

Viva Questions

,

shortcuts and tricks

,

mock tests for examination

;