Lesson 16 - Centroid of an Area

Lesson 16 - Centroid of an Area - TOPIC APPLICATIONS...

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TOPIC APPLICATIONS CENTRIODS OF PLANE AREA
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The mass of a physical body is a measure of the quantity of the matter in it, whereas the volume of the body is a measure of the space it occupies. If the mass per unit volume is the same throughout the body is said to be homogeneous or to have constant density. It is highly desirable in physics and mechanics to consider a given mass as concentrated at a point, called its center of mass (also, its center of gravity). For a homogeneous body, this point coincides with its geometric center or centroid. For example, the center of mass of a homogeneous rubber ball coincides with the centroid (center) of the ball considered as a geometric solid (a sphere). DISCUSSION The centroid of a rectangular sheet of paper lies midway between the two surfaces but it may well be considered as located on one of the surfaces at the intersection of the two diagonals. Then the center of mass of a thin sheet coincides with the centroid of the sheet considered as a plane area.
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The Moment M L of a Plane Region with respect to a line L is the product of the area and the directed distance of the centroid from the line. The moment of a composite region with respect to a line is the sum of the moments of the individual sub-regions with respect to the line. The moment of a plane region with respect to a coordinate axis may be found as follows: 3.Evaluate the definite integral of the product in step 2 and apply the fundamental theorem.
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