![]() ![]() The product means the multiplication of( x*y) by the area. When we take the distance X to the y-axis, the distance y to the X-axis to estimate the product of inertia we will use the product of A*x*y. The product of inertia is the sum of the product of each elementary area by its corresponding distances referred to in these two axes. While the second moment of area about the X-axis will be the product of A*Y^2. The second moment of area about the y-axis will be the product of A*X^2. While the Moment of inertia or the second moment of area about an axis is the product of each elementary area by the square of the normal distance from that axis. This is the definition of the first moment of area about the x-axis and also about the y-axis. The distance should be perpendicular to the axis since we are estimating moments. The statical moment or the first moment of area is the product of each elementary area by the normal distance. The difference between the first moment of area and the second moment of area. In this sketch, the areas are represented by forces that create moments about the Y-axis. This is the case if we consider the areas as forces then we can get the x bar of these forces. Then X bar value can be estimated by dividing the first moment of the five areas by the total of these areas. The sum of areas from A1 to A5 is =At, if we consider that each area has a horizontal distance from its CG to an arbitrary axis y, Again the sum of the first area of all of the five areas will be equal to At*Xbar. In case that, we use X-axis at any location, still the first moment of the area is valid if we replace the area by force. The y bar value can be estimated by dividing the first moment of the five areas by the total of these areas, so Y-bar=(A1 y1+A2y2+A3 圓+A4y4+A5*y5)/A is called y bar. The sum of the first moment of area, that is, (A1*y1+A2*y2+A3*圓+A4*y4+A5*y5)=A*YA, or YA is called y bar. The center of gravity is the at which the resultant of all elementary areas passes, according to the next slide, if we have five areas each area has a CG distance to an arbitrary axis, we call it axis X, the distance from Cg of A1 is y1, and the distance from CG of area 2 is y2, he distance from Cg of A1 is y1, and the distance from CG of area 32 is 圓. The second definition for the first moment of AREA. The total value of the area will be=ΣAi, for i=1 to i=5. As we can see from the next slide, if we have an enclosed area by a boundary line, it can be represented by Five areas A1, A2, A3, A4, A5. ![]() The same principle, as we have done with the resultant force, can be applied in the case of areas. The area enclosed by a boundary is the sum of all elementary areas enclosed by that boundary line. The first application for integration is how to estimate the area and CG of an object. The first definition for area determination. ![]() Y- bar is the distance for the resultant Force F from the external arbitrary axis = the moment produced from the Five forces/ F. If the distance from the outer axis to the first force F1 is X1, and similarly, the distance from this axis to the second force F2 is X2, and X3 is the distance to the third force F3, etc. To get the y bar of these forces, we draw a horizontal axis X. The sum of forces represented by the symbol Σ can be written as F=ΣFi for I from 1 to 5. If we have a series of forces, please refer to statics, to get the resultant F of these forces F1, F2, F3, F4, F5. T here is a similarity between forces and the areas. This is a brief description of the content of the lecture. If we have a series of forces, we can get the resultant and the point of application, likewise, we can estimate the CG of an area and estimate the first moment of the area. A discussion of the similarity between areas and forces, how to represent an enclosed area by a group of areas, the sum of these areas will be the same as of the arbitrary area. The lecture includes important definitions for the area determination and first moment of area and second moment of area and the product of inertia. What is the First moment of area? Topics are included in the illustration. The difference between the first moment of area and the second moment of area.The second definition for the first moment of AREA.The first definition for area determination.The similarity between areas and forces.Topics are included in the illustration.
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