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bjectives Objectives ± To discuss the concept of center of gravity and centroid for a body of arbitrary shape. o show how to locate the ntroid r a line an area or ± To show how to locate the centroid for a line, an area, or a volume. ± To determine the resultant of distributed loading. ± To determine the moment of inertia & polar moment of inertia for an area. 6-2
erms Terms 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. he entroid incides with the G f a body only if the material e ce t o d co c des t t e CG 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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1 CG for a System of Particles 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
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. gy 6-5

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2CG/ entroid f Body rigid body is made of an infinite 6.2 CG/Centroid of a Body A rigid body is made of an infinite number of particles. To apply the same principles to the body, it is necessary to use integration rather than a discrete summation of terms. We get the coordinates of G by simply replacing Σ 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.
2CG/ entroid f Body 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 eed to be calculated if the area is a flat 6-7 need to be calculated if the area is a flat surface or the line lies on a single plane.

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mportant points qn 6) are called the rst moments of area Important points 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. Where a shape has two or three axes of symmetry, the centroid lies at the intersection of these axes.
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