Determine the moment of inertia for the shaded area about...

Key Concepts

Area Moment of Inertia

The area moment of inertia is a geometric property of a cross-section that quantifies its resistance to bending or deflection about an axis. It is defined as the integral of the square of the distance of each differential area element from the axis of interest, capturing how the area is distributed relative to that axis.

Integration of Differential Area Elements

To compute the moment of inertia of an area, one divides the area into infinitesimally small elements and sums (integrates) the contributions of each element, which are determined by multiplying the square of the distance from the element to the axis by the differential area. This integration process is fundamental in arriving at an accurate measure of the area moment of inertia.

Coordinate System and Limits of Integration

Choosing an appropriate coordinate system is crucial when setting up the integral for the moment of inertia. The selection of variables and the definition of the limits of integration must accurately describe the geometry of the area in question, ensuring that the integration covers every part of the area contributing to the moment of inertia.

Transcript

00:01 Hi everyone here we have to find moment of inertia of shaded area about the x -axis.

00:08 This is your x -axis and this is the y -axis.

00:12 So find moment of inertia of shaded area about x -axis.

00:19 This one.

00:28 Let us start solving it.

00:30 Given ceded area has to be splited into three segments and moment of inertia can be obtained about x -axis can be obtained of each segment by applying parallel axis theorem.

01:26 Consider this is to be third segment, this is to be second segment.

01:38 This complete one is fourth segment and this is, sorry, first segment and this is the fourth minus because this is the this segment has to be removed from the given system so this is negative one each segment having the centroid c3 c2 c1 c4 if we write the coordinate of each i am writing here c1 its y bar is half into six you have to follow the figure that is three inch for c2 by bar c2 is 6 plus 1 third of 3 inch that is 7 inch c3 yc3 is equal to 2 third of 9 inch that is 6 inch and c4 which is negative so yc4 having negative value minus 3 inch now for each segment we will find moment of inertia about x -axis for each segment as tabulated segment area of each segment in inch square perpendicular distance from the y -axis of each segment in inch moment of inertia about center of centroid in inch to the power four product of area into a square square of perpendicular distance from the y axis and moment of inertia about x -axis.

05:16 Unit will be inch to the power 4.

05:19 Here also inch to the power 4.

05:22 Writing for value for each segment as shown in the figure.

05:26 Area you have to follow the figure...

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