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

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 i x of shaded area about x the basic concept is same moment of inertia about x axis for each segment can be calculated using the parallel axis theorem segments are mentioned as the figure see the segments this is the segment one complete this is the segment two and this is segment three this is the segment which has to be removed from the existing system so it having the negative centroid remembering it now we will calculate area of each segment perpendicular distance of each segment from the y axis moment of inertia of each segment about centroid product of area into square of perpendicular distance for each segment and then moment of inertia of the segment about x -axis so we are making the table here segment area of each segment we are measuring in millie square perpendicular distance from y axis in millimeter moment of inertia about centroid in millimeter to the power four area into a square of perpendicular distance and i x i first segment having the area 200 into 3 you have to follow the figure simple calculation is there perpendicular distance is by 150 millimeter.

03:26 Moment of inertia of area about the center at 112 of 200 into 3 square.

03:37 Product of area into square of perpendicular distance 1 .35 into 10 to the power 9 and moment of inertia is 1 .8 into 10 to the power 9.

03:51 Here i am writing the unit millimeter to the power 4 segment 2.

04:00 Area is half 1 .1.

04:06 150 into 300 perpendicular distance is 100 moment of inertia about the centroid area product of area and perpendicular distance square of perpendicular distance i'm sorry 0 .225 into 10 to the power 9 and moment of inertia about x axis 0 .3375 into 10 to the power 9 third area of the circle but it is negative because this part has to be removed from the given system...

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