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

Key Concepts

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.

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.

Transcript

00:01 In this question, you have to determine what is the moment of inertia of this shaded area when it's rotating about the x -axis.

00:09 So this area will rotate about this axis.

00:14 Okay, keep that in mind.

00:16 Now, in order to evaluate the moment of inertia, you have to use this equation.

00:21 And this equation tells us that the moment of inertia is equal to the integral of y squared element of area.

00:28 So all we have to do in this kind of question is determine what is the expression for the element of area.

00:35 To do that, it's nice to begin by drawing what would be an element of area in this situation.

00:41 And there are two possibilities.

00:43 We can draw a horizontal element of area, which would be something like that.

00:48 Or we could draw a vertical element of area, which would be something like that.

00:53 So in order to decide if you are using a horizontal element of area or a horizontal element of area, vertical element of area, you have to take a look here at the expression for the moment of inertia.

01:04 In this expression, the variable is y.

01:07 So you would like to have something that has a dy in it.

01:13 And what have a dy in it? this version of the element of area or this version? well, it happens to be this version, because in this version, you have dy and then you have the x coordinate.

01:28 Expression for the element of area is the following.

01:31 Da is equals to x, d .y.

01:34 And that's it.

01:35 So let me raise the other element of area, which will be something like y, d .x, and we are not interested in that situation and continue serving the equation from here.

01:46 Okay, it would be very nice to draw that element of area in the figure so that you can have a better visualization of the situation.

01:53 So let me draw it.

01:55 So an element of area would be something like this.

02:01 Okay, and this would be red.

02:04 So let me paint it red in order to be more clear about what i'm saying.

02:10 Okay, so what are the measures of this element of area? as you can see, you have a height here and that height is the y and you have a width right here and that width as you can guess is just x.

02:25 So the moment of inertia is given by the integral of y square and d -a, which happens to be x, d -y.

02:35 So now you have two variables.

02:36 How can you get rid of one of those two variables? it happens that you have two choices.

02:42 You can solve this equation for x and then substitute that expression for x here, or you can just plug in y squared here and then work with an interval over x.

02:56 You can do whatever you want.

02:58 You have these two choices.

03:00 Go with serving this equation for x and then plug in here.

03:04 So solving that equation for x is nothing outside of this word...

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