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Objectives T di th t f t f it d To discuss the concept of center of gravity and centroid for a body of arbitrary shape. To show how to locate the centroid for a line an area or To show how to locate the centroid for a line, an area, or a volume. T d t i th lt t f di t ib t d l di To determine the resultant of distributed loading. To determine the moment of inertia & polar moment of inertia for an area. 6-2
Terms C t f G it (CG) i th i t hi h l t th lt t Centre of Gravity (CG) is the point which locates the resultant weight of a system of particles or a body. Centroid C is the point which defines the geometric center of an object. This term is used in connection with figures like lines, areas and volumes. The centroid coincides with the CG of a body only if the material e ce t o d co c des t t e o a body o y t e ate a composing the body is uniform or homogenous (density is constant). 6-3

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6 1 CG for a System of Particles C id t f ti l i 6.1 CG for a System of Particles Consider a system of n particles in space. The weights of each particle (in z direction) can be replaced by a single resultant Weight W R = Σ W acting at point G. To find the x coordinates of G, take moments about the y-axis Similarly we can sum the moments about the x axis to find the y, z coordinates of G 6-4
6 1 CG for a System of Particles 6.1 CG for a System of Particles The coordinates of G for a system of particles are: Eqn 6.1 represents a balance between the sum of the moments of all the particles of the system and the moment of the resultant weight for the system. 6-5

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6 2 CG/Centroid of a Body A rigid body is made of an infinite 6.2 CG/Centroid of a Body number of particles. To apply the same principles to the body, it is necessary to se integ ation athe than a disc ete use integration rather than a discrete summation of terms. We get the coordinates of G by simply l i Σ b d W b dW i (6 1) replacing by and W by dW in (6.1), If the body is made of homogenous material of specific weight γ , we can replace dW by γ dV (or γ dA/ γ dL). The resulting formulas 6-6 define the centroids of a volume, area or line.
6 2 CG/Centroid of a Body V l R l i dW b dV 6.2 CG/Centroid of a Body Volume: Replacing dW by dV Area: Replacing dW by dA Line: Replacing dW by dL In (6.6) and (6.7), only two coordinates of C need to be calculated if the area is a flat 6-7 surface or the line lies on a single plane.

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Important points 1 In Eqn (6 6) are called the first moments of area 1. In Eqn (6.6), are called the first moments of area about the y and the x axis, respectively. Its values may be positive, negative or zero depending on the coordinate system used. 2. Where a shape has an axis of symmetry, its centroid lies on that axis.
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