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

Key Concepts

Moment of Inertia

This is a physical quantity that measures the resistance of a body or a shape to angular acceleration about a given axis. It is calculated by integrating or summing the products of each infinitesimal mass or area element with the square of its distance from the axis of rotation, reflecting how distribution of mass or area influences rotational dynamics.

Integration

Integration is the fundamental calculus tool used to compute the moment of inertia by summing the contributions from all differential elements within the shape. In the context of area calculations, it involves setting up an integral that accumulates the squared distances of differential area elements relative to the axis of rotation.

Differential Element of Area

In the calculation of area moments of inertia, the shape is divided into infinitesimally small parts, each represented as a differential area element (dA). The moment contribution of each element is determined by multiplying dA by the square of its distance from the specified axis, and integrating these contributions gives the total moment of inertia.

Axis of Rotation

The axis of rotation specifies the line about which the moment of inertia is calculated. In problems concerning the y-axis, distances in the integration are measured perpendicular to this axis. The selection of this axis influences the limits of integration and the form of the distance function within the integral.

Transcript

00:01 Hello everyone here we have to calculate moment of inertia of shaded area about y -axis so this one is the shaded area so from the given area circular area of radius 2 inch has to be removed so this moment of inertia can be calculated by dividing the given shaded area into segments so this is this complete square is one segment this is second segment and this one is third segment.

00:47 It having the center c1, it having the centroid c2 and it having the centroid c3.

01:04 C1 centroid is at x coordinate of it is xc1 having the value half of 6 inch.

01:21 So it having the value 3 inch c2 having the centroid, y coordinate, x coordinate of it having the value one third of six inch that is two inch and x coordinate of centroid c3 is one third of six inch that is two inch this is negative mass sorry negative area from the shaded area this portion has to be removed so it is negative it having the centroid c4 and c4 has having the value x c4 3 inch.

02:18 Now moment of inertia can be calculated of each segment can be calculated by using parallel axis theorem iy to be i bar y plus a into d square x.

03:30 So for each segment here we are calculating area perpendicular distance from the axis moment of inertia about the centroid product of area into square of perpendicular distance and moment of inertia about y -axis so here i am tabulating for each segment segment area of the segment in inch square perpendicular distance of centroid of the given segment from the x -axis an inch moment of inertia of the given segment about the centroid product of area of the segment and square of perpendicular distance from the x -axis and moment of inertia about y -axis segment one its area is six into six inch perpendicular distance from the x -axis is 3.

04:50 Its moment of inertia is 112 of 6 xx2.

04:56 Product of area into square of its perpendicular distance, 324.

05:03 So moment of inertia of the segment 1 about y -axis...

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