Mechanics of Solids PRESENTATION ON CENTROID BY DDC 22:- Ahir Devraj DDC 23:- DDC 24:- Pravin Kumawat DDC 25:- Hardik K. Ramani DDC 26:- Hiren Maradiya 1

Centroid 2

Defination :-  It is defined as a point about which the entire line, area or volume is assumed to be concentrated.  It is related to distribution of length, area & volume.  Centroid word is used to represent centre of :- line, arc, square, circle etc. 3

Centre of mass If, instead of weight of object, the mass of the object is considered, the procedure for finding centre of mass (the point, where whole mass is concentrated) remains the same. We know that, W = m*g (1) where, W = total wt. of the body m = total mass of the body g = gravitational acceleration 4

Centroid and First Moments of Areas and Lines Centroid of an areaCentroid of a line 5

First Moments of Areas and Lines 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’. 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. 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. 6

Geometric Properties of An Area and Volume 7

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Example:-1 Calculate centroid of a line or rod by first principle. 9

Centroids of Common Shapes of Areas 10

Centroids of Common Shapes of Lines 11

Determination of Centroids by Integration Double integration to find the first moment may be avoided by defining dA as a thin rectangle or strip. 12

Theorems of Pappus-Guldinus 13 Surface of revolution is generated by rotating a plane curve about a fixed axis. Area of a surface of revolution is equal to the length of the generating curve times the distance traveled by the centroid through the rotation.

Theorems of Pappus-Guldinus 14 Body of revolution is generated by rotating a plane area about a fixed axis. Volume of a body of revolution is equal to the generating area times the distance traveled by the centroid through the rotation.

Center of Gravity of a 3D Body: Centroid of a Volume 15 Center of gravity GResults are independent of body orientation, For homogeneous bodies,

Centroids of Common 3D Shapes 16

Composite 3D Bodies 17 Moment of the total weight concentrated at the center of gravity G is equal to the sum of the moments of the weights of the component parts. For homogeneous bodies,